Calculus in Non-Integer-Dimensional Space: Tool for Fractal Physics

Abstract

Integration in non-integer-dimensional spaces (NIDS) is actively used in quantum field theory, statistical physics, and fractal media physics. The integration over the entire momentum space with non-integer dimensions was first proposed by Wilson in 1973 for dimensional regularization in quantum field theory. However, self-consistent calculus of integrals and derivatives in NIDS and the vector calculus in NIDS, including the fundamental theorems of these calculi, have not yet been explicitly formulated. The construction of precisely such self-consistent calculus is the purpose of this article. The integral and differential operators in NIDS are defined by using the generalization of the Wilson approach, product measure, and metric approaches. To derive the self-consistent formulation of the NIDS calculus, we proposed some principles of correspondence and self-consistency of NIDS integration and differentiation. In this paper, the basic properties of these operators are described and proved. It is proved that the proposed operators satisfy the NIDS generalizations of the first and second fundamental theorems of standard calculus; therefore, these NIDS operators form a calculus. The NIDS derivative satisfies the standard Leibniz rule; therefore, these derivatives are integer-order operators. The calculation of the NIDS integral over the ball region in NIDS gives the well-known equation of the volume of a non-integer dimension ball with arbitrary positive dimension. The volume, surface, and line integrals in D-dimensional spaces are defined, and basic properties are described. The NIDS generalization of the standard vector differential operators (gradient, divergence, and curl) and integral operators (the line and surface integrals of vector fields) are proposed. The NIDS generalizations of the standard gradient theorem, the divergence theorem (the Gauss–Ostrogradsky theorem), and the Stokes theorem are proved. Some basic elements of the calculus of differential forms in NIDS are also proposed. The proposed NIDS calculus can be used, for example, to describe fractal media and the fractal distribution of matter in the framework of continuum models by using the concept of the density of states.

Contents
1.	Introduction	3
2.	NIDS Integral Calculus	7
	2.1.      Non-Integer-Dimensional Space...............................................................................................................................................................................	7
	2.1.1.      Stillinger Approach.......................................................................................................................................................................................	7
	2.1.2.      Palmer–Stavrinou Approach........................................................................................................................................................................	8
	2.1.3.      Product Measure and Metric Approaches..................................................................................................................................................	8
	2.1.4.      Wilson Approach...........................................................................................................................................................................................	9
	2.1.5.      Fractal Sets and Analysis on Fractals..........................................................................................................................................................	10
	2.2.      Principles of NIDS Integral Calculus.........................................................................................................................................................................	10
	2.3.      Integration in Non-Integer-Dimensional Space: Wilson Approach........................................................................................................................	15
	2.3.1.      Wilson Approach...........................................................................................................................................................................................	15
	2.3.2.      Generalization of the Wilson Approach......................................................................................................................................................	19
	2.3.3.      NIDS Integration Axioms..............................................................................................................................................................................	20
	2.4.      NIDS Integrals of a Single Variable and Their Properties.......................................................................................................................................	24
	2.4.1.      Function Space and NIDS Integral..............................................................................................................................................................	25
	2.4.2.      Fulfillment of the First and Second Correspondence Principles..............................................................................................................	26
	2.4.3.      Scaling Property of Single Variable NIDS Integration................................................................................................................................	28
	2.4.4.      NIDS Length of a Straight Line Segment......................................................................................................................................................	29
	2.4.5.      Translation Property of Single-Variable NIDS Integration........................................................................................................................	31
	2.4.6.      Translation Property of NIDS Segment Lengths..........................................................................................................................................	32
	2.4.7.      Property of the Inequality of NIDS Segment Lengths.................................................................................................................................	33
	2.4.8.      Additivity Property of NIDS Integral and NIDS Segment Lengths............................................................................................................	34
	2.4.9.      Interrelation of the Properties of Additivity and Inequality of NIDS Lengths..........................................................................................	35
	2.5.      Integration in Non-Integer-Dimensional Space: Product Measure Approach.........................................................................................................	38
	2.6.      Examples of NIDS Integrals in D-Dimensional Space.................................................................................................................................................	45
	2.7.      NIDS Integration Satisfies Correspondence Principles................................................................................................................................................	47
	2.8.      Product Measure and Induced Metric Approaches......................................................................................................................................................	51
	2.8.1.      Product Measure Approach and Correspondence Principle.........................................................................................................................	51
	2.8.2.      Metric Approach and Universal Metric Consistency Principle.....................................................................................................................	52
	2.8.3.      Two Approaches to Define Metric Tensors in NIDS........................................................................................................................................	55
	2.9.      Line NIDS Integrals of Scalar Function and NIDS Length..........................................................................................................................................	61
	2.9.1.      Line NIDS Integration in the Metric Approach...............................................................................................................................................	61
	2.9.2.      Line NIDS Integration in the Product Measure Approach............................................................................................................................	65
	2.9.3.      Physical Interpretation of the NIDS Length......................................................................................................................................................	69
	2.9.4.      Kronecker Delta in D-Dimensional Space..........................................................................................................................................................	70
	2.10.      Surface NIDS Integral of Scalar Function and NIDS Area..........................................................................................................................................	71
	2.10.1.      Surface NIDS Integrals in W = W1 × W2.............................................................................................................................................................	71
	2.10.2.      Surface NIDS Integrals in W = W1 × W2 × W3 in the Metric Approach............................................................................................................	73
	2.10.3.      Surface NIDS Integrals in W = W1 × W2 × W3 in the Product Measure Approach.........................................................................................	76
	2.10.4.      Surface NIDS Integrals in W = W1 × … × Wn in the Product Measure Approach...........................................................................................	83
	2.10.5.      Fulfillment of the Correspondence Principle in the Product Measure Approach.....................................................................................	87
3.	NIDS Differential Calculus	91
	3.1.      Principles of NIDS Differential Calculus.........................................................................................................................................................................	91
	3.2.      NIDS Derivatives of a Single Variable.............................................................................................................................................................................	92
	3.3.      NIDS Derivative via Limit................................................................................................................................................................................................	97
	3.4.      Fundamental Theorems of NIDS Calculus.....................................................................................................................................................................	98
	3.5.      NIDS Integration by Parts................................................................................................................................................................................................	99
4.	NIDS Vector Calculus	101
	4.1.      Principles of NIDS Vector Calculus..............................................................................................................................................................................	101
	4.2.      NIDS Differential Vector Operators in the Metric Approach......................................................................................................................................	102
	4.3      Line and Surface NIDS Integral of Vector Fields.........................................................................................................................................................	105
	4.3.1.      Line NIDS Integral of Vector Fields...............................................................................................................................................................	105
	4.3.2.      Surface NIDS Integral of Vector Fields.........................................................................................................................................................	112
	4.4.      Fundamental Theorems of NIDS Vector Calculus.....................................................................................................................................................	119
	4.4.1.      NIDS Gradient Theorem and NIDS Gradient Operators............................................................................................................................	119
	4.4.2.      NIDS Green Theorem......................................................................................................................................................................................	125
	4.4.3.      NIDS Stokes Theorem and NIDS Curl Operators........................................................................................................................................	128
	4.4.4.      NIDS Gauss–Ostrogradsky Theorem and NIDS Divergences....................................................................................................................	134
	4.4.5.      NIDS Green First and Second Identities........................................................................................................................................................	139
	4.5.      Properties of NIDS Vector Differential Operators.....................................................................................................................................................	140
	4.5.1.      NIDS Gradient, Divergence, and Curl Operators.......................................................................................................................................	140
	4.5.2.      Properties of First-Order Vector NIDS Operators......................................................................................................................................	141
	4.5.3.      Second-Order Vector NIDS Operators.........................................................................................................................................................	147
	4.6.      NIDS Vector Operators in Orthogonal Coordinate Systems....................................................................................................................................	150
	4.6.1.      NIDS Integration in Orthogonal Coordinates.............................................................................................................................................	150
	4.6.2.      NIDS Differentiation in Orthogonal Coordinates.........................................................................................................................................	153
	4.6.3.      NIDS Gradient in Cylindrical Coordinates..................................................................................................................................................	154
	4.6.4.      NIDS Gradient in Spherical Coordinates......................................................................................................................................................	157
	4.6.5.      NIDS Divergence in Cylindrical Coordinates.............................................................................................................................................	159
	4.7.      NIDS Derivative with Respect to the NIDS Volume.............................................................................................................................................	161
	4.7.1.      Definition of the NIDS Derivative with Respect to the NIDS Volume...................................................................................................	161
	4.7.2.      Fundamental Theorem for Volume NIDS Derivative.............................................................................................................................	162
	4.7.3.      NIDS Divergence, Gradient, and Curl Operator via the Volume NIDS Derivative................................................................................	163
	4.8.      Differential Forms in D-Dimensional Space................................................................................................................................................................	167
	4.8.1.      Definition and Example of NIDS Differential Forms.............................................................................................................................	168
	4.8.2.      Exterior Derivative of NIDS Differential Forms....................................................................................................................................	171
	4.8.3.      General NIDS Stokes Theorem for NIDS Differential Forms................................................................................................................	177
	4.8.4.      Maxwell’s Equations in D-Dimensional Space.......................................................................................................................................	177
	4.9.      Some Comments on the Application of NIDS Calculus in Fractal Physics...........................................................................................................	178
5.	Conclusions ...................................................................................................................................................................................................................	181
A.	Appendix A ...................................................................................................................................................................................................................	182
B.	References .....................................................................................................................................................................................................................	187

1. Introduction

The use of non-integer-dimensional space (NIDS) in physics actually began in 1964 after the creation of the dimensional regularization method for statistical physics [1]. NIDS is used in statistical physics when describing critical phenomena [2,3,4]. In 1974, Wilson and Kogut [2] used the NIDS to describe both critical phenomena in classical statistical mechanics and quantum field theory.

The use of non-integer-dimensional space in quantum field theory actually began in 1972 after the creation of the dimensional regularization method. Dimensional regularization was proposed in the 1972 papers by Giambiagi and Bollini [5] and by ’t Hooft and Veltman [6,7] for regularizing integrals in quantum field theory by the analytic continuation of space–time dimensions [8].

For the first time, the concept of a linear (vector) space of non-integer dimension and integration in such spaces was proposed in the article [9] by Wilson in 1973. Note that this concept was not directly related to fractal sets. It should be also noted that a fractal set cannot be considered as a linear (vector) space; therefore, it cannot be considered as a linear space with non-integer dimensions. Note that the term “fractal” was introduced by Mandelbrot in 1975 and became widely known with the publication of his book Fractal Geometry of Nature in 1977. Then, in the 1977 paper [4], NIDS and dimensional regularization were used to describe the physics of some crystals that can be macroscopically considered as fractal media.

In the Wilson paper [9,10], the principles (properties, axioms) of integrations in NIDS were formulated for the entire NIDS. The consideration of only the integration of NIDS over the entire space is motivated by the aim of the Wilson article [9], which was devoted to the construction of the dimensional renormalization for quantum field theory in momentum space. The use of integration over the bounded regions of momentum space represents a separate and independent regularization called the “cutoff regularization” [11,12,13] that has not been related to dimensional regularization in quantum field theory, at least until now. In addition, the differential operators in NIDS are not considered in Wilson paper [9].

Currently, the Wilson approach to the integration in NIDS is actively used in quantum field theory [9,10]. Following Wilson’s 1973 article, an axiomatic basis for a space with non-integer dimension was considered by Stillinger in the 1977 work [14]. Unfortunately, Stillinger assumes that the NIDS is not a linear (vector) space. In ordinary vector space, the linear combination of the space vectors is again the element of this space. Stillinger rejects this property for the non-integer-dimensional spaces. In the Stillinger paper [14], a generalization of the scalar Laplace operators is proposed. Other differential operators in NIDS are not considered in Stillinger paper [14]. The further active construction of various differential operators in NIDS was started twenty years ago by the papers [15,16] by using the product of integration measures and differential NIDS vector calculus [17,18,19].

The scalar Laplace operators, which are suggested in [14,15] for NIDS, have a wide application in physics and mechanics. The Stillinger form of the Laplacian was first applied by He [20,21,22], where the Schrodinger equation in NIDS is used, and anisotropic solids are considered an isotropic NIDS system, the dimension of which is determined by the degree of anisotropy. Using NIDS, Thilagam describes the stark shifts of excitonic complexes in quantum wells [23], exciton–phonon interaction in fractional-dimensional space [24], and blocking effects in quantum wells [25]. Matos-Abiague describes particles in NIDS [26,27,28], Bose-like oscillator in NIDS [29], parabolic-confined polarons [30], quantum well [31,32,33], and exitons semiconductor heterostructures by Reyes-Gomez, Matos-Abiague, Perdomo-Leiva et al. in [34]. Quantum mechanical models with NIDS were proposed by Palmer and Stavrinou [15], Lohe and Thilagam [35], Awoga and Ikot in [36], Ahmad, Zubair and Younis in [37], and Pena, Garca-Martnez, Garca-Ravelo, and Morales in [38]. NIDS is used to describe the algebraic properties of the Weyl-ordered polynomials for the momentum and position operators [39,40] and the correspondent coherent states [41]. The Schrodinger equation with the Stillinger form of the Laplacian in NIDS is considered by Eid, Muslih, Baleanu, and Rabei in [42,43], Rabei, Al-Masaeed, Muslih, and Baleanu in [44], Muslih and Agrawal in [45,46], and Calcagni, Nardelli, and Scalisi in [47]. The fractional Schrodinger equation with NIDS is considered by Martins, Ribeiro, Evangelista, Silva, and Lenzi in [48] and by Sandev, Petreska, and Lenzi in [49]. The scalar field on NIDS is considered by Trinchero [50], and the fractional diffusion equation in non-integer-dimensional space and its solutions are suggested in [51]. The gravity in NIDS is described by Sadallah, Muslih, and Baleanu in [52,53] and by Calcagni in [54,55,56]. The electromagnetic fields in NIDS are considered by Muslih, Saddallah, and Baleanu in [57,58], by Zubair, Mughal, and Naqvi in their papers [59,60,61,62], book [63], and others [64,65]. The Klein–Gordon equation in NIDS is considered in [66], the Weyl–Dirac equation in NIDS for graphene in [67], and the nonlinear Schrodinger equation in NIDS [68].

In fractal physics, the first application of NIDS to fractal systems and media was proposed by Le Guillou and Zinn-Justin in the 1977 paper [4] to describe the physics of crystals’ fractal structure, conidered as fractal media. Many different approaches and methods are used for the mathematical description of fractal media (see the review [69] and references therein). The most promising approach is the non-integer-dimensional space approach that uses continuous (continuum) models with the concept of the density of states. The continuous models of fractal media (CMFM) allow us to use integral and differential operators in linear (vector) spaces. The continuous medium model approach was first proposed and applied in the author’s works from 2005 to 2007. First, CMFM were proposed for continuum mechanics and hydrodynamics in [70,71,72], solid mechanics in [73,74], the electromagnetic fields on fractal media in [75,76,77], the magnetohydrodynamics of fractal media in [78], the gravitational field of the fractal distribution of particles in [79], the Ginzburg–Landau equation for fractal media in [80], the Fokker–Planck equation for fractal media in [81], and the Chapman–Kolmogorov equation for fractal media in [82]. The statistical mechanics of fractal distributions is described in [16,83,84] and physical kinetics in [85,86]. Most of these CMFM were later combined in Chapters 1–7 of the book [87]. From 2007, various CMFM have been actively developed by Ostoja-Starzewski including the thermomechanics of fractal media in [88,89], the thermoelasticity of fractal media in [90], turbulence in [91], extremum and variational principles for fractal porous media in [92,93], and electromagnetism on anisotropic fractals in [94]. Then, Ostoja-Starzewski, Li, Demmie, and Joumaa described fractal media and fractal solids in [95,96,97], waves in fractal media in [96,98,99], the micropolar continuum mechanics of fractal media in [100,101], and the acoustic-elastodynamic of fractal media in [102]. The continuum homogenization of fractal media is an important basis of the NIDS approach, which is considered by Ostoja-Starzewski, Li, Joumaa, and Demmie in [97,103].

From 2018, various CMFM have been developed by Mashayekhi, Stanisauskis, Pahari, Miles, Hussaini, Oates, Mehnert, and Steinmann in theoretical models and the experimental validation of viscoelasticity in fractal and non-fractal media [104,105], excluded volume effects in viscoelasticity in polymers [106], fractal order effects in soft elastomers [107], and entropy dynamics in fractal and fractional order viscoelasticity to elastomers [108].

The first self-consistent vector calculus of differential operators in NIDS was proposed only in the 2014 and 2015 papers [18,19], where the differential and integral NIDS operators were suggested. This allowed us to eliminate some shortcomings of the NIDS models of fractal media proposed in the 2004–2007 papers by using differential and integral operators in the NIDS proposed in [18,19]. In 2014–2018, the vector calculus in NIDS was applied to the fractal electrodynamics [109,110], the steady flow of fractal fluid [111,112], the elasticity of fractal material [113], the heat transfer in fractal materials [114], the acoustic waves in fractal media [115], quantum mechanics and statistical physics [116], Born–Infeld electrodynamics in fractional space [117], and the thermoelasticity of fractal media [118].

However, in the articles [18,19] and papers of 2015–2018, only some elements of vector calculus in NIDS were given and used. For example, these articles did not define line and surface NIDS integrals of scalar and vector functions in the general case. These articles also did not prove the NIDS generalization of the classical fundamental theorems of vector calculus such as the gradient theorem, the Gauss–Ostrogradsky (divergence) theorem, the Green theorem, and the Stokes theorem.

In this connection, the obvious task arises of constructing a self-consistent calculus of integrals and derivatives in NIDS and self-consistent NIDS vector calculus.

Let us use an example to explain what is meant by the self-consistency of the NIDS calculus.

1.

The NIDS divergence must be defined such that the NIDS Gauss–Ostrogradsky theorem is satisfied for the surface and volume NIDS integrals of vector fields.

2.

These surface NIDS integrals of vector fields must be defined through the surface NIDS integrals of scalar fields.

3.

The surface NIDS integrals of scalar fields must be defined such that the NIDS integration over a hypersphere should give the well-known value for the area of the D-dimensional sphere for any non-integer and integer positive values of dimensions $D>0$ of the NIDS.

Requirements of this kind are called principles in this work, and they will be used for self-consistent and correct definition of integral and differential operators of the NIDS calculus.

Three following types of principles will be proposed.

1.

The first type of principles ensures the self-consistency of the NIDS integral calculus.

2.

The second type of principles ensures the self-consistency of the NIDS differential calculus.

3.

The third type of principles ensures the mutual consistency between the integral and differential NIDS calculi.

The principles are used to formulate the NIDS calculus, since there are many works, which propose only individual NIDS operators or elements of the calculus that are not related to each other. As a result, these NIDS operators do not form any calculus, and their application leads to mathematical errors and to inadequate results in applied sciences.

This work consists of three following basic sections. Due to the length of the article, we will provide below a brief outline of the construction and description of the NIDS calculus. More detailed information can be seen in the table of contents.

• Section 2 formulates the principles and provides definitions of integral operators in NIDS, including the volume, surface, and line NIDS integrals of scalar functions. The basic properties of the NIDS integral operators are proved. The sequence of steps is as follows.

  1.

  Principles of NIDS integral calculus.

  2.

  Wilson approach for entire NIDS.

  3.

  Generalization for bounded regions.

  4.

  Product measure approach (PMA).

  5.

  Volume NIDS integrations.

  6.

  Difference between PMA and metric approach (MA).

  7.

  Line and surface NIDS integrals of scalar fields in PMA and MA.

  Note that the volume NIDS integration in terms of the product measure and metric approaches coincide. The implementation of the principles of the integral calculus of NIDS is proposed first for one variable and then for the general case.
• Section 3 formulates the principles and provides definitions of differential operators in NIDS, including proofs of the basic properties of these operators. The NIDS generalizations of the classical fundamental theorems of calculus are proved. The sequence of steps is as follows.

  1.

  Principles of NIDS differential calculus.

  2.

  NIDS differentiation of a single variable.

  3.

  Fundamental theorems of NIDS calculus.

• Section 4 formulates the principles and definitions of integral and differential operators of the vector calculus in NIDS, including NIDS generalizations of divergence, gradient, and curl operators. The surface and line NIDS integrals of vector fields are defined. The differential operators in NIDS are defined, and the basic properties of these operators are proved. The NIDS generalization of the classical fundamental theorems of vector calculus such as the gradient theorem, the Gauss–Ostrogradsky (divergence) theorem, Green theorem, and the Stokes theorem are proved. Some elements of the NIDS calculus of differential k-forms are proposed. The sequence of steps is as follows.

  1.

  Principles of NIDS vector calculus.

  2.

  NIDS differential vector operators in MA.

  3.

  Line and surface NIDS integrals of vector fields in PDA and MA.

  4.

  Fundamental theorems of NIDS vector calculus in PDA and MA.

  5.

  NIDS differential vector operator in PMA.

  6.

  Properties of the NIDS differential vector operator in PDA and MA.

  7.

  NIDS operators in orthogonal coordinate systems.

  8.

  NIDS differentiation with respect to NIDS volume.

  9.

  Calculus of NIDS differential forms.

In Section 5, a brief conclusion is given.

2. NIDS Integral Calculus

2.1. Non-Integer-Dimensional Space

In this subsection, we briefly discuss the concept of a non-integer-dimensional space (NIDS), which is also called D-dimensional space and fractional-dimensional space. This concept and its applications have been discussed in various works. Let us describe some basic approaches to formulating the NIDS concept and calculus in NIDS.

2.1.1. Stillinger Approach

In the Stillinger approach, proposed in the 1977 paper [14], it is assumed that NIDS contains points ${\mathbf{x}}_1$, ${\mathbf{x}}_2$, …, and has a topological structure that is specified by the following axioms.

Axiom 1. NIDS is a metric space.

Axiom 2. NIDS is dense in itself.

Axiom 3. NIDS is metrically unbounded.

Axiom 1 means that NIDS is an ordered pair $(W,r)$ consisting of a set W and a function $r:W\times W\to \mathbb{R}$ that is called metric, and $r({\mathbf{x}}_1,{\mathbf{x}}_2)$ is called the distance between the points ${\mathbf{x}}_1$ and ${\mathbf{x}}_2$ of NIDS. The function $r({\mathbf{x}}_1,{\mathbf{x}}_2)$ must satisfy the following conventional properties (axioms) of the metric.

(a)

The non-negativity property: the distance between two different points is non-negative $r({\mathbf{x}}_1,{\mathbf{x}}_2)⩾0$.

(b)

The symmetry property: the distance from ${\mathbf{x}}_1$ to ${\mathbf{x}}_2$ is equal to the distance from ${\mathbf{x}}_2$ and ${\mathbf{x}}_1$, i.e., $r({\mathbf{x}}_2,{\mathbf{x}}_1)=r({\mathbf{x}}_1,{\mathbf{x}}_2)$.

(c)

The zero distance property: the distance from a point to itself is equal to zero $r({\mathbf{x}}_1,{\mathbf{x}}_2)=0$, if and only if ${\mathbf{x}}_1={\mathbf{x}}_2$.

(d)

The triangle in equality: $r({\mathbf{x}}_1,{\mathbf{x}}_2)+r({\mathbf{x}}_2,{\mathbf{x}}_3)⩾r({\mathbf{x}}_1,{\mathbf{x}}_3)$.

Axiom 1 also means that the existence of a metric allows one to construct neighborhoods of a given positive radius around each NIDS point. Axiom 2 means that every such neighborhood about an arbitrary point ${\mathbf{x}}_1$ of the NIDS contains at least one other point ${\mathbf{x}}_2$. Axioms 1 and 2 together require that NIDS contains an infinite number of points. Axiom 3 means that, for every points ${\mathbf{x}}_1$ of NIDS and any positive real number $R>0$, there exists a point ${\mathbf{x}}_{\mathbf{2}}$ such that $r({\mathbf{x}}_1,{\mathbf{x}}_2)>R$.

The Stillinger approach has significant shortcomings.

• First, the Stillinger approach did not give strict definitions of the integrals and derivatives in NIDS, for which NIDS analogs of fundamental theorems are satisfied.
• Second, the scalar Laplace operator proposed by Stillinger is not a scalar product of two NIDS del operators.
• Third, the Stillinger approach does not define first-order vector differential NIDS operators that can be interpreted as NIDS generalizations of the standard gradient, divergence, and curl operator.
• Fourth, the NIDS analogs of the fundamental theorems of the standard vector calculus such as the gradient theorem, the Gauss–Ostrogradsky theorem, and the Stokes theorem are not proven.
• Fifth, Stillinger [14] assumes that the NIDS is not a vector space [14]. In ordinary Euclidean space and vector space, the linear combination of the space vectors is again an element of this space. Stillinger rejected this property for the non-integer-dimensional spaces. The Stillinger justification was the following. Since a vector space must have a finite integer (or infinite) number of basis vectors, and this number is considered as the dimension of the space, then NIDS cannot be considered as a vector space [14].

Due to these shortcomings, the Stillinger approach cannot be considered as consistent for constructing the calculus of integrals and derivatives in NIDS.

2.1.2. Palmer–Stavrinou Approach

The use of the product measure approach to formulate the NIDS concept was proposed by Palmer and Stavrinou in the 2004 paper [15]. The Palmer–Stavrinou approach [15] has the same shortcomings as the Stillinger approach, since the Palmer and Stavrinou paper [15] is based on the Stillinger approach.

2.1.3. Product Measure and Metric Approaches

The product measure approach has also been proposed in works [16,77,78,87,96]. Then, this approach was developed in the 2014 paper [18], where the NIDS vectors’ differential operators were proposed to describe anisotropic fractal media by using NIDS (fractional-dimensional space). In the product measure approach, each non-integer-dimensional space $W_k$ has own dimension ${\alpha }_k\in (0,1]$ and is measured on measurable space $(W_k,{\mathcal{S}}_k)$, where ${\mathcal{S}}_k$ is the $\sigma$-algebra on $W_k$. The product of the spaces $W_1\times \dots \times W_n$ and the product of the measures on $(W_k,{\mathcal{S}}_k)$ with $k=1,\dots ,n$ allows us to actually define some anisotropic non-integer-dimensional space (anisotropic NIDS) with the total dimension $D={\alpha }_1+\dots +{\alpha }_n$. Anisotropy of a non-integer-dimensional space means that its properties, namely its dimensions, can be different in different directions.

One can state that the first generalization of the standard vector calculus to NIDS was proposed in [18,19]. In these papers, the NIDS del operator, the NIDS gradient, the NIDS divergence, and the NIDS curl operator were proposed. Note that in the 2014 paper [18], in addition to the product measure approach, the metric approach was also proposed for formulating differential NIDS vector calculus. However, this paper did not describe in detail the self-consistent NIDS calculus of derivatives and integrals. The NIDS analogs of the standard vector calculus theorems such as the gradient theorem, the Gauss–Ostrogradsky theorem, and the Stokes theorem are not proven in [18] for the suggested NIDS gradient, NIDS divergence, and NIDS curl operator in the product measure approach and in the metric approach.

Despite this, various applications of the NIDS vector calculus operators in physics were suggested. For example, models have been proposed to describe the flow of fractal fluid [111,112], the elasticity of fractal media [19,113], fractal electrodynamics [19,109,110], heat transfer in fractal media [19,114], acoustic waves in fractal media [115], quantum mechanics and statistical physics [116], Born–Infeld electrodynamics in fractional space [117], and the thermoelasticity of fractal media [118].

In fact, the product measure approach and the metric approach in the form prosed in the 2014 paper [18] is a generalization of the Wilson approach, where it is assumed that NIDS is a vector space.

2.1.4. Wilson Approach

The Wilson approach, which was proposed in the 1973 paper [9], assumes that NIDS is the vector space (the Euclidean space). In the Wilson approach, the NIDS is actually infinite-dimensional, and the dimension of this space is determined precisely by the operation of the NIDS integration [10]. Wilson proposed a definition of the integration in NIDS (the NIDS integration), imposing restrictions (properties) that this integration must satisfy [9,10]. The axioms proposed by Stillinger for NIDS remain in the Wilson approach, because NIDS is considered a vector space.

Note that the Wilson approach has become generally accepted in modern physics, including quantum field theory and the statistical physics of phase transitions.

Let us note some disadvantages of the Wilson approach to formulating the NIDS calculus of integrals and derivatives.

• A disadvantage of the Wilson approach to NIDS is the fact that it is usually limited to consideration of NIDS integration. It can be said that NIDS differential calculus is practically not considered. As a result, the fundamental theorems of the NIDS calculus were not formulated and were not proved.
• The definition of the NIDS generalizations of the classical vector differential operators such as gradient, divergence, and curl were not proposed. As a consequence, the NIDS analogs of vector calculus theorems such as the gradient theorem, the Gauss theorem, and the Stokes theorem were not proved.
• The Wilson axioms are formulated only for the integration over all NIDS [9,10]. This is due to the fact that Wilson’s article [9] is devoted to the construction of dimensional regularization for quantum field theory in momentum space. The use of integration over the bounded regions of the momentum space is a separate and independent method of regularization [11,12,13] that is called cutoff regularization. Therefore, it is not necessary to consider bounded regions in NIDS for dimensional regularization.

These shortcomings of the Wilson approach can be eliminated in the formulation of the NIDS integral calculus by generalizing and extending this approach to bounded regions of NIDS and using the product of measures.

The purpose of this work is to construct a self-consistent NIDS calculus of integrals and derivatives in NIDS. In the formulation of NIDS integration, we will use the generally accepted Wilson approach [9,10], the product measure approach, and the metric approach proposed in [18]. In this paper, the Wilson approach will be extended to the product of measurable NIDS and the measured product. The construction of the NIDS calculus will be based on principles that will ensure the self-consistency of the integral and differential NIDS calculi themselves and principles that ensure the mutual consistency of these NIDS calculi. The Stillinger axioms for NIDS remain in our approach, because the NIDS is considered a vector space, similar to the Wilson approach. The Wilson axioms for the integration over all NIDS also remain in our approach but will be extended to the case of bounded regions in NIDS.

The Wilson approach, the product measure approach, the metric approach, and the proposed principles, which will be used to build the self-consistent NIDS calculus, are discussed in more detail in the following subsections.

2.1.5. Fractal Sets and Analysis on Fractals

There is a field of research close to NIDS calculus. This field of research is calculus on fractal sets that is usually called “Analysis on Fractals”. In this article, the “Analysis on Fractals” will not be discussed and considered. In this remark, we will limit ourselves to just a few comments and references to basic works.

Firstly, fractals sets are sets of points in the Euclidean space that have a fractional metric dimension (for example, in the sense of Hausdorff or Minkowski) or a metric dimension that is different from the topological dimension (for example, see the books by Falconer [119], Mattila [120], Aleksandrov and Pasynkov [121], Engelking [122], and Pesin [123]). Many fractal sets can be described as metric spaces, since suitable distance functions (metrics), which satisfy the axioms of a metric space, can be defined for these fractal sets.

Secondly, fractal sets are usually not vector spaces. A vector space is defined as a set of elements (vectors) for which the operations of addition and multiplication by a scalar are defined, satisfying certain axioms. Many fractal sets are characterized by the properties of self-similarity and not by the linear properties required for a vector space [124,125].

Thirdly, integration and differentiation on fractal sets are generally different from integration and differentiation in Euclidean space or on smooth manifolds. It requires the use of special definitions that take into account the fractal structure of the fractal set and its Hausdorff dimension. Instead of conventional measures, which are used in the Riemann and Lebesgue integration, some special measures adapted to fractal geometry are used.

There are many works on “Analysis on Fractals”. For example, the analysis on fractals is described in books [126,127] and papers [128,129,130], the calculus on the Cantor sets in [131], the Laplace operators on fractals in [132], the fractal generalization of the Gauss, Green, and Stokes theorems in [133], and the differential equations on fractals in the book [134].

Note that the “Analysis on Fractals” was not formulated in a way that could be successfully applied in quantum field theory for regularization instead of dimensional regularization and in other branches of modern physics.

It should also be noted that fractal media cannot be considered as fractal sets, since the fractal properties of these media are manifested only at scales limited both from above and from below [18,69].

2.2. Principles of NIDS Integral Calculus

In this paper, we will call “principles” those basic conditions that allow us to define the NIDS integration and NIDS differentiation in such a way that they form a self-consistent calculus in NIDS. Three following types of principles will be proposed.

1.

The first type of principles ensures the self-consistency of the NIDS integral calculus.

2.

The second type of principles ensures the self-consistency of the NIDS differential calculus.

3.

The third type of principles ensures the mutual consistency between the integral and differential NIDS calculi.

Based on these principles, a self-consistent NIDS calculus will be constructed in this paper. The principles of the first type are considered in this section. The principles of the second and third types are discussed in the following sections.

The approach to formulating the NIDS calculus based on principles is due to the fact that there are many works that propose only individual NIDS operators or elements of the calculus that are not related to each other. As a result, these NIDS operators do not form any calculus. Therefore, their application leads to mathematical errors and to inadequate results in applied sciences.

In this section, we can formulate the first type of principles for constructing the NIDS integral calculus and define integration in NIDS. Note that only volume NIDS integrals are considered in these subsections. The NIDS integrations along curved lines and over surfaces, as well as the principles for these integrals, will be considered in the first few subsections.

For the NIDS integral in D-dimensional space W, we will use the notation

$$I_{\Omega }^{D}f\left(\mathbf{x}\right)={\int }_{\Omega }f\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x})$$

(1)

that will be interpreted as the integrals of function $f\left(\mathbf{x}\right)$ over a region $\Omega \subseteq W$ of the D-dimensional space W with $D\in (0,n]$, where $n\in \mathbb{N}$, and $d{\mu }_n(D,\mathbf{x})$ denotes an integral measure. The index $n\in \mathbb{N}$, which is used in Equation (1), will be explained in the next subsection devoted to the product of measure of NIDS and the product measure approach. The introduced notation (1) for the NIDS integrals will allow us to formulate the principles not only in words but also in symbolic form.

The first principle is the principle of correspondence of the NIDS integration to standard integration in the spaces with finite integer dimensions $n\in \mathbb{N}$.

Principle 1 

(First Correspondence Principle of NIDS Integral Calculus). For the integer dimension $D=n\in \mathbb{N}$, the NIDS integral should give the standard integral (the Riemann or Lebesgue integrals) in the n-dimensional Euclidean space.

For the regions $\Omega \subseteq W$ of the NIDS with dimension $D\in (0,n]$, the NIDS integration should be defined such that the property

$$\underset{\epsilon \to 0+}{lim}{I}_{\Omega }^{n-\epsilon }f\left(\mathbf{x}\right)=\underset{\epsilon \to 0+}{lim}{\int }_{\Omega }f\left(\mathbf{x}\right)d{\mu }_n(n-\epsilon ,\mathbf{x})={\int }_{\Omega }f\left(\mathbf{x}\right)d{\mu }_n(n,\mathbf{x})$$

(2)

holds for all $n\in \mathbb{N}$ and all $f\left(\mathbf{x}\right)\in C(\Omega )$, where $\epsilon =n-D>0$ and

$$d{\mu }_n(n,\mathbf{x}):=\prod _{k=1}^{n}d{x}_k$$

(3)

with $n\in \mathbb{N}$.

The second correspondence principle of NIDS integral calculus is the principle that ensures the basic property of the non-integer dimensionality for the NIDS integrals. From the physical point of view, the space can be called D-dimensional space, if the volume of the ball increases by the factor ${\lambda }^D$ with a non-integer value of D, when the ball radius R increases by the factor $\lambda >1$, [18,19]. Therefore, the use of the volume NIDS integration to calculate the volume $V_D\left({\Omega }_B\right)$ of the D-dimensional ball region

$${\Omega }_B=\{\mathbf{x}\in W:|\mathbf{x}|⩽R,R>0\},$$

(4)

where R will be called the radius of the ball ${\Omega }_B$, should give the well-known expression

$$V_D\left({\Omega }_B\right)={\displaystyle \frac{2{\pi }^{D/2}}{D\Gamma (D/2)}}{R}^D,$$

(5)

for all $D\in (0,n]$ and all $R>0$.

To formulate the second principle, we should define the concept of the bounded region in NIDS. The main property of the NIDS bounded region is that all distances between its points must be finite.

Definition 1. 

Let Ω be a region of the NIDS, in which $r({\mathbf{x}}_1,{\mathbf{x}}_2)$ is the distance between the points ${\mathbf{x}}_1$ and ${\mathbf{x}}_2$ of this NIDS.

Then, the region Ω is called the bounded region, if the condition

$$r({\mathbf{x}}_1,{\mathbf{x}}_2)<\infty$$

(6)

holds for all two points ${\mathbf{x}}_1$ and ${\mathbf{x}}_2$ that belong to the region Ω.

Let us give the formulation of the principle that provides the property of the non-integer dimensionality for the NIDS integrals.

Principle 2 

(Second Correspondence Principle of NIDS Integral Calculus). For the D-dimensional space W with non-integer values of the dimension $D\in (0,n]$, the calculation of the volume NIDS integrals of the function $f\left(\mathbf{x}\right)={\chi }_{\Omega }\left(\mathbf{x}\right)$ should give the well-known expressions of the volumes for the D-dimensional bounded regions $\Omega \subset W$ by the equation

$${\int }_W{\chi }_{\Omega }\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x})=V_D(\Omega ),$$

(7)

where ${\chi }_{\Omega }\left(\mathbf{x}\right)$ is the characteristic function of the region Ω, i.e., ${\chi }_{\Omega }\left(\mathbf{x}\right)=1$ if $\mathbf{x}\in \Omega$, and ${\chi }_{\Omega }\left(\mathbf{x}\right)=0$ if $\mathbf{x}\notin \Omega$.

In a particular case, the calculation of the volume NIDS integrals of the function $f\left(\mathbf{x}\right)={\chi }_{{\Omega }_B}\left(\mathbf{x}\right)$ over the ball region ${\Omega }_B:=\{\mathbf{x}:|\mathbf{x}|⩽R,R>0\}$, where $|x|:=\sqrt{r(\mathbf{x},\mathbf{x})}$, should give the well-known expression of the volume for the D-dimensional ball. This principle means that the equation

$${\int }_W{\chi }_{{\Omega }_B}\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x})=V_D\left({\Omega }_B\right):={\displaystyle \frac{2{\pi }^{D/2}}{D\Gamma (D/2)}}{R}^D$$

(8)

must be satisfied for all $D\in (0,n]$, all $n\in \mathbb{N}$, and all $R>0$.

Let us give some remarks about Principle 2.

Remark 1. 

In the Wilson approach to the formulation of the NIDS integration, the fulfillment of the scaling property for the NIDS integrals is assumed. Therefore, the scaling property does not impose any restrictions on the normalization of the NIDS integral, and this property does not replace the second correspondence principle. In other words, the scaling property for the NIDS integration over D-dimensional ball region ${\Omega }_B$ with radius R requires the dependence of the volume $V_D\left({\Omega }_B\right)$ of this region on $R^D$ in the form

$$V_D\left({\Omega }_B\right)=B_n\left(D\right)R^D.$$

(9)

The scaling property does not impose any restrictions on the coefficient $B_n\left(D\right)$. The first correspondence principle actually requires only that this coefficient coincide with the standard expressions

$$B_n\left(n\right)={\displaystyle \frac{2{\pi }^{n/2}}{n\Gamma (n/2)}},$$

(10)

for all integer dimensions $D=n\in \mathbb{N}$. The second correspondence principle gives restrictions on the coefficient $B_n\left(D\right)$.

For this reason, it is useful to formulate the second correspondence principle explicitly as a condition for the self-consistency of the formulation of NIDS integration. In other words, it is proposed to postulate that the NIDS integration over the D-dimensional ball region ${\Omega }_B:=\{\mathbf{x}:|\mathbf{x}|⩽R\}$ with the radius $R>0$ should give

$${\int }_{{\Omega }_B}{\chi }_{{\Omega }_B}\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x})={\displaystyle \frac{2{\pi }^{D/2}}{D\Gamma (D/2)}}{R}^D,$$

(11)

which must be satisfied for all values of $D>0$.

Note that in the Wilson approach [9,10], the normalization condition is postulated for the value of the NIDS integral of the exponent $exp(-{\mathbf{x}}^2)$ over the entire D-dimensional space. As a normalization condition, Wilson proposed the equation

$${\int }_{W}d{\mu }_n(D,\mathbf{x})exp(-{\mathbf{x}}^2)={\pi }^{D/2},$$

(12)

which must be satisfied for all $D>0$ (see Equation A7 in paper [9] or Equation 4.1.8 in [10]).

In the general case, the second correspondence principle of NIDS integral calculus (Principle 2) means that the NIDS integration should allow one to calculate the D-dimensional volumes of the D-dimensional ball ${\Omega }_B$, parallelepiped ${\Omega }_P$, cylinder ${\Omega }_C$, ellipsoid $W_E$, and other bounded regions in NIDS by the equation

$${\int }_W{\chi }_{\Omega }\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x})=V_D(\Omega )$$

(13)

for $\Omega ={\Omega }_B$, $\Omega ={\Omega }_P$, $\Omega ={\Omega }_C$, and others.

Note that NIDS can be characterized by the power-law relation for the volume of D-dimensional parallelepiped, ball, and cylinder in NIDS.

Example 1. 

For the first example, the NIDS integration of the D-dimensional parallelepiped region ${\Omega }_P:\{\mathbf{x}:0⩽x_k⩽l_{k}k=1,\dots ,n\}$ must be derived by Equation (7), and the volume NIDS integral must give

$$V_D\left({\Omega }_P\right)=P_n({\alpha }_1,\dots ,{\alpha }_n)\prod _{k=1}^n{\left(l_k\right)}^{{\alpha }_k},$$

(14)

where the coefficient $P_n({\alpha }_1,\dots ,{\alpha }_n)$ must satisfy the condition $P_n(1,\dots ,1)=1$; that is, it must be equal to one when all ${\alpha }_k$ are equal to one. The parameter ${\alpha }_k\in (0,1]$ is the non-integer dimension along $X_k$-axis, $k=1,\dots ,n$. The parameter ${\alpha }_k$ ($k=1,\dots ,n$) describes how to increase the volume of the region ${\Omega }_P$ in the case of increasing the size of the parallelepiped ${\Omega }_P$ along the $X_k$-axis, when the sizes along other axes do not change. One can state that the value

$$L_k\left({\alpha }_k\right)=P_1\left({\alpha }_k\right){\left(l_k\right)}^{{\alpha }_k}$$

(15)

can be interpreted as the length of the edge of the region ${\Omega }_P$ in the NIDS with the dimension ${\alpha }_k\in (0,1]$.

Let us consider the parallelepiped region $\lambda {\Omega }_P:=\{\mathbf{x}:0⩽x_k⩽\lambda l_k,k=1,\dots ,n\}$, in which all parameters $l_k$ were increased by $\lambda >1$ times. Then, the volume of this region is increased as

$$V_D\left(\lambda {\Omega }_P\right)=P_n({\alpha }_1,\dots ,{\alpha }_n)\prod _{k=1}^n{\left(\lambda l_k\right)}^{{\alpha }_k}={\lambda }^D{P}_n({\alpha }_1,\dots ,{\alpha }_n)\prod _{k=1}^n{l}_k^{{\alpha }_k}={\lambda }^D{V}_D\left({\Omega }_P\right),$$

(16)

where

$$D=\sum _{k=1}^n{\alpha }_k\in (0,n].$$

(17)

The sum (17) can be called the “total dimension” (“volume dimension”) of the NIDS. The property (16) describes the scaling property that means the arbitrariness (non-integrity) of the dimension $D\in (0,n]$.

Example 2. 

For the second example, the NIDS integration of the D-dimensional ball ${\Omega }_B=\{\mathbf{x}:|\mathbf{x}|⩽R,R>0\}$ must be derived by Equation (7), and the volume NIDS integral must give

$$V_D\left({\Omega }_B\right)=B_n({\alpha }_1,\dots ,{\alpha }_n)R^D,$$

(18)

where

$$B_n({\alpha }_1,\dots ,{\alpha }_n):={\displaystyle \frac{2{\pi }^{D/2}}{D\Gamma (D/2)}}={\displaystyle \frac{{\pi }^{D/2}}{\Gamma (D/2+1)}},$$

(19)

with $D={\alpha }_1+\dots +{\alpha }_n\in (0,n]$. For integer values of D, Equations (18) and (19) give the well-known expression

$$V_1\left({\Omega }_B\right)=2R,V_2\left({\Omega }_B\right)=\pi R^2,$$

(20)

$$V_3\left({\Omega }_B\right)={\displaystyle \frac{4\pi }{3}}{R}^3,V_4\left({\Omega }_B\right)={\displaystyle \frac{{\pi }^2}{2}}{R}^4,$$

(21)

where we use $\Gamma (3/2)=\sqrt{\pi }/2$, $\Gamma \left(2\right)=1$, and $\Gamma (5/2)=3\sqrt{\pi }/4$.

If we denote the NIDS ball region $\lambda {\Omega }_B$, in which the radius R was increased by $\lambda >1$ times $\lambda {\Omega }_B:=\{\mathbf{x}:|\mathbf{x}|⩽\lambda R\}$, then the NIDS volume is increased as

$$V_D\left(\lambda {\Omega }_B\right)={\displaystyle \frac{2{\pi }^{D/2}}{D\Gamma (D/2)}}{\left(\lambda R\right)}^D={\lambda }^D{\displaystyle \frac{2{\pi }^{D/2}}{D\Gamma (D/2)}}{R}^D={\lambda }^D{V}_D\left({\Omega }_B\right),$$

(22)

for all $\lambda >1$ and all $D>0$.The property (22) describes the scaling property that means the arbitrariness (non-integrity) of the dimension $D\in (0,n]$.

Example 3. 

For third example, the NIDS integration of the D-dimensional cylinder region ${\Omega }_C:=\{\mathbf{x}:0⩽x_n⩽\lambda l_k,{\mathbf{x}}^2-x_n^2⩽R^2,R>0\}$ must be derived by Equation (7), and the volume NIDS integral must give

$$V_D\left({\Omega }_C\right)=C_n({\alpha }_1,\dots ,{\alpha }_n)R^d{\left(l_n\right)}^{{\alpha }_n},$$

(23)

where

$$d:=\sum _{k=1}^{n-1}{\alpha }_k,D:=d+{\alpha }_n,$$

(24)

$$C_n({\alpha }_1,\dots ,{\alpha }_n)=B_{n-1}({\alpha }_1,\dots ,{\alpha }_{n-1})P_1\left({\alpha }_n\right),$$

(25)

and R is the radius of the cylinder base, if the cylinder axis is parallel to the $X_n$-axis.

For the cylinder region $\lambda {\Omega }_C$, in which the radius R and the length of the cylinder $l_n$ are increased by $\lambda >1$ times, then the volume is increased as

$$V_D\left(\lambda {\Omega }_C\right)=C_n({\alpha }_1,\dots ,{\alpha }_n){\left(\lambda R\right)}^d{\left(\lambda l\right)}^{{\alpha }_n}={\lambda }^{d+{\alpha }_n}{V}_D\left({\Omega }_C\right)={\lambda }^D{V}_D\left({\Omega }_C\right),$$

(26)

for all $\lambda >1$ and all $D>0$. The property (26) describes the scaling property that means the arbitrariness (non-integrity) of the dimension $D\in (0,n]$.

As a result, these examples demonstrate that the volumes of the D-dimensional regions $\Omega$ such as the ball ${\Omega }_B$, parallelepiped ${\Omega }_P$, and cylinder ${\Omega }_C$ in the D-dimensional space satisfy the scaling property

$$V_D(\lambda \Omega )={\lambda }^D{V}_D(\Omega ),$$

(27)

if all linear sizes of the D-dimensional regions $\Omega$ are increased by $\lambda >1$ times, where $D\in (0,n]$. One can say that this property is a characteristic property for the wide class of NIDS. Therefore, it can be assumed that, in applications, the dimension of NIDS can be “measured” and calculated by the equation

$$D={log}_{\lambda }\left({\displaystyle \frac{V_D(\lambda \Omega )}{V_D(\Omega )}}\right)$$

(28)

in the physics of fractal media, for example.

Let us give the third principle called the self-consistency principle of NIDS integration. This principle is necessary for the self-consistency of the NIDS integral calculus in the form of the relationship between multiple NIDS integrals and iterated NIDS integrals for some cases. This principle expresses the requirement of the existence of the possibility of calculating multiple NIDS integrals through the calculation of iterated NIDS integrals.

Principle 3 

(Self-Consistency Principle of NIDS Integration). In order for NIDS integral calculus to be self-consistent, the NIDS generalization of the Fubini theorem must exist for multiple NIDS integrals. For the multiple NIDS integrals over the product of the $D_k$-dimensional spaces $W_k$ with $D_k={\alpha }_k\in (0,1]$, a way to calculate through the iterated NIDS integrals must exist.

For $D_k$-dimensional spaces with $D_k={\alpha }_k\in (0,1]$, $k=1,2$, the NIDS integration over the D-dimensional space $W:=W_1\times W_2$ with $D={\alpha }_1+{\alpha }_2\in (0,2]$ of function $f\left(\mathbf{x}\right)$ should satisfy the equation

$${\int }_{W}f\left(\mathbf{x}\right)d{\mu }_2(D,\mathbf{x})={\int }_{W_1}d{\mu }_1({\alpha }_1,x_1){\int }_{W_2}d{\mu }_1({\alpha }_2,x_2)f(x_1,x_2)=$$

$${\int }_{W_2}d{\mu }_1({\alpha }_2,x_2){\int }_{W_1}d{\mu }_1({\alpha }_1,x_1)f(x_1,x_2),$$

(29)

where $x_1\in W_1$, $x_2\in W_2$, and $\mathbf{x}=(x_1,x_2)\in W$.

Remark 2. 

This principle is related to the importance of the NIDS generalization of the Fubini theorem for the self-consistency of the NIDS integral calculus and for applications of this calculus. The classical Fubini theorem is a fundamental element of the classical calculus that connects two types of integrals and specifies a way to calculate multiple integrals through the calculation of iterated integrals. The existence of the Fubini theorem for the NIDS integration makes it possible to reduce the calculation of the multiple NIDS integrals to the calculation of iterated NIDS integrals.

Remark 3. 

The third principle leads to the necessity of using products of measures and products of non-integer-dimensional spaces. This requirement of Principle 3 leads to the need to consider the product measure approach to NIDS integral calculus.

This approach will be discussed after describing the Wilson approach and extending this approach to bounded regions of the NIDS by using the second correspondence principle of NIDS integral calculus (Principle 2).

2.3. Integration in Non-Integer-Dimensional Space: Wilson Approach

2.3.1. Wilson Approach

The definition of the NIDS integration can be based on the Wilson approach [9,10]. In this approach, the NIDS is considered as an infinite-dimensional space ${\mathcal{H}}^D$, and the dimension D of this space is determined precisely by integration operation [9,10].

The space with non-integer dimension must be infinite-dimensional space [9,10].

In the Wilson approach [9,10], infinite-dimensional vectors $\mathbf{x}\in {\mathcal{H}}^D$ are described by infinite sequenced of components $\mathbf{x}=(x_1,x_2,\dots )$ such that

$$\mathbf{x}=\sum _{k=1}^{\infty }{x}_k{\mathbf{e}}_k,$$

(30)

for all $\mathbf{x}\in {\mathcal{H}}^D$. The system $\left\{{\mathbf{e}}_k\right\}$ of elements forms a semi-basis; that is, each element $\mathbf{x}\in \mathcal{H}$ is uniquely representable (30). The term semi-basis is used rather than basis because the convergence of the sum ${\sum }_k^{\infty }{x}_k^2$ is not required [9], p. 2926. The scalar product in the infinite-dimensional space ${\mathcal{H}}^D$ is

$$(\mathbf{x},\mathbf{y})=\sum _{k=1}^{\infty }{x}_k{y}_k,$$

(31)

and

$$|\mathbf{x}|=\sqrt{(\mathbf{x},\mathbf{x})},$$

(32)

for all $\mathbf{x}\in W$.

In works [9,10], the space ${\mathcal{H}}^D$ is called infinite-dimensional Euclidean space [10]. One can say that Wilson considers an infinite-dimensional semi-separable Hilbert space as such an infinite-dimensional space, $\mathcal{H}={\mathcal{H}}^D$. The term “semi-separable” is used, since in the Wilson paper [9], p. 2926, there are no convergence requirements in the space. For the space ${\mathcal{H}}^D$, the complete set of components $\left\{x_k\right\}$ of the set $\mathbf{x}$ is not explicitly indicated, and the convergence of the sum ${\sum }_k^{\infty }{x}_k^2$ is not required. Wilson [9] uses only the partition $\mathbf{x}$ into a finite set of components $x_1,\dots ,,x_m$, which form the vector ${\mathbf{x}}_{\Vert }$, and a perpendicular vector ${\mathbf{x}}_{\perp }$. This will lead to the fact that vectors $\mathbf{x}$ in the function $f\left(\mathbf{x}\right)$ will lie in some m-dimensional subspace [10].

In paper [9], Wilson considered properties (axioms, principles) that must be imposed on the NIDS integration of functions $f\left(\mathbf{x}\right)$ in order to regard it as D-dimensional integration. These properties (axioms) are natural and necessary in applying the NIDS integrations in physics, including quantum field theory and the theory of fractal media.

In the Wilson paper [9,10], the properties (axioms, principles) were formulated for the NIDS integration over the entire NIDS. The consideration of only the entire NIDS is motivated by the aim of the Wilson paper [9], which was devoted to the construction of the dimensional regularization for quantum field theory in momentum space [10]. The use of integration over the bounded regions of the momentum space is the separate independent regularization called the “cutoff regularization” [11,12,13], which has not been related to dimensional regularization in quantum field theory, at least until now.

Wilson proposed the properties of the integration in D-dimensional space that were formulated as follows [9].

1.

The property of linearity of the NIDS integration over the entire D-dimensional space

$$\int \left(a{f}_1\left(\mathbf{x}\right)+b{f}_2\left(\mathbf{x}\right)\right)d^D\mathbf{x}=a\int f_1\left(\mathbf{x}\right)d^D\mathbf{x}+b\int f_2\left(\mathbf{x}\right)d^D\mathbf{x}$$

(33)

must be satisfied for all real numbers $a,b\in \mathbb{R}$ and all $D>0$.

2.

The translation property of the NIDS integration over the entire D-dimensional space

$$\int f(\mathbf{x}+\mathbf{u})d^D\mathbf{x}=\int f\left(\mathbf{x}\right)d^D\mathbf{x}$$

(34)

must be satisfied for all fixed vectors $\mathbf{u}$ of the D-dimensional space and all $D>0$.

Note that, in this property, it is assumed that

$$d^D(\mathbf{x}+\mathbf{u})=d^D\mathbf{x}.$$

(35)

3.

The scaling property of the NIDS integration over the entire D-dimensional space

$$\int f\left(\lambda \mathbf{x}\right)d^D\mathbf{x}={\lambda }^{-D}\int f\left(\mathbf{x}\right)d^D\mathbf{x}$$

(36)

must be satisfied for all $\lambda >0$ and all $D>0$.

Remark 4. 

Collins in [10] proposed rotation covariance as an additional property of the NIDS integration in the entire D-dimensional space. In fact, such a property means that, if the function has the form if $f\left(\mathbf{x}\right)=f(|\mathbf{x}|)$ (or if $f\left(\mathbf{x}\right)=f\left({\mathbf{x}}^2\right)$), then integration over the entire D-dimensional space W can be represented in the form

$$\int f(|\mathbf{x}|)d^D\mathbf{x}=B\left(D\right){\int }_0^{\infty }f\left(x\right)x^{D-1}dx,$$

(37)

where $x:=|\mathbf{x}|=\sqrt{{\mathbf{x}}^2}$, and $B\left(D\right)$ is a coefficient that depends only on $D>0$.

In addition to the three properties, Wilson proposed a normalization condition for integration in the entire D-dimensional space [9,10] in the form of the value of the NIDS integral of the exponent

$$\int exp(-{\mathbf{x}}^2)d^D\mathbf{x}={\pi }^{D/2}$$

(38)

that should be satisfied for all $D>0$.

Remark 5. 

The linearity property holds for any integration. The properties of invariance under translations and rotations are the main properties of Euclidean space. The scaling property is the main property of D-dimensionality. The scaling property was proposed to ensure that the volume in the D-dimensional space has the dependence ${|\mathbf{x}|}^D$, [9], p. 2924.

Three properties proposed by Wilson ensure that the NIDS integration is unique up to normalization [9,10]. The proof of the uniqueness theorem is given in Section 4.1 of the book [10], p. 55. The properties and normalization proposed by Wilson for the NIDS integration over the entire NIDS uniquely defines the NIDS integrals for any real $D>0$. In fact, the NIDS integration is defined by the special form of the standard measure of integration in the integer-dimensional space ${\mathbb{R}}^n$, and normalization uniquely defines the NIDS integrals for any real $D>0$.

Remark 6. 

The space with a non-integer dimension must be infinite-dimensional space [9,10]. The reason for the infinite dimensionality of the vector space is that the NIDS integral with $D\in (1,2)$ can be used not only in theories and mathematical models with a space with dimension $n=2$ but also for theories and models with any higher dimension (for example, for $n=3$, $n=4$, or $n=8$).

In general, functions $f\left(\mathbf{x}\right)$, which are integrated in NIDS, could be any function of the components of the vector argument $f\left(\mathbf{x}\right)=f(x_1,x_2,\dots )$. In general, we can have infinitely many vectors ${\mathbf{y}}_j$ with $j\in \mathbb{N}$. For example, if we use rotationally covariant functions, then the scalar function is a function only of the scalar products of the vectors $\mathbf{x}$ and ${\mathbf{y}}_j$ in the form

$$f=f({\mathbf{x}}^2,({\mathbf{y}}_1,\mathbf{x}),\dots ,{\mathbf{y}}_1^2,({\mathbf{y}}_2,\mathbf{x}),\dots ,{\mathbf{y}}_2^2,\dots ).$$

(39)

The vectors ${\mathbf{y}}_1,{\mathbf{y}}_2,\dots$ in functions (39) can be interpreted as “external” vectors [10]. The vector space must be large enough to be able to consider models with all $n\in \mathbb{N}$. Since $n\in \mathbb{N}$ is arbitrary, we should use infinite dimension, in general.

In order to obtain a realization of the vectors $\mathbf{x},{\mathbf{y}}_1,{\mathbf{y}}_2,\dots$, it is assumed that they are vectors in standard vector space of the infinite dimension $\mathcal{H}$.

To define the NIDS integral of the scalar function $f\left(\mathbf{x}\right)$, one can use the finite-dimensional subspace $E^m$, with $dim{E}^m=m\in \mathbb{N}$ in the infinite-dimensional space $\mathcal{H}$, which contains all the vectors ${\mathbf{y}}_j$. In this case, we can represent the vector $\mathbf{x}$ in the form

$$\mathbf{x}={\mathbf{x}}_{\Vert }+{\mathbf{x}}_{\perp }=\sum _{k=1}^m{x}_k{\mathbf{e}}_k+{\mathbf{x}}_{\perp },$$

(40)

where ${\mathbf{x}}_{\Vert }$ is the component of the vector $\mathbf{x}$ in the subspace $E^m$ (${\mathbf{x}}_{\Vert }\in E^m$), and ${\mathbf{x}}_{\perp }$ is the orthogonal component of the vector $\mathbf{x}$, where the scalar product $({\mathbf{x}}_{\Vert },{\mathbf{x}}_{\perp })=0$. The subspace $E^m$, in which the ${\mathbf{y}}_j$ lies, is spanned by an orthonormal basis ${\mathbf{e}}_k$, with $k=1,\dots ,m$.

Then, the NIDS integral over $\mathbf{x}\in {\mathcal{H}}^D$ is defined as a sequential action of two integrations. The first is the NIDS integration over the $(D-m)$-dimensional space ${\mathcal{H}}^{D-m}$ with respect to the variable ${\mathbf{x}}_{\perp }$. The second is the ordinary (standard) m-dimensional integration in the space $E^m$ with respect to the variable ${\mathbf{x}}_{\Vert }$, ($m\in \mathbb{N}$). As a result, the NIDS integration is defined as

$${\int }_{{\mathcal{H}}^D}f\left(\mathbf{x}\right)d^D\mathbf{x}={\int }_{E^m}{d}^m{\mathbf{x}}_{\Vert }{\int }_{{\mathcal{H}}^{D-m}}f\left(\mathbf{x}\right)d^{D-m}{\mathbf{x}}_{\perp },$$

(41)

where

$$d^m{\mathbf{x}}_{\Vert }=\prod _{k=1}^{m}d{x}_k.$$

(42)

In Equation (41), we use the following notations.

• $\mathcal{H}={\mathcal{H}}^D$ is the D-dimensional space with $D>0$.
• $E^m$ is the m-dimensional lineal space (Euclidean space) with $m\in \mathbb{N}$.
• ${\mathcal{H}}^{D-m}$ is the $(D-m)$-dimensional space.

Since $f\left(\mathbf{x}\right)$ does not depend on the direction of the vector ${\mathbf{x}}_{\perp }$ [10], we have

$$f\left(\mathbf{x}\right)=f({\mathbf{x}}_{\Vert },{\mathbf{x}}_{\perp })=f({\mathbf{x}}_{\Vert },|{\mathbf{x}}_{\perp }|)=f({\mathbf{x}}_{\Vert },x_{\perp }),$$

(43)

where $x_{\perp }=|{\mathbf{x}}_{\perp }|$. Therefore, we get

$${\int }_{{\mathcal{H}}^{D-m}}f\left(\mathbf{x}\right)d^{D-m}{\mathbf{x}}_{\perp }=$$

$${\displaystyle \frac{2{\pi }^{\beta /2}}{\Gamma \left(\beta /2\right)}}{\int }_0^{\infty }f\left(\mathbf{x}\right)|{\mathbf{x}}_{\perp }{|}^{\beta -1}d|{\mathbf{x}}_{\perp }|={\displaystyle \frac{2{\pi }^{\beta /2}}{\Gamma \left(\beta /2\right)}}{\int }_0^{\infty }f\left(\mathbf{x}\right)x_{\perp }^{\beta -1}d{x}_{\perp },$$

(44)

where $\beta =D-m$.

As a result, Wilson derived [9,10] the definition of D-dimensional integration in terms of the standard integration in integer-dimensional space

$${\int }_{{\mathcal{H}}^D}f\left(\mathbf{x}\right)d^D\mathbf{x}={\displaystyle \frac{2{\pi }^{\beta /2}}{\Gamma \left(\beta /2\right)}}{\int }_{E^m}\prod _{k=1}^{m}d{x}_k{\int }_0^{\infty }f\left(\mathbf{x}\right)x_{\perp }^{\beta -1}d{x}_{\perp },$$

(45)

where $\beta =D-m$.

It is important to emphasize that the result is independent of the choice of the “parallel” subspace $E^m$ of the vectors ${\mathbf{x}}_{\Vert }$. The m-dimensional “parallel” subspace of vectors ${\mathbf{x}}_{\Vert }$ was chosen with only one condition, namely that it contains all external vectors ${\mathbf{y}}_j$. If necessary, it is possible to extend this subspace to include extra dimensions [10]. The different choices of the “parallel” subspace have no effect on the value of the NIDS integral (see Section 4 in [10]).

In the particular case of Equation (45), with $m=0$ and $f\left(\mathbf{x}\right)=f(|\mathbf{x}|)$, the D-dimensional integration over the entire NIDS with dimension $\beta =D$ in terms of the standard integration can be represented as

$${\int }_{{\mathcal{H}}^D}f(|\mathbf{x}|)d^D\mathbf{x}={\displaystyle \frac{2{\pi }^{D/2}}{\Gamma (D/2)}}{\int }_0^{\infty }f\left(x\right)x^{D-1}dx,$$

(46)

where $x=|\mathbf{x}|$. In the Wilson paper [9], Equation (46) is given as Equation (A9).

As a result, in the Wilson approach, the integration in the D-dimensional space with $D>0$ is represented by the standard integration in the integer-dimensional space.

2.3.2. Generalization of the Wilson Approach

A rigorous construction of a generalization of the Wilson approach will be described in following subsections within the framework of the product measure approach. This subsection provides some comments on the Wilson approach and the direction of its generalization.

(1A) Let us use the analytic continuation method that allows one to define a function and operators on a new domain based on its values on some initial subset. Using the analytical continuation of the right side of Equation (46) with respect to the variable x into the region of negative values, we obtain the functional

$$I_W^{D}f\left(x\right):={\int }_{-\infty }^{+\infty }f\left(x\right)c_1(D,x)dx,$$

(47)

where $x\in {\mathbb{R}}^1$, and

$$c_1(D,x):={\displaystyle \frac{{\pi }^{D/2}}{\Gamma (D/2)}}{|x|}^{D-1},$$

(48)

with $D>0$. This allows x to be interpreted as a coordinate in the space ${\mathbb{R}}^1$.

Note that if $f\left(x\right)=f(|x|)$, then Equation (47) describes integration in the D-dimensional space by Equation (46), since

$$I_W^{D}f\left(x\right):={\displaystyle \frac{2{\pi }^{D/2}}{\Gamma (D/2)}}{\int }_0^{+\infty }f\left(x\right)x^{D-1}dx.$$

(49)

(1B) If $D=1$, then $c_1(D,x)=1$, and Equation (49) is the standard integration over the entire one-dimensional space

$$I_W^{1}f\left(x\right):={\int }_{-\infty }^{+\infty }f\left(x\right)dx.$$

(50)

It is obvious that the integral $I_W^{D}f\left(x\right)$ with $D=n\in \mathbb{N}$ and $n>1$ cannot give the integration over the entire n-dimensional space ${\mathbb{R}}^n$, since

$${\displaystyle \frac{{\pi }^{n/2}}{\Gamma (n/2)}}{\int }_{-\infty }^{+\infty }f\left(x\right)x^{n-1}dx\ne {\int }_{{\mathbb{R}}^n}f(x_1,\dots ,x_n)d^n\mathbf{x},$$

(51)

for $n>1$.

(1C) Let us use the analytical continuation of right side of the Equation (45) with respect to the variable x, and we can consider the integral

$$I_W^{\{{\alpha }_1,\dots ,{\alpha }_m,\alpha \}}f(x_1,\dots ,x_m,x):={\int }_{{\mathbb{R}}^m}\left(I_W^{\alpha }f(x_1,\dots ,x_m,x)\right)(x_1,\dots ,x_m)\prod _{k=1}^{m}d{x}_k=$$

$${\int }_{{\mathbb{R}}^m}{\int }_{-\infty }^{+\infty }f(x_1,\dots ,x_m,x)c_1(\alpha ,x)dx\prod _{k=1}^{m}d{x}_k,$$

(52)

where ${\alpha }_1=\dots ={\alpha }_m=1$, $\alpha \in (0,1]$, and $I_W^{\alpha }f(x_1,\dots ,x_m,x)$ is NIDS integration with respect to the variable x that is defined by Equation (47), $D=\alpha +m$, with $\alpha >0$ and $m\in \mathbb{N}$. Then, the integral (52) gives the standard integration in the integer-dimensional space ${\mathbb{R}}^n$, if $\alpha =1$ and $m=n-1$. In this case, Equation (52) can be written as

$$I_W^{\{{\alpha }_1,\dots ,{\alpha }_{n-1},\alpha \}}f(x_1,\dots ,x_n):={\int }_{{\mathbb{R}}^n}f(x_1,\dots ,x_n)d^n\mathbf{x},$$

(53)

where ${\alpha }_1=\dots ={\alpha }_{n-1}=\alpha =1$, ${\alpha }_n=\alpha$, $x_n:=x$, $\mathbf{x}=(x_1,\dots ,x_n)$, and

$$D=\sum _{k=1}^n{\alpha }_k,$$

(54)

where ${\alpha }_1=\dots ={\alpha }_{n-1}=1$, and ${\alpha }_n=\alpha \in (0,1]$.

In fact, this approach, which is based on Equation (52) and, therefore, on Equation (53), distinguishes only one direction in the space ${\mathbb{R}}^n$ (the direction along the $X_n$ axis), along which, the spatial dimension is a non-integer number. In all other directions along the other coordinate axes $(X_1,\dots ,X_{n-1})$, the spatial dimensions are integer and equal to one.

(2) There is another way to define integration, for which the first correspondence principle of NIDS integral calculus (Principle 1) would be satisfied. We can consider the integration in the form

$$I_W^{\left\{\alpha \right\}}f(x_1,\dots ,x_n):=I_{W_1}^{{\alpha }_1}\dots I_{W_n}^{{\alpha }_n}f(x_1,\dots ,x_n),$$

(55)

where $\left\{\alpha \right\}=({\alpha }_1,\dots ,{\alpha }_n)$, and $D={\alpha }_1+\dots +{\alpha }_n$, with ${\alpha }_k\in (0,1]$ for all $k=1,\dots ,n$. This way is called the product measure approach. To define the multiple NIDS integration, one can consider the NIDS as the product of the measured non-integer-dimensional spaces $W_k$ with dimensions ${\alpha }_k\in (0,1]$. In this case, along each orthogonal coordinate axis $X_k$, $k=1,\dots ,n$, the space $W_k$ has its own non-integer dimension ${\alpha }_k\in (0,1]$, i.e., $dim\left(W_k\right)={\alpha }_k$. This product measure approach is described in the next subsection (see Section 2.5).

(3) Note that using Equation (47), Equation (55) can be written as

$$I_W^{\left\{\alpha \right\}}f(x_1,\dots ,x_n):={\int }_{{\mathbb{R}}^n}f(x_1,\dots ,x_n)d{\mu }_n(\left\{\alpha \right\},\mathbf{x}),$$

(56)

where

$$d{\mu }_n(\left\{\alpha \right\},\mathbf{x}):=c_n(\left\{\alpha \right\},\mathbf{x})\prod _{k=1}^{n}d{x}_k,$$

(57)

$$c_n(\left\{\alpha \right\},\mathbf{x}):=\prod _{k=1}^n{c}_1({\alpha }_k,x_k),$$

(58)

and

$$c_1({\alpha }_k,x_k)={\displaystyle \frac{{\pi }^{{\alpha }_k/2}}{\Gamma ({\alpha }_k/2)}}{|x_k|}^{{\alpha }_k-1},$$

(59)

with ${\alpha }_k\in (0,1]$, $\mathbf{x}=(x_1,\dots ,x_n)$, and $D={\alpha }_1+\dots +{\alpha }_n\in (0,n]$.

Further, we will frequently use notation $I_W^D$, $d{\mu }_n(D,\mathbf{x})$, $c_n(D,\mathbf{x})$ instead of notation $I_W^{\left\{\alpha \right\}}$, $d{\mu }_n(\left\{\alpha \right\},\mathbf{x})$, $c_n(\left\{\alpha \right\},\mathbf{x})$, respectively.

In this work, we propose a general approach to formulation integration in NIDS, which can be considered as a generalization of the Wilson approach and the product measure approach. In this general approach, we use some axioms and principles that allow us to impose restrictions on the representation of integration in non-integer dimension space in the form of standard integration with the non-standard measure $d{\mu }_n(\left\{\alpha \right\},\mathbf{x})$. The properties of this measure will be determined by the axioms of integration proposed in the next subsection and the integration principles described in Section 2.2.

2.3.3. NIDS Integration Axioms

The definition of the NIDS integration can be based on the Wilson approach [9,10]. In the Wilson approach, the integration in the D-dimensional space with $D>0$ is represented by the standard integration in the integer-dimensional space.

In this subsection, we consider the properties (axioms) of these standard integrals and some measures ($d{\mu }_n(D,\mathbf{x})$), which allow us to consider and interpret these standard integrals as integrals over D-dimensional space with $D\in (0,n]$ and $n\in \mathbb{N}$. These axioms are connected with properties proposed in the Wilson paper [9] for integration in non-integer-dimensional space.

To define the NIDS integral, the basic properties are proposed as axioms that must be imposed on the NIDS integration in order to regard it as D-dimensional integration. The proposed axioms are natural and necessary in applying the NIDS integrations in physics, including quantum field theory and the theory of fractal media.

In this section, the integral over the entire vector (linear) space W will be denoted in the form

$$I_W^{D}f\left(\mathbf{x}\right):={\int }_{W}f\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x}),$$

(60)

where W will be called the D-dimensional space with $D\in (0,n]$ and the integral (60) will be called the NIDS integral, if the integration (60) satisfies the following axioms.

Note that $d^D\mathbf{x}\ne d{\mu }_n(D,\mathbf{x})$, since $d^D\mathbf{x}$ is the symbolic notation of the measure in the infinite-dimensional (D-dimensional) space ${\mathcal{H}}^D$, and $d{\mu }_n(D,\mathbf{x})$ is a measure in integer-dimensional (n-dimensional) space W.

In the Wilson paper [9,10], the properties (axioms, principles) were formulated for the entire NIDS. To construct the integral calculus in NIDS, it is important to formulate these axioms for the bounded domains of non-integer-dimensional space. In the particular case, it is important for application in the physics of fractal media and systems.

If $\Omega$ is a bounded region of the D-dimensional space W with $D\in (0,n]$, where $n\in \mathbb{N}$, then the NIDS integration of function $f\left(\mathbf{x}\right)$ over the region $\Omega \subseteq W$ can be described as

$${\int }_{\Omega }f\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x})={\int }_{W}f\left(\mathbf{x}\right){\chi }_{\Omega }\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x}),$$

(61)

where on the right side of Equation (61), the integration occurs over the entire space W, and ${\chi }_{\Omega }\left(\mathbf{x}\right)$ is the characteristic function of the region $\Omega$ such that ${\chi }_{\Omega }\left(\mathbf{x}\right)=1$ if $\mathbf{x}\in \Omega$, and ${\chi }_{\Omega }\left(\mathbf{x}\right)=0$ if $\mathbf{x}\notin \Omega$.

The axioms of the NIDS integration (60) in the space W can be formulated in the following form.

Axiom 1 

(Linearity Property). The property of linearity of the NIDS integration over the entire D-dimensional space W in the form

$${\int }_W\left(\sum _{j=1}^m{a}_k{f}_j\left(\mathbf{x}\right)\right)d{\mu }_n(D,\mathbf{x})=\sum _{j=1}^m{a}_j\left({\int }_W{f}_j\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x})\right)$$

(62)

should be satisfied for all real numbers $a_j\in \mathbb{R}$.

For the bounded region $\Omega \subset W$ of the D-dimensional space W, the linearity of the NIDS integration

$${\int }_{\Omega }\left(\sum _{j=1}^m{a}_k{f}_j\left(\mathbf{x}\right)\right)d{\mu }_n(D,\mathbf{x})=\sum _{j=1}^m{a}_j\left({\int }_{\Omega }{f}_j\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x})\right)$$

(63)

should also hold for all real numbers $a_j\in \mathbb{R}$.

Axiom 2 

(Translation Property). The translation property of the NIDS integration over the entire D-dimensional space W in the form

$${\int }_{W}f(\mathbf{x}+\mathbf{u})d{\mu }_n(D,\mathbf{x}+\mathbf{u})={\int }_{W}f\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x})$$

(64)

should be satisfied for all fixed vectors $\mathbf{u}$ of the D-dimensional space W.

For the bounded region $\Omega \subset W$, the translation property of the NIDS integration

$${\int }_{T_{\mathbf{u}}^{-1}\Omega }f(\mathbf{x}+\mathbf{u})d{\mu }_n(D,\mathbf{x}+\mathbf{u})={\int }_{\Omega }f\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x})$$

(65)

should be satisfied for all fixed vectors $\mathbf{u}$ of the D-dimensional space W, where $T_{\mathbf{u}}\Omega$ is a bounded region of the NIDS W obtained by the parallel translations $T_{\mathbf{u}}:\mathbf{x}\to \mathbf{x}+\mathbf{u}$ and $T_{\mathbf{u}}^{-1}:\mathbf{x}\to \mathbf{x}-\mathbf{u}$ of the region Ω. Note that, for the entire D-dimensional space, W $T_{\mathbf{u}}W=W$.

Equation (66) can be written as

$${\int }_{W}f(\mathbf{x}+\mathbf{u})d{\mu }_n(D,\mathbf{x})={\int }_{W}f\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x}-\mathbf{u}).$$

(66)

Note that the translation property (64) can be represented in the form

$${\int }_W\widehat{f}(\mathbf{x}+\mathbf{u})d{\mu }_n(n,\mathbf{x})={\int }_W\widehat{f}\left(\mathbf{x}\right)d{\mu }_n(n,\mathbf{x})$$

(67)

for all fixed vectors $\mathbf{u}$, where the function $\widehat{f}\left(\mathbf{x}\right)$ is defined as

$$\widehat{f}\left(\mathbf{x}\right):=c_n(D,\mathbf{x})f\left(\mathbf{x}\right)$$

(68)

and is used further. For example, the functions $\widehat{f}\left(\mathbf{x}\right)$ describe the 0-form in the calculus of NIDS differential forms (see Section 4.8).

Axiom 3 

(Additivity Property). The additivity property of the NIDS integration over the bounded regions ${\Omega }_1\subset W$ and ${\Omega }_2\subset W$, such that

$${\Omega }_1\bigcap {\Omega }_2=\varnothing$$

(69)

in the form

$$I_{{\Omega }_1\bigcup {\Omega }_2}^{D}f\left(\mathbf{x}\right)=I_{{\Omega }_1}^{D}f\left(\mathbf{x}\right)+I_{{\Omega }_2}^{D}f\left(\mathbf{x}\right)$$

(70)

should be satisfied for the functions $f\left(\mathbf{x}\right)$ integrable in the domain $\Omega :={\Omega }_1\bigcup {\Omega }_2$ and all $D>0$ of the D-dimensional space W.

It should be emphasized that Axiom 3 is needed for the NIDS integration over finite domains. While the additivity property is somewhat obvious, there are some questions here. For example, there is the incompatibility between the additivity property and the property of translation invariance for the NIDS integration over the bounded regions. A detailed discussion of this issue is given in Section 2.4.9 and in the Theorem Incompatibility of Additivity and Length Equality (Theorem 4).

Axiom 4 

(Rotation Covariance Property). Collins [10] proposed the rotation covariance as an additional axiom for the NIDS integration in the entire D-dimensional space. In fact, such an axiom means that, if the function has the form if $f\left(\mathbf{x}\right)=f(|\mathbf{x}|)$, (or if $f\left(\mathbf{x}\right)=f\left({\mathbf{x}}^2\right)$), then integration over the entire D-dimensional space W can be represented in the form

$${\int }_{W}f(|\mathbf{x}|)d{\mu }_n(D,\mathbf{x})=B\left(D\right){\int }_0^{\infty }f\left(x\right)x^{D-1}dx,$$

(71)

where $x:=|\mathbf{x}|$, and $B\left(D\right)$ is the coefficient that depends on $D>0$.

For the bounded spherically symmetric region ${\Omega }_S\subset W$ such that

$${\Omega }_S:=\{\mathbf{x}\in W:|\mathbf{x}|\in {\Omega }_R\subset [0,R]\},$$

(72)

this axiom can be represented for the functions $f\left(\mathbf{x}\right)=f(|\mathbf{x}|)$ in the form

$${\int }_{{\Omega }_s}f(|\mathbf{x}|)d{\mu }_n(D,\mathbf{x})=B\left(D\right){\int }_{{\Omega }_R}f\left(x\right)x^{D-1}dx,$$

(73)

for all $D\in (0,n]$, all $n\in \mathbb{N}$, and all $R>0$. As will be shown further in Section 2.7 (see Theorem 6), this axiom is satisfied by virtue of the second correspondence principle of NIDS integral calculus (Principle 2), where

$$B\left(D\right)={\displaystyle \frac{2{\pi }^{D/2}}{\Gamma (D/2)}},$$

(74)

for all $D>0$.

Axiom 5 

(Scaling Property). The scaling property of the NIDS integration over the entire D-dimensional space W in the form

$${\int }_{W}f\left(\lambda \mathbf{x}\right)d{\mu }_n(D,\mathbf{x})={\lambda }^{-D}{\int }_{W}f\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x})$$

(75)

should be satisfied for all $\lambda \in {\mathbb{R}}_{+}$ and all $D\in {\mathbb{R}}_{+}$.

The scaling property of the NIDS integration over the bounded region $\Omega \subset W$ is described by the equation

$${\int }_{\Omega }f\left(\lambda \mathbf{x}\right)d{\mu }_n(D,\mathbf{x})={\lambda }^{-D}{\int }_{\lambda \Omega }f\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x}),$$

(76)

where $\lambda \Omega$ is the bounded region of the NIDS obtained by changing all sizes of this region W by λ times ($\lambda >0$).

Axiom 6 

(Correspondence Property/Normalization Condition). As an axiom, one can also consider the second correspondence principle of NIDS integral calculus (Principle 2) or/and the Wilson normalization condition. The second correspondence Principle 2 is used for the bounded regions of the NIDS. This principle will be formulated and discussed in the following subsections.

For the entire NIDS, the normalization condition is the Wilson condition for the value of the NIDS integral of the exponent

$${\int }_{W}d{\mu }_n(D,\mathbf{x})exp(-{\mathbf{x}}^2)=A\left(D\right).$$

(77)

The Wilson normalization condition postulates the requirement that the normalization coefficient be $A\left(D\right)$. It is known that, for integer values of dimensions $D=n\in \mathbb{N}$, the coefficient is given by the value $A\left(n\right)={\pi }^{n/2}$. The Wilson normalization condition postulates that the NIDS integration over the entire D-dimensional space W must satisfy the equation

$${\int }_{W}d{\mu }_n(D,\mathbf{x})exp(-{\mathbf{x}}^2),={\pi }^{D/2}$$

(78)

for all $D>0$, [9].

For bounded regions, we proposed the second correspondence principle of NIDS integral calculus (Principle 2) that postulates that the property

$${\int }_{W}d{\mu }_n(D,\mathbf{x}){\chi }_{{\Omega }_B}\left(\mathbf{x}\right)=V_D\left({\Omega }_B\right)={\displaystyle \frac{2{\pi }^{D/2}}{D\Gamma \left(D\right)/2}}{R}^D$$

(79)

is satisfied, where ${\chi }_{{\Omega }_B}\left(\mathbf{x}\right)$ is the characteristic function of the D-dimensional ball region ${\Omega }_B:=\{\mathbf{x}:|\mathbf{x}|⩽R,R>0\}$, i.e., ${\chi }_{{\Omega }_B}\left(\mathbf{x}\right)=1$ if $\mathbf{x}\in {\Omega }_B$, and ${\chi }_{{\Omega }_B}\left(\mathbf{x}\right)=0$ if $\mathbf{x}\notin {\Omega }_B$. Principle 2 will be discussed in detail in the following subsections.

Let us give an example of a standard integral that satisfies these axioms. As an example, we can consider the NIDS integral

$$I_W^{\alpha }f\left(x\right):={\int }_{-\infty }^{+\infty }f\left(x\right)d{\mu }_1(\alpha ,x),$$

(80)

where

$$d{\mu }_1(\alpha ,x)=c_1(\alpha ,x)dx={\displaystyle \frac{{\pi }^{\alpha /2}}{\Gamma (\alpha /2)}}{|x|}^{\alpha -1}dx,$$

(81)

which is used in the Wilson approach to describe the integration in the D-dimensional space by the equation

$$\int f(|\mathbf{x}|)d^D\mathbf{x}=I_W^{D}f(|x|),$$

(82)

with $D=\alpha \in (0,1]$. Note that $d^D\mathbf{x}\ne d{\mu }_1(D,\mathbf{x})$, since $d^D\mathbf{x}$ is the symbolic notation of the measure in the infinite-dimensional (D-dimensional) space ${\mathcal{H}}^D$, and $d{\mu }_1(D,\mathbf{x})$ is a measure in integer-dimensional (one-dimensional) space W, with $D=\alpha \in (0,1]$.

Proofs that all these axioms are satisfied for integral (80) are proved in the next subsection.

Remark 7. 

In fact, the proposed axioms define the NIDS integration by the special form of the standard measure of integration in the integer-dimensional space ${\mathbb{R}}^n$. Incorporation of the correspondence principle into the axioms (see Axiom 6) allows us to uniquely define the NIDS measure and the NIDS integrals for any real $D>0$.

For applications, the uniqueness theorem is not sufficient. It is necessary to have explicit equations that represent the NIDS integral through the standard integrals in the integer-dimensional Euclidean space. Using these explicit equations, one can prove the properties of the NIDS integration that were postulated by Axioms 1–6.

Furthermore, this explicit representation of the NIDS integration should allow us to define the NIDS derivatives. Using the consistency principles of the third type, which require the mutual consistency between the NIDS integration and the NIDS differentiation, the exact representation of the NIDS integrals ensures the self-consistent definitions of the NIDS derivatives. The consistency principles require the existence of the NIDS generalization of the standard first and second fundamental theorems of standard calculus. In addition, consistency principles are represented by the NIDS generalized analogs of the gradient theorem, the Gauss–Ostrogradsky theorem (the divergence theorem), and the Stokes theorem, which must be satisfied. These principles allow us to formulate a self-consistent NIDS calculus. These consistency principles and fundamental theorems of NIDS calculus are proposed in the next section.

2.4. NIDS Integrals of a Single Variable and Their Properties

The single-variable NIDS calculus is the calculus of integrals and derivatives in the D-dimensional space $W=W_1$ with $D=\alpha \in (0,1]$.

At the beginning of the subsection, it should be emphasized that for single-variable NIDS integral calculus, the metric and product measure approaches coincide.

It should also be emphasized that the concepts of the NIDS length, the NIDS area, and the NIDS volume coincide in the D-dimensional space $W=W_1$ with $D=\alpha \in (0,1]$. As a result, these concepts coincide in the single-variable NIDS integral calculus considered below.

Let us consider the NIDS integration for the D-dimensional space with $D=\alpha \in (0,1]$.

Definition 2. 

Let $W=W_1$ be D-dimensional space with $D=\alpha \in (0,1]$, and let Ω be a region of W.

Then, if $c_1(\alpha ,x)f\left(x\right)\in C(\Omega )$ (or $f\left(x\right)\in C(\Omega )$), and $\Omega :=\{x\in W:a⩽x⩽b\}$, the NIDS integration over the region Ω is defined by the equation

$$I_{\Omega }^{\alpha }f\left(x\right):={\int }_a^{b}f\left(x\right)d{\mu }_1(\alpha ,x)={\int }_a^{b}f\left(x\right)c_1(\alpha ,x)dx={\displaystyle \frac{{\pi }^{\alpha /2}}{\Gamma (\alpha /2)}}{\int }_a^{b}f\left(x\right){|x|}^{\alpha -1}dx,$$

(83)

where $x\in [a,b]$, and $\alpha \in (0,1]$.

It is obvious that the linearity property (axiom 1) of the single-variable NIDS integration over the bounded region $\Omega =[a,b]\subset W$ of the D-dimensional space $W=W_1$ is satisfied in the form

$${\int }_a^b\left(\sum _{j=1}^m{a}_k{f}_j\left(x\right)\right)d{\mu }_1(\alpha ,x)=\sum _{j=1}^m{a}_j\left({\int }_a^b{f}_j\left(x\right)d{\mu }_1(\alpha ,x)\right)$$

(84)

for all real numbers $a_j\in \mathbb{R}$, $m\in \mathbb{N}$. The linear property of the NIDS integrals is a corollary of the linearity of the integral over a region in the space of integer dimension, if ${|x|}^{\alpha -1}{f}_j\left(x\right)\in C(\Omega )$ for all $j=1,\dots ,m$. Below, it will be shown that the we can use the condition $f_j\left(x\right)\in C(\Omega )$.

Let us consider some other properties of the NIDS integrals and the fulfillment of the correspondence principles of NIDS integral calculus (Principles 1 and 2).

2.4.1. Function Space and NIDS Integral

Let us note that the NIDS integral (83) of the function $f\left(x\right)\in C(\Omega )$ is the improper integral if $a⩽0⩽b$, since $c_1(\alpha ,x)=c\left(\alpha \right){|x|}^{\alpha -1}$ is discontinuous on $[a,b]$.

Theorem 1. 

Let $W=W_1$ be the D-dimensional space with $D=\alpha \in (0,1]$, and let $\Omega \subseteq W$.

If $f\left(x\right)\in C(\Omega )$ for $\Omega =[a,b]$, then the NIDS integral of the function $f\left(x\right)$ is convergent, and the inequality

$${\int }_a^b{c}_1(\alpha ,x)|f\left(x\right)|dx<\infty$$

(85)

holds for all closed bounded intervals $\Omega =[a,b]$.

Proof. 

The proof is based on the Boundedness Theorem and Comparison Theorem [135], pp. 159, 400.

(1) The theorem of the boundedness of a continuous function (Boundedness Theorem) [135], p. 159, states that if a function is continuous on a closed bounded interval, then it is bounded on this interval. Therefore, if $f\left(x\right)\in C(\Omega )$ for the bounded region $\Omega$, then $f\left(x\right)$ is bounded in $\Omega$; that is, there exists a positive number M such that $|f(x)|⩽M$ for all $x\in \Omega$. As a result, we get the inequality

$$|c_1(\alpha ,x)f\left(x\right)|=c_1(\alpha ,x)|f\left(x\right)|⩽{\displaystyle \frac{c_1\left(\alpha \right)M}{{|x|}^{1-\alpha }}},$$

(86)

where $1-\alpha \in [0,1)$, since $\alpha \in (0,1]$.

(2) The Comparison Theorem for improper integrals allows one to determine the convergence of an improper integral by comparing it with another integral whose convergence is known.

The Comparison Theorem for improper integrals states the following ([135], p. 400). Let $h\left(x\right)$ and $g\left(x\right)$ be defined on the interval $[a,\omega )$ and integrable on any closed integrals $[a,c)\subset [a,\omega )$. If $0⩽h\left(x\right)⩽g\left(x\right)$ for all $x\in [a,\omega )$, then the inequality

$${\int }_a^{\omega }h\left(x\right)dx⩽{\int }_a^{\omega }g\left(x\right)dx$$

(87)

holds, and these integrals converge.

In our case,

$$h\left(x\right)=c_1(\alpha ,x)|f\left(x\right)|,g\left(x\right)={\displaystyle \frac{c_1\left(\alpha \right)M}{{|x|}^{1-\alpha }}},$$

(88)

and $\omega =0$. Using Equation (86), we get that the equality $0⩽h\left(x\right)⩽g\left(x\right)$ holds for all $x\in [a,b]$.

(3) Let us consider two cases: $a⩽0<b$ and $a<0⩽b$. The condition $\alpha \in (0,1]$ leads to the fact that the exponent $1-\alpha$ in the denominator of the fraction is less than one. Then, using inequalities (87), (88), and the condition $\alpha \in (0,1]$, we get

$${\int }_a^{\omega }{c}_1(\alpha ,x)|f\left(x\right)|dx⩽{\int }_a^{\omega }{\displaystyle \frac{c_1\left(\alpha \right)M}{{|x|}^{1-\alpha }}}dx=$$

$${\displaystyle \frac{{\pi }^{\alpha /2}M}{\Gamma (\alpha /2)}}{\int }_a^{\omega }{|x|}^{\alpha -1}dx={\displaystyle \frac{{\pi }^{\alpha /2}M}{\alpha \Gamma (\alpha /2)}}\left({\omega }^{\alpha }+{|a|}^{\alpha }\right)<\infty .$$

(89)

As a result, we get

$${\int }_a^{\omega }{c}_1(\alpha ,x)|f\left(x\right)|dx⩽{\displaystyle \frac{{\pi }^{\alpha /2}M}{\alpha \Gamma (\alpha /2)}}{|a|}^{\alpha }<\infty ,$$

(90)

where we use $\omega =0$ and $a<0$.

Similarly, we obtain

$${\int }_{\omega }^b{c}_1(\alpha ,x)|f\left(x\right)|dx⩽{\displaystyle \frac{{\pi }^{\alpha /2}M}{\alpha \Gamma (\alpha /2)}}{b}^{\alpha }<\infty ,$$

(91)

where we use $\omega =0$ and $b>0$.

Using Equations (90) and (91) and

$${\int }_a^b{c}_1(\alpha ,x)|f\left(x\right)|dx=$$

$${\int }_a^{\omega }{c}_1(\alpha ,x)|f\left(x\right)|dx+{\int }_{\omega }^b{c}_1(\alpha ,x)|f\left(x\right)|dx⩽{\displaystyle \frac{{\pi }^{\alpha /2}M}{\alpha \Gamma (\alpha /2)}}\left(b^{\alpha }+{|a|}^{\alpha }\right)<\infty ,$$

(92)

we get inequality (85) for closed bounded intervals $\Omega =[a,b]$. □

As a result, we can use the functions $f\left(x\right)\in C(\Omega )$ with $\Omega =[a,b]$ for the NIDS integration of single variables.

2.4.2. Fulfillment of the First and Second Correspondence Principles

Let us consider the fulfillment of the first correspondence principle of NIDS integral calculus (Principle 1). According to this principle, the following theorem must be satisfiied. The proof of this theorem proves that correspondence Principle 1 holds for the single-vatiable NIDS integrals (83).

Theorem 2. 

Let $W=W_1$ be the D-dimensional space with $D=\alpha \in (0,1]$, and let $\Omega \subseteq W$.

Then, the single-variable NIDS integration of the functions $f\left(x\right)\in C(\Omega )$ over a domain $\Omega =[a,b]$ of the space $W=W_1$ with $D=\alpha \in (0,1]$ satisfies the property

$$\underset{\alpha \to 1-}{lim}{I}_{\Omega }^{\alpha }f\left(x\right)=\underset{\epsilon \to 0+}{lim}{I}_{\Omega }^{1-\epsilon }f\left(x\right)={\int }_{\Omega }f\left(x\right)dx,$$

(93)

for all $f\left(x\right)\in C\left(\right[a,b\left]\right)$, where $\epsilon =1-D=1-\alpha >0$.

Proof. 

The fulfillment of this principle is easily proved by the following equations. Let the parameter $\beta >0$ satisfy the inequalities

$$a<-\beta <0<\beta <b.$$

(94)

Then, we have

$$\underset{\alpha \to 1-}{lim}{I}_{\Omega }^{\alpha }f\left(x\right)=\underset{\alpha \to 1-}{lim}{\int }_a^{b}f\left(x\right)d{\mu }_1(\alpha ,x)=\underset{\alpha \to 1-}{lim}{\int }_a^{b}f\left(x\right)c_1(\alpha ,x)dx=$$

$$\underset{\alpha \to 1-}{lim}{\int }_{+\beta }^{b}f\left(x\right)c_1(\alpha ,x)dx+\underset{\alpha \to 1-}{lim}{\int }_{-\beta }^{+\beta }f\left(x\right)c_1(\alpha ,x)dx+\underset{\alpha \to 1-}{lim}{\int }_a^{-\beta }f\left(x\right)c_1(\alpha ,x)dx=$$

$${\int }_{+\beta }^{b}f\left(x\right)\left(\underset{\alpha \to 1-}{lim}{c}_1(\alpha ,x)\right)dx+\underset{\alpha \to 1-}{lim}{\int }_{-\beta }^{+\beta }f\left(x\right)c_1(\alpha ,x)dx+{\int }_a^{-\beta }f\left(x\right)\left(\underset{\alpha \to 1-}{lim}{c}_1(\alpha ,x)\right)dx=$$

$${\int }_{+\beta }^{b}f\left(x\right)dx+\underset{\alpha \to 1-}{lim}{\int }_{-\beta }^{+\beta }f\left(x\right)c_1(\alpha ,x)dx+{\int }_a^{-\beta }f\left(x\right)dx,$$

(95)

where we use

$$\underset{\alpha \to 1-}{lim}{c}_1(\alpha ,x)={\displaystyle \frac{{\pi }^{1/2}}{\Gamma (1/2)}}=1,$$

(96)

for $x\ne 0$, and $\Gamma (1/2)=\sqrt{\pi }$.

Using the property $c_1(\alpha ,x)⩾0$ and the theorem about triangle inequality for integrals in the form

$$\left|{\int }_{-\beta }^{+\beta }{c}_1(\alpha ,x)f\left(x\right)dx\right|⩽{\int }_{-\beta }^{+\beta }|c_1(\alpha ,x)f\left(x\right)|dx={\int }_{-\beta }^{+\beta }{c}_1(\alpha ,x)|f\left(x\right)|dx,$$

(97)

and inequality (92) with $a=-\beta <0$, $b=\beta >0$ and $\alpha \in (0,1)$ in the form

$${\int }_{-\beta }^{+\beta }{c}_1(\alpha ,x)|f\left(x\right)|dx⩽{\displaystyle \frac{{\pi }^{\alpha /2}M}{\alpha \Gamma (\alpha /2)}}2{\beta }^{\alpha }<\infty ,$$

(98)

we obtain

$$\left|{\int }_{-\beta }^{+\beta }{c}_1(\alpha ,x)f\left(x\right)dx\right|⩽{\int }_{-\beta }^{+\beta }{c}_1(\alpha ,x)|f\left(x\right)|dx⩽{\displaystyle \frac{2{\pi }^{\alpha /2}M}{\alpha \Gamma (\alpha /2)}}{\beta }^{\alpha }.$$

(99)

Therefore, we obtain

$$\underset{\beta \to 0+}{lim}\left|{\int }_{-\beta }^{+\beta }{c}_1(\alpha ,x)f\left(x\right)dx\right|⩽\underset{\beta \to 0+}{lim}{\displaystyle \frac{2{\pi }^{\alpha /2}M}{\alpha \Gamma (\alpha /2)}}{\beta }^{\alpha }=0,$$

(100)

for $\alpha \in (0,1]$.

As a result, Equation (95) for $\beta \to 0+$ gives

$$\underset{1-\epsilon \to 1-}{lim}{\int }_a^{b}f\left(x\right)c_1(1-\epsilon ,x)dx={\int }_0^{b}f\left(x\right)dx+{\int }_a^{0}f\left(x\right)dx={\int }_a^{b}f\left(x\right)dx.$$

(101)

□

Theorem 2 proves that for the single-variable NIDS integration the first correspondence principle of NIDS integral calculus (Principle 1) is satisfied, and the single-variable NIDS integral at the limit $\alpha \to 1-$ coincides with the standard integral in the integer-dimensional space $\mathbb{R}$.

Let us consider the fulfillment of the second correspondence principle of NIDS integral calculus (Principle 2).

Theorem 3. 

For the D-dimensional space $W=W_1$ with $D=\alpha \in (0,1]$, the calculation of the volume NIDS integrals of the function $f\left(x\right)={\chi }_{{\Omega }_B}\left(x\right)$ over the ball region

$${\Omega }_B:=\{x:|x|⩽R,R>0\}$$

(102)

gives the well-known expression of the volume for the D-dimensional ball

$${\int }_W{\chi }_{{\Omega }_B}\left(x\right)d{\mu }_n(\alpha ,x)=V_{\alpha }\left({\Omega }_B\right):={\displaystyle \frac{2{\pi }^{\alpha /2}}{\alpha \Gamma (\alpha /2)}}{R}^{\alpha },$$

(103)

for all $\alpha \in (0,1]$ and all $R>0$.

Proof. 

Let us prove Equation (103). Using the NIDS integration over domain (102), we obtain

$${\int }_W{\chi }_{{\Omega }_B}\left(x\right)d{\mu }_n(\alpha ,x)={\int }_{{\Omega }_B}d{\mu }_n(\alpha ,x)={\int }_{-R}^R{c}_1(\alpha ,x)dx=$$

$${\displaystyle \frac{{\pi }^{\alpha /2}}{\Gamma (\alpha /2)}}{\int }_{-R}^R{|x|}^{\alpha -1}dx={\displaystyle \frac{2{\pi }^{\alpha /2}}{\Gamma (\alpha /2)}}{\int }_0^R{x}^{\alpha -1}dx={\displaystyle \frac{2{\pi }^{\alpha /2}}{\alpha \Gamma (\alpha /2)}}{R}^{\alpha }$$

(104)

for all $\alpha \in (0,1]$ and all $R>0$. □

Theorem 3 proves that the second correspondence principle of NIDS integral calculus (Principle 2) is satisfied for the single-variable NIDS integral. Therefore, Theorem 3 states that Axiom 6 for single-variable NIDS integration holds for all $D\in (0,1]$.

Remark 8. 

Let us note that the Wilson normalization condition for the D-dimensional space $W=W_1$ with $D=\alpha \in (0,1]$ in the form

$${\int }_{-\infty }^{+\infty }d{\mu }_1(\alpha ,x)exp(-x^2)={\pi }^{D/2}$$

(105)

is satisfied for all $D=\alpha \in (0,1]$. This fact is proved in Theorem 7.

2.4.3. Scaling Property of Single Variable NIDS Integration

Axiom 5 requires that the scaling property to be satisfied for the NIDS integration over the entire D-dimensional space W. Therefore, the single-variable NIDS integration (83) should satisfy the scaling property. For the NIDS integration over the entire D-dimensional space $W=W_1$ with $D=\alpha \in (0,1]$, this property must be satisfied in the form

$${\int }_{-\infty }^{+\infty }f\left(\lambda x\right)d{\mu }_1(\alpha ,x)={\lambda }^{-\alpha }{\int }_{-\infty }^{+\infty }f\left(s\right)d{\mu }_1(\alpha ,s)$$

(106)

for all $\lambda \in {\mathbb{R}}_{+}$ and all $\alpha \in (0,1]$.

To prove property (106), it is sufficient to use the variable $s=\lambda x$ and the definition of the single-variable NIDS measure

$$d{\mu }_1(\alpha ,x)=d{\mu }_1(\alpha ,{\lambda }^{-1}s)={\displaystyle \frac{{\pi }^{\alpha /2}}{\Gamma (\alpha /2)}}{|{\lambda }^{-1}s|}^{\alpha }=$$

$${\lambda }^{-\alpha }{\displaystyle \frac{{\pi }^{\alpha /2}}{\Gamma (\alpha /2)}}{|s|}^{\alpha }={\lambda }^{-\alpha }d{\mu }_1(\alpha ,s),$$

(107)

which proves this property for $f\left(x\right)\in C\left(\mathbb{R}\right)$.

Let us consider the bounded region $\Omega \subset W$. For the NIDS integration over the bounded region $\Omega$ of the D-dimensional space $W=W_1$ with $D=\alpha \in (0,1]$, the NIDS scaling property has the form

$${\int }_{\Omega }f\left(\lambda x\right)d{\mu }_1(\alpha ,x)={\lambda }^{-\alpha }{\int }_{\lambda \Omega }f\left(s\right)d{\mu }_1(\alpha ,s),$$

(108)

where $\lambda \Omega$ is the bounded region of the NIDS obtained by changing all sizes of this region W by $\lambda$ times ($\lambda >0$).

Let us prove this property for the bounded region $\Omega =[a,b]$ in the form of the finite interval $[a,b]$. In this case, $\lambda \Omega =[\lambda a,\lambda b]$. For the NIDS integration over the bounded region $\Omega =[a,b]$, the scaling property

$${\int }_a^{b}f\left(\lambda x\right)d{\mu }_1(\alpha ,x)={\lambda }^{-\alpha }{\int }_{\lambda a}^{\lambda b}f\left(s\right)d{\mu }_1(\alpha ,s)$$

(109)

must be satisfied for all $\lambda >0$ and all $\alpha \in (0,1]$. Using variable $s=\lambda x$ with $\lambda >0$, we get

$${\int }_{\Omega }f\left(\lambda x\right)d{\mu }_1(\alpha ,x)={\int }_a^{b}f\left(\lambda x\right)d{\mu }_1(\alpha ,x)=$$

$${\int }_{\lambda a}^{\lambda b}f\left(s\right)d{\mu }_1(\alpha ,{\lambda }^{-1}s)=$$

$${\lambda }^{-\alpha }{\int }_{\lambda a}^{\lambda b}f\left(s\right)d{\mu }_1(\alpha ,s)={\lambda }^{-\alpha }{\int }_{\lambda \Omega }f\left(s\right)d{\mu }_1(\alpha ,s),$$

(110)

where we use

$$d{\mu }_1(\alpha ,{\lambda }^{-1}s)={\lambda }^{-\alpha }d{\mu }_1(\alpha ,s),$$

(111)

and $\lambda >0$.

The scaling property (109) is satisfied for all $\lambda >0$, if $f\left(x\right)\in C\left(\right[\lambda a,\lambda b\left]\right)$ for all $\lambda >0$. One can consider the fulfillment of the scale property on some finite region of scale change. For example, scaling property (109) can be satisfied for all $\lambda (0,{\lambda }_{sup})$, if $f\left(x\right)\in C\left([{\lambda }_{sup}a,{\lambda }_{sup}b]\right)$.

2.4.4. NIDS Length of a Straight Line Segment

Let us consider the concept of the NIDS length of a straight line segment in the D-dimensional space $W=W_1$ with $D=\alpha \in (0,1]$.

Let A and B be points of the D-dimensional space $W=W_1$ with $D=\alpha \in (0,1]$, where coordinates of points A and B will be designated as $x_A$ and $x_B$, respectively. The NIDS length of the line segment $[A,B]$ in the space W is described by the single-variable NIDS integral

$$L_D[A,B]={\int }_{x_A}^{x_B}{c}_1(\alpha ,x)dx=c\left(\alpha \right)sgn(x_B-x_A){\int }_{x_A}^{x_B}{|x|}^{\alpha -1}dx,$$

(112)

where $sgn\left(x\right)$ is the sign function ($sgn\left(x\right)=1$ if $x>0$, $sgn\left(x\right)=-1$ if $x<0$, and $sgn\left(x\right)=0$ if $x=0$). In Equation (112), the integral can be written as

$${\int }_{x_A}^{x_B}{|x|}^{\alpha -1}dx={\int }_{sgn\left(x_A\right)|x_A|}^{sgn\left(x_B\right)|x_B|}{|x|}^{\alpha -1}dx=$$

$${\int }_{sgn\left(x_A\right)|x_A|}^{sgn\left(x_B\right)|x_B|}{|x|}^{\alpha -1}dx={\int }_0^{sgn\left(x_B\right)|x_B|}{|x|}^{\alpha -1}dx+{\int }_{sgn\left(x_A\right)|x_A|}^0{|x|}^{\alpha -1}dx=$$

$$sgn\left(x_B\right){\int }_0^{sgn\left(x_B\right)|x_B|}{|x|}^{\alpha -1}d|x|+sgn\left(x_A\right){\int }_{sgn\left(x_A\right)|x_A|}^0{|x|}^{\alpha -1}d|x|=$$

$$sgn\left(x_B\right){\int }_0^{sgn\left(x_B\right)|x_B|}{\xi }^{\alpha -1}d\xi -sgn\left(x_A\right){\int }^{sgn{\left(x_A\right)}_0|x_A|}{\xi }^{\alpha -1}d\xi =$$

$$sgn\left(x_B\right){\displaystyle \frac{|x_B{|}^{\alpha }}{\alpha }}-sgn\left(x_A\right){\displaystyle \frac{|x_A{|}^{\alpha }}{\alpha }},$$

(113)

where we use $\xi =|x|$.

Substitution of Equation (113) into Equation (112) gives the length of the segment $AB$ in the D-dimensional space with $D=\alpha \in (0,1]$ in the form

$$L_D[A,B]:={\displaystyle \frac{{\pi }^{\alpha /2}}{\alpha \Gamma (\alpha /2)}}sgn(x_B-x_A)\left(sgn\left(x_B\right)|x_B{|}^{\alpha }-sgn\left(x_A\right){|x_A|}^{\alpha }\right).$$

(114)

For integer-dimensional space $\alpha =1$, we get the standard equation

$$L_{\alpha =1}[A,B]=sgn(x_B-x_A)\left(sgn\left(x_B\right)|x_B{|}^{\alpha }-sgn\left(x_A\right){|x_A|}^{\alpha }\right)=$$

$$sgn(x_B-x_A)\left(x_B-x_A\right)=|x_B-x_A|,$$

(115)

since $sgn\left(x\right)|x|=x$, and $\Gamma (1/2)=\sqrt{\pi }$.

If $x_B>x_A>0$, Equation (114) gives

$$L_{\alpha }[A,B]:={\displaystyle \frac{{\pi }^{\alpha /2}}{\alpha \Gamma (\alpha /2)}}\left(x_B^{\alpha }-x_A^{\alpha }\right).$$

(116)

If $x_B>0>x_A$, Equation (114) is

$$L_{\alpha }[A,B]:={\displaystyle \frac{{\pi }^{\alpha /2}}{\alpha \Gamma (\alpha /2)}}\left(x_B^{\alpha }+{|x_A|}^{\alpha }\right).$$

(117)

If $0>x_B>x_A$, Equation (114) has the form

$$L_{\alpha }[A,B]:={\displaystyle \frac{{\pi }^{\alpha /2}}{\alpha \Gamma (\alpha /2)}}\left(|x_A{|}^{\alpha }-{|x_B|}^{\alpha }\right).$$

(118)

Note that $L_{\alpha }^{\ast }[A,B]=L_{\alpha }[A,B]$ in the single-variable NIDS calculus.

Note that, if we use $c_1\left(\alpha \right)=2{\pi }^{\alpha /2}/\Gamma (\alpha /2)$ instead of $c_1\left(\alpha \right)={\pi }^{\alpha /2}/\Gamma (\alpha /2)$, then we get $L_{\alpha =1}=2|x_B-x_a|$, which violates the correspondence principle, since it gives a value twice as large.

2.4.5. Translation Property of Single-Variable NIDS Integration

Axiom 2 requires that the translation property of the NIDS integration be over the entire D-dimensional space W.

(1) The translation property of the single-variable NIDS integration (83) over the entire NIDS is represented in the form of the translation invariance

$${\int }_{-\infty }^{+\infty }\widehat{f}(x+u)d{\mu }_1(1,x)={\int }_{-\infty }^{+\infty }f(x+u)d{\mu }_1(\alpha ,x+u)={\int }_{-\infty }^{+\infty }f\left(s\right)d{\mu }_1(\alpha ,s)$$

(119)

that should be satisfied for all fixed numbers $u\in \mathbb{R}$, where

$$\widehat{f}\left(x\right):=f(x+u)c_1(\alpha ,x).$$

(120)

Note that Equation (119) can be written as

$${\int }_{-\infty }^{+\infty }f(x+u)d{\mu }_1(\alpha ,x)={\int }_{-\infty }^{+\infty }f\left(s\right)d{\mu }_1(\alpha ,s-u).$$

(121)

The proofs of Equations (119) and (121) are realized by the change in the variable $s=x+u$.

(2) For the bounded region in the form of the finite interval $\Omega =[a,b]$, the translation property can be represented as

$${\int }_{T_u^{-1}\Omega }\widehat{f}\left(T_{u}x\right)d{\mu }_1(1,x)={\int }_{T_u^{-1}\Omega }f\left(T_{u}x\right)d{\mu }_1(\alpha ,T_{u}x)={\int }_{\Omega }f\left(x\right)d{\mu }_1(\alpha ,x),$$

(122)

which should be satisfied for all fixed numbers $u\in \mathbb{R}$. The proof of Equation (122) is also realized by the change in the variable $s=x+u$.

For the finite interval $\Omega =[a,b]$, using that $T_u^{-1}\Omega =T_{-u}\Omega =T_{-u}[a,b]=[a-u,b-u]$ and $T_{u}x=x+u$, Equation (122) has the form

$${\int }_{a-u}^{b-u}\widehat{f}(s+u)ds={\int }_{a-u}^{b-u}f(s+u)c_1(\alpha ,s+u)ds={\int }_a^{b}f\left(x\right)c_1(\alpha ,x)dx,$$

(123)

which is proved by the change in the variable $x=s+u$. For $u=a$, Equation (123) takes the form

$${\int }_0^{b-a}f(s+a)c_1(\alpha ,s+a)ds={\int }_a^{b}f\left(x\right)d{\mu }_1(\alpha ,x),$$

(124)

where $b>a$.

(3) In the particular case, property (122) can be considered as the natural requirement that the length of a segment does not change, when the variable is shifted by a constant vector. The fact that the length of a segment does not change when a variable is changed is described by the equation

$$L_{\alpha }[a,b]:={\int }_a^{b}d{\mu }_{(}\alpha ,x)={\int }_{a-u}^{b-u}d{\mu }_{(}\alpha ,s+u).$$

(125)

Note that property (125) should not be confused with the property of independence of length from points in space. In the D-dimensional space $W=W_1$ with $D=\alpha \in (0,1)$, we have the inequality

$${\int }_a^{b}d{\mu }_{(}\alpha ,x)\ne {\int }_{a-u}^{b-u}d{\mu }_{(}\alpha ,s)$$

(126)

that gives

$$L_{\alpha }[a,b]\ne L_{\alpha }[a-u,b-u]$$

(127)

for all $\alpha \in (0,1)$ and $u\ne 0$, since $d{\mu }_{(}\alpha ,s+u)\ne d{\mu }_{(}\alpha ,s)$. Note that, for integer-dimensional space, $d{\mu }_1(1,s+u)=d{\mu }_{(}1,s)$. Inequalities (126) and (127) distinguish NIDS from the space of integer dimensions. This distinction is discussed in more detail in the following subsections.

2.4.6. Translation Property of NIDS Segment Lengths

The property of the inequality of NIDS segment lengths (property 2) leads to another property of the segment lengths that distinguishes space with non-integer dimensions from the standard spaces with integer dimensions.

Let us prove the following property of the of NIDS segment length.

Property 1 

(Translation Property of NIDS Segment Lengths). Let W be the D-dimensional space with $D=\alpha \in (0,1]$ and A, B be points of W with the coordinates $x_A=a$ and $x_B=b$, such that $b>a>0$, and $T_u$ is the shift operator with $u⩾-a$, such that

$$T_{u}A=A^{\prime },T_{u}B=B^{\prime },$$

(128)

where $x_A^{\prime }=a+u$ and $x_B^{\prime }=b+u$ are the coordinates of the points $A^{\prime }$ and $B^{\prime }$ of W, respectively.

Then, the NIDS lengths (114) of the segments $AB$ and $A^{\prime }{B}^{\prime }$ are not equal:

$$L_{\alpha }\left[AB\right]\ne L_{\alpha }\left[A^{\prime }{B}^{\prime }\right],$$

(129)

$$L_{\alpha }[a,b]\ne L_{\alpha }[a+u,a+u],$$

(130)

for all $\alpha \in (0,1)$ and $u\ne 0$.

Proof. 

Inequality (130) means that

$${\int }_a^{b}d{\mu }_1(\alpha ,x)\ne {\int }_{a+u}^{b+u}d{\mu }_1(\alpha ,x),$$

(131)

if $\alpha \in (0,1)$. Using the equations

$$L_{\alpha }\left[AB\right]={\int }_a^{b}d{\mu }_1(\alpha ,x)={\int }_a^b{c}_1(\alpha ,x)dx={\displaystyle \frac{{\pi }^{\alpha /2}}{\alpha \Gamma (\alpha /2)}}\left(b^{\alpha }-a^{\alpha }\right),$$

(132)

$$L_{\alpha }\left[A^{\prime }{B}^{\prime }\right]={\int }_{a+u}^{b+u}d{\mu }_1(\alpha ,x)={\int }_{a+u}^{b+u}{c}_1(\alpha ,x)dx={\displaystyle \frac{{\pi }^{\alpha /2}}{\alpha \Gamma (\alpha /2)}}\left({(b+u)}^{\alpha }-{(a+u)}^{\alpha }\right),$$

(133)

inequality (130) can be represented in the form

$$b^{\alpha }-a^{\alpha }\ne {(b+u)}^{\alpha }-{(a+u)}^{\alpha }.$$

(134)

For $u=-a$, inequality (134) has the form

$$b^{\alpha }-a^{\alpha }\ne {(b-a)}^{\alpha },$$

(135)

which holds for $b>a>0$ and $\alpha \in (0,1)$. □

Remark 9. 

Note that Property 1 does not contradict the translation property of the single-variable NIDS integration (83), if the NIDS integral is defined by Equation (83). The translation property (123) holds in the form

$${\int }_a^{b}d{\mu }_1(\alpha ,x)={\int }_{a+u}^{b+u}d{\mu }_1(\alpha ,x-u).$$

(136)

Equation (136) is proved by the change in the variable. Using $s=x-u$, the right hand side of Equation (136) takes the form

$${\int }_{a+u}^{b+u}d{\mu }_1(\alpha ,x-u)={\int }_{a+u}^{b+u}{c}_1(\alpha ,x-u)dx={\int }_a^b{c}_1(\alpha ,s)ds={\int }_a^{b}d{\mu }_1(\alpha ,s).$$

(137)

Redesignating the integration variable, we obtain (136).

Remark 10. 

Note that similar properties have the optical length (or optical path length) in optically inhomogeneous media [136,137]. In general, the optical path lengths have the property

$$L_{OPL}[x_A,x_B]\ne L_{OPL}[x_A+u,x_B+u].$$

(138)

In the framework of the proposed physical interpretation, the D-dimensional space with $D=\alpha \in (0,1]$ is analogous to an optically inhomogeneous one-dimensional medium, in which the refractive index is equal to $n_{op}\left(x\right)=c_1(\alpha ,x)$.

2.4.7. Property of the Inequality of NIDS Segment Lengths

Let us consider the property of the single-variable NIDS integration (83) that distinguishes it from standard integration in integer-dimensional space.

Property 2 

(Property of the Inequality of NIDS Segment Lengths). Let $W=W_1$ be the D-dimensional space with $D=\alpha \in (0,1]$ and A, B, and C be points of W with the coordinates $x_A$, $x_B$, and $x_C$, such that $x_B>x_C>x_A⩾0$, where the point C has the coordinate $x_C=(x_A+x_B)/2$.

Then, the NIDS lengths (114) of the segments $AC$ and $CB$ are not equal:

$$L_{\alpha }[x_A,x_C]\ne L_{\alpha }[x_C,x_B]$$

(139)

for the case $\alpha \in (0,1)$.

Proof. 

The violation of the equality of segment lengths in the form (139) can be proved by contradiction. Using Equation (114), the length of the segments are

$$L_{\alpha }[x_A,x_C]={\alpha }^{-1}c\left(\alpha \right)\left(x_C^{\alpha }-x_A^{\alpha }\right),$$

(140)

$$L_{\alpha }[x_C,x_B]={\alpha }^{-1}c\left(\alpha \right)\left(x_B^{\alpha }-x_C^{\alpha }\right).$$

(141)

If $L_{\alpha }[x_A,x_C]=L_{\alpha }[x_C,x_B]$, then the equation

$$x_C^{\alpha }-x_A^{\alpha }=x_B^{\alpha }-x_C^{\alpha },$$

(142)

where $x_B>x_C>x_A>0$, should be satisfied for all $\alpha \in (0,1]$. Equation (142) can written as

$$2{\left({\displaystyle \frac{x_B+x_A}{2}}\right)}^{\alpha }=x_B^{\alpha }+x_A^{\alpha }.$$

(143)

If $x_A=0$, the condition (143) takes the form

$$2^{1-\alpha }{x}_B^{\alpha }=x_B^{\alpha },$$

(144)

which holds for $x_B>0$, only if $\alpha =1$. □

Remark 11. 

Note that similar properties have the optical length (or optical path length) in optically inhomogeneous media [136,137]. The NIDS length in the non-integer-dimensional space with dimension $\alpha \in (0,1]$ can be interpreted as the optical path length in one-dimensional media with the refractive index equal to $n_{op}\left(x\right)=c_1(\alpha ,x)$.

If the point C has coordinate $x_C=(x_A+x_B)/2$, then the optical path lengths of the segments $AC$ and $CB$ are not equal:

$$L_{OPL}[x_A,x_C]\ne L_{OPL}[x_C,x_B],$$

(145)

but the additivity property of these length holds:

$$L_{OPL}[x_A,x_C]+L_{OPL}[x_C,x_B]=L_{OPL}[x_A,x_B].$$

(146)

In the framework of the proposed physical interpretation, a space with non-integer dimension $D\in (0,n]$ is analogous to an optically inhomogeneous n-dimensional medium, in which the refractive index is equal to $n_{op}\left(\mathbf{x}\right)=c_n(D,\mathbf{x})$.

2.4.8. Additivity Property of NIDS Integral and NIDS Segment Lengths

Axiom 2 requires the additivity property of the NIDS integration over the bounded regions of the D-dimensional space W. Let us prove this property for single-variable NIDS integrations.

Property 3 

(Additivity Property of NIDS Segment Lengths). Let W be the D-dimensional space with dimension $D=\alpha \in (0,1]$ and A, B, and C be points of W with the coordinates $x_A$, $x_B$, and $x_c$, such that $x_B>x_C>x_A⩾0$.

Then, the sum of the single-variable NIDS integrals of the function $f\left(x\right)\in C[x_A,x_B]$ over the segments $AC$ and $CB$ are equal to the NIDS integrals over the segments $AB$:

$${\int }_{x_A}^{x_C}f\left(x\right)d{\mu }_1(\alpha ,x)+{\int }_{x_C}^{x_B}f\left(x\right)d{\mu }_1(\alpha ,x)={\int }_{x_A}^{x_B}f\left(x\right)d{\mu }_1(\alpha ,x),$$

(147)

for all $\alpha \in (0,1]$ and all $x_B>x_C>x_A⩾0$.

In a particular case, for the NIDS lengths of the segments $A_1{A}_2$ defined as

$$L_{\alpha }[x_1,x_2]:={\int }_{x_1}^{x_2}f\left(x\right)d{\mu }_1(\alpha ,x),$$

(148)

where $x_2>x_1$, and $\alpha \in (0,1]$, the additivity property is satisfied. The sum of the NIDS lengths (148) of the segments $AC$ and $CB$ are equal to the NIDS length of the segments $AB$:

$$L_{\alpha }[x_A,x_C]+L_{\alpha }[x_C,x_B]=L_{\alpha }[x_A,x_B],$$

(149)

for all $\alpha \in (0,1]$ and all $x_B>x_C>x_A⩾0$.

Proof. 

To prove the additivity property of the NIDS integrals in the form (147), we use the function $\widehat{f}\left(x\right)=c_1(\alpha ,x)f\left(x\right)$ and the additivity property of the standard single-variable integrals in the integer-dimensional space:

$${\int }_{x_A}^{x_C}f\left(x\right)d{\mu }_1(\alpha ,x)+{\int }_{x_C}^{x_B}f\left(x\right)d{\mu }_1(\alpha ,x)=$$

$${\int }_{x_A}^{x_C}f\left(x\right)c_1(\alpha ,x)dx+{\int }_{x_C}^{x_B}f\left(x\right)c_1(\alpha ,x)dx=$$

$${\int }_{x_A}^{x_C}f\left(x\right)c_1(\alpha ,x)dx+{\int }_{x_C}^{x_B}f\left(x\right)c_1(\alpha ,x)dx=$$

$${\int }_{x_A}^{x_C}\widehat{f}\left(x\right)dx+{\int }_{x_C}^{x_B}\widehat{f}\left(x\right)dx=$$

$${\int }_{x_A}^{x_B}\widehat{f}\left(x\right)dx={\int }_{x_A}^{x_B}f\left(x\right)c_1(\alpha ,x)dx={\int }_{x_A}^{x_B}f\left(x\right)d{\mu }_1(\alpha ,x),$$

(150)

where $f\left(x\right)\in C[x_A,x_B]$. Equation (150) holds for all $\alpha \in (0,1]$, if $f\left(x\right)\in C[x_A,x_B]$, where $x_B>x_C>x_A⩾0$.

As a result, using the function $f\left(x\right)=1$ for all $x\in [x_A,x_B]$, Equation (150) gives that Equation (149) holds for all $\alpha \in (0,1]$, if $f\left(x\right)\in C[x_A,x_B]$, where $x_B>x_C>x_A⩾0$. □

2.4.9. Interrelation of the Properties of Additivity and Inequality of NIDS Lengths

Let us note some relationship between the additivity property and the inequality of segment lengths in NIDS. One can state an incompatibility of the additivity and lengths equality in the NIDS integrations. In general, one can state that the additivity property of the NIDS integration over the bounded regions is the incompatibility with translational symmetry and translational invariance in some sense. Translational symmetry of a geometric object means that the translation operation does not change the object. Translational invariance under spatial translation essentially means that different points in space are not distinguished. Let us now consider some explanations for this incompatibility.

Let us assume that we change the definition of the line NIDS integral on a finite integral such that the property of the equality of NIDS segment lengths is satisfied. Then, the additivity property of this NIDS integral and the NIDS segment length will be violated. To demonstrate this fact, let us prove the following theorem.

Theorem 4 

(Incompatibility of the Additivity and Length Equality). Let us consider D-dimensional space $W=W_1$ with $D=\alpha \in (0,1]$. Let us also assume that the NIDS integration on the finite interval $[a,b]$ is defined by the equation

$$I_{[a,b]}^{\alpha ,NA}f\left(x\right):={\int }_a^{b}f\left(x\right)d{\mu }_1(\alpha ,x-a),$$

(151)

where we use $d{\mu }_1(\alpha ,x-a)$ instead of $d{\mu }_1(\alpha ,x)$. Integral (151) will be called the non-additive NIDS integral. Using the integral (151), the NIDS segment length will be defined as

$$L_{\alpha }^{NA}[x_1,x_2]=I_{[x_1,x_2]}^{\alpha ,NA}1={\int }_{x_1}^{x_2}d{\mu }_1(\alpha ,x-x_1),$$

(152)

where $x_1<x_2$.

(1) Then, the property of the equality of the NIDS segment length is satisfied. The non-additive NIDS length of the two segments $AC$ and $CB$ with $x_C=(x_A+x_B)/2$ are equal:

$$L_{\alpha }^{NA}[x_A,x_C]=L_{\alpha }^{NA}[x_C,x_B],$$

(153)

for all $\alpha \in (0,1]$, where $x_B>x_C>x_A$.

(2) Then, the additivity property of the NIDS integral (151) is violated

$$I_{[a,c]}^{\alpha ,NA}f\left(x\right)+I_{[c,b]}^{\alpha ,NA}f\left(x\right)\ne I_{[a,b]}^{\alpha ,NA}f\left(x\right),$$

(154)

for $\alpha \in (0,1)$, where $a<c<b$.

(3) Then, for the NIDS segment length (152), the additivity property is also violated

$$L_{\alpha }^{NA}[x_A,x_C]+L_{\alpha }^{NA}[x_C,x_B]\ne L_{\alpha }^{NA}[x_A,x_B],$$

(155)

for $\alpha \in (0,1)$, where $x_A<x_C<x_B$.

Proof. 

Let us assume that the NIDS integration on the finite interval $[a,b]$ is defined by Equation (151), where we use $d{\mu }_1(\alpha ,x-a)$ instead of $d{\mu }_1(\alpha ,x)$. Then, the non-additive NIDS length is defined by Equation (152). Let us consider points A, B, and C of the space W with coordinates $x_A$, $x_B$, and $x_C$, such that $x_C=(x_A+x_B)/2$, and $x_B>x_C>x_A$.

(1) For these definitions, the non-additive NIDS length of the two segments $AC$ and $CB$ are equal (153) for the case $\alpha \in (0,1]$, since

$${\int }_{x_C}^{x_B}f\left(x\right)d{\mu }_1(\alpha ,x-x_C)={\int }_0^{x_B-x_C}f\left(s\right)d{\mu }_1(\alpha ,s),$$

(156)

$${\int }_{x_A}^{x_C}f\left(x\right)d{\mu }_1(\alpha ,x-x_A)={\int }_0^{x_C-x_A}f\left(s\right)d{\mu }_1(\alpha ,s),$$

(157)

and $x_C-x_A=x_B-x_C$. As a result, the NIDS length (152) satisfies the property of equality of segment lengths (153).

(2) Let us prove that the additivity property of the NIDS integration (151) is violated, and inequality (154) holds.

Note that the operator (151) can be written as

$$I_{[a,b]}^{\alpha ,NA}f\left(x\right):={\int }_a^{b}f\left(x\right)d{\mu }_1(\alpha ,x-a)={\int }_a^{b}f\left(x\right)c_1(\alpha ,x-a)dx=$$

$${\displaystyle \frac{{\pi }^{\alpha /2}}{\Gamma (\alpha /2)}}{\int }_a^{b}f\left(x\right){|s-a|}^{\alpha -1}ds={\displaystyle \frac{{\pi }^{\alpha /2}\Gamma \left(\alpha \right)}{\Gamma (\alpha /2)}}\left(I_{RL,b-}^{\alpha }f\right)\left(a\right),$$

(158)

where $I_{RL,b-}^{\alpha }$ is the right-sided Riemann–Liouville fractional integral of the order $\alpha \in (0,1]$ that is defined [138], p. 69, as

$$\left(I_{RL,b-}^{\alpha }f\right)\left(x\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_x^b{(s-x)}^{\alpha -1}f\left(s\right)ds,$$

(159)

where $a<x<b$ and $0<\alpha <1$. Note that the Riemann–Liouville fractional integral is a non-additive operator.

Using representation (158), inequality (154) can be represented as

$$\left(I_{RL,c-}^{\alpha }f\right)\left(a\right)+\left(I_{RL,b-}^{\alpha }f\right)\left(c\right)\ne \left(I_{RL,b-}^{\alpha }f\right)\left(a\right),$$

(160)

for all $\alpha \in (0,1)$ and non-zero functions $f\left(x\right)\in C\left(\right[a,b\left]\right)$, where $b>c>a$.

Let us prove the inequality (160) by contradiction. We will assume that the following equality is satisfied

$$\left(I_{RL,c-}^{\alpha }f\right)\left(a\right)+\left(I_{RL,b-}^{\alpha }f\right)\left(c\right)=\left(I_{RL,b-}^{\alpha }f\right)\left(a\right),$$

(161)

for all $\alpha \in (0,1)$ and non-zero functions $f\left(x\right)\in C\left(\right[a,b\left]\right)$, where $b>c>a$.

Using Equation 2.1.18 of the property 2.1 in [138], p. 71, in the form

$$\left(I_{RL,c-}^{\alpha }{(c-s)}^{\beta -1}\right)\left(x\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_x^c{(s-x)}^{\alpha -1}{(c-s)}^{\beta -1}ds={\displaystyle \frac{\Gamma \left(\beta \right)}{\Gamma (\alpha +\beta )}}{(c-x)}^{\alpha +\beta -1},$$

(162)

where $\alpha >0$ and $\beta >0$. Equation (162) can be used to calculate the non-additive NIDS integrals for the functions ${(c-x)}^{\beta -1}$. Using Equation (162), Equation (161) with $f\left(x\right)={(c-x)}^{\beta -1}$ with $\beta >0$ takes the form

$${\int }_a^b{\displaystyle \frac{{(x-a)}^{\alpha -1}}{\Gamma \left(\alpha \right)}}{(c-x)}^{\beta -1}dx+{\displaystyle \frac{\Gamma \left(\beta \right)}{\Gamma (\alpha +\beta )}}{(c-b)}^{\alpha +\beta -1}={\displaystyle \frac{\Gamma \left(\beta \right)}{\Gamma (\alpha +\beta )}}{(c-a)}^{\alpha +\beta -1}.$$

(163)

Using the variable $z=(x-a)/(c-a)$, the integral in Equation (163) can be written as

$${\int }_a^b{\displaystyle \frac{{(x-a)}^{\alpha -1}}{\Gamma \left(\alpha \right)}}{(c-x)}^{\beta -1}dx={\displaystyle \frac{{(c-a)}^{\alpha +\beta -1}}{\Gamma \left(\alpha \right)}}{\int }^{(b-a)/(c-a)}{z}^{\alpha -1}{(1-z)}^{\beta -1}=$$

$${\displaystyle \frac{{(c-a)}^{\alpha +\beta -1}}{\Gamma \left(\alpha \right)}}B\left({\displaystyle \frac{b-a}{c-a}},\alpha ,\beta \right),$$

(164)

where $B\left(x,\alpha ,\beta \right)$ is an incomplete beta function. If $x=1$, the function $B\left(x,\alpha ,\beta \right)$ is the beta function

$$B(\alpha ,\beta ):={\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{\Gamma (\alpha +\beta )}}.$$

(165)

Then, using Equation (164), Equation (163) can be written as

$${\displaystyle \frac{{(c-a)}^{\alpha +\beta -1}}{\Gamma \left(\alpha \right)}}B\left({\displaystyle \frac{b-a}{c-a}},\alpha ,\beta \right)={\displaystyle \frac{\Gamma \left(\beta \right)}{\Gamma (\alpha +\beta )}}\left({(c-a)}^{\alpha +\beta -1}-{(c-b)}^{\alpha +\beta -1}\right).$$

(166)

Equation (166) can be represented in the form

$$I\left({\displaystyle \frac{b-a}{c-a}},\alpha ,\beta \right)=1-{\left({\displaystyle \frac{c-b}{c-a}}\right)}^{\alpha +\beta -1},$$

(167)

where $\alpha \in (0,1]$, $\beta >0$, $0<c-b<c-a$, and

$$I\left(x,\alpha ,\beta \right)={\displaystyle \frac{B(x,\alpha ,\beta )}{B(\alpha ,\beta )}}$$

(168)

is the regularized incomplete beta function.

Using the fact that the regularized incomplete beta function is a cumulative distribution function of the beta distribution, we get

$$I\left(x,\alpha ,\beta \right)\in [0,1],$$

(169)

for all $x,\alpha ,\beta >0$. Note that

$$1-{\left({\displaystyle \frac{c-b}{c-a}}\right)}^{\alpha +\beta -1}<0,$$

(170)

if $\alpha +\beta <1$, where $0<c-b<c-a$. We get that the left side of the Equation (168) is positive, and the right side is negative when $\beta \in (0,1-\alpha )$.

As a result, the NIDS integration (151) of the function $f\left(x\right)={(b-x)}^{\beta -1}$ violates the additivity property for $\alpha \in (0,1)$, if $\beta \in (0,1-\alpha )$.

(3) In the particular case $f\left(x\right)=1$ for $x\in [a,b]$, this property (154) leads to a violation of the additivity of the NIDS length (152) of a straight line segment. In the particular case $\beta =1$, we have $f\left(x\right)={(b-x)}^{\beta -1}=1$ for $x\in [a,b]$, and Equation (162) takes the form

$$\left(I_{RL,b-}^{\alpha }1\right)\left(x\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_x^b{(s-x)}^{\alpha -1}ds={\displaystyle \frac{\Gamma \left(1\right)}{\Gamma (\alpha +1)}}{(b-x)}^{\alpha }.$$

(171)

If the function $f\left(x\right)=1$ for all $x\in [a,b]$, the inequality (161) can be represented as

$$\left(I_{RL,c-}^{\alpha }1\right)\left(a\right)+\left(I_{RL,b-}^{\alpha }1\right)\left(c\right)=\left(I_{RL,b-}^{\alpha }1\right)\left(a\right),$$

(172)

where $b>c>a$, and $\alpha \in (0,1]$. Using Equation (171), Equation (172) takes the form

$${(c-a)}^{\alpha }+{(b-c)}^{\alpha }={(b-a)}^{\alpha },$$

(173)

where $b>c>a$. Equation (173) holds only if $\alpha =1$. Therefore, we get

$${(c-a)}^{\alpha }+{(b-c)}^{\alpha }\ne {(b-a)}^{\alpha },$$

(174)

for all $\alpha \in (0,1)$, where $b>c>a$. Therefore, the NIDS length (152) violates the additivity property

$$L_{\alpha }^{NA}[x_A,x_C]+L_{\alpha }^{NA}[x_C,x_B]\ne L_{\alpha }^{NA}[x_A,x_B],$$

(175)

where $b>c>a⩾0$.

We proved that the NIDS integration (151) and the NIDS length (152) are non-additive. □

As a result, it can be stated that the additivity property and the property of the equality of segment lengths (153) are incompatible properties (mutually contradictory) properties of the NIDS integration in a certain sense.

The laws of physics are translation-invariant under spatial translation, if they do not distinguish between different points in space. Therefore, one can state that the property of the inequality of segment lengths means that we distinguish points in space with non-integer dimensions in some sense.

In this sense, the additivity property and property of translational invariance are incompatible (mutually contradictory) properties of NIDS integration in non-integer-dimensional space.

2.5. Integration in Non-Integer-Dimensional Space: Product Measure Approach

A product measure approach was proposed in [15,16] (see also, [77,78,87,96]) to describe anisotropic fractal media by using non-integer-dimensional spaces (fractional-dimensional spaces). Then, this approach was developed in paper [18], where the NIDS vectors differential operators were proposed. In this approach, each non-integer-dimensional space $W_k$ along each orthogonal coordinate axis has its own non-integer dimension ${\alpha }_k\in (0,1]$. The product measure approach allows us to actually define some anisotropic non-integer-dimensional space (anisotropic NIDS). Anisotropy of a non-integer-dimensional space means that its properties, namely its dimensions, are different in different directions.

Mathematically, this approach is based on the concept of the product of measures and the product of the measured NIDS. One can state that the product measure approach, which is described below, is a generalization of the Wilson approach in a certain sense. Note that the Wilson approach represents the NIDS integration via the standard integration in integer-dimensional space with a special form of measure. In fact, for Axioms 1–6 for entire space and bounded domains together, the second correspondence principle of NIDS integral calculus (Principle 2) defines a special form of the usual measure of integration in space. As a result, NIDS integration in the D-dimensional space is represented through the standard integration in the integer-dimensional space with this special form of measure.

Note that Principle 3 imposes self-consistency conditions on the NIDS integral calculus. According to this principle, in order for the calculus of NIDS integrals to be self-consistent, the multiple NIDS integrals over the product of the $D_k$-dimensional spaces $W_k$ with $D_k={\alpha }_k\in (0,1]$ must be connected to the iterated NIDS integrals. In fact, Principle 3 imposes a condition on the NIDS integral calculus in the form of the existence of a NIDS analog of the Fubini theorem for multiple NIDS integrals.

Let us generalize some notions of the measure theory [139,140] for the non-integer-dimensional spaces. Let us give the NIDS generalizations of the measurable space (the Borel space) and measured space [140].

Definition 3. 

Let W be a NIDS with the dimension $D>0$, with $\mathcal{S}$ as a set of subsets of W. Then, the set $\mathcal{S}$ is called a σ-algebra on W, if the following conditions are satisfied:

(a) 

$⌀\in \mathcal{S}$;

(b) 

If $\Omega \in \mathcal{S}$, then $W\setminus \Omega \in \mathcal{S}$;

(c) 

If $\{{\Omega }_k\in \mathcal{S}\}$ is a sequence of $\mathcal{S}$, then ${\bigcup }_{k=1}^{\infty }{\Omega }_k\in \mathcal{S}$.

An ordered pair $(W,\mathcal{S})$, consisting of the D-dimensional space W and the σ-algebra $\mathcal{S}$ on W, is called the measurable NIDS.

Definition 4. 

Let W be a NIDS with dimension $D\in (0,n]$, with $\mathcal{S}$ as a σ-algebra on W. Then, the measure on the measurable NIDS $(W,\mathcal{S})$ is a function ${\mu }_n:\mathcal{S}\mapsto [0,\infty ]$, such that

(a) 

${\mu }_n(D,⌀)=0$;

(b) 

${\mu }_n\left(D,{\bigcup }_{k=1}^{\infty }{\Omega }_k\right)={\sum }_{k=1}^{\infty }{\mu }_n(D,{\Omega }_k)$, for every disjoint set sequence ${\Omega }_k\in \mathcal{S}$.

An ordered triple $(W,\mathcal{S},{\mu }_n)$, consisting of D-dimensional space W, σ-algebra $\mathcal{S}$ on W, and measure ${\mu }_n$ on $(W,\mathcal{S})$, is called the measures NIDS (or the D-dimensional measure space).

Remark 12. 

In standard measure theory, one should not confuse the concepts of measurable space and measure space [140]. Similar to the standard theory, one should also not confuse the concepts of the measurable NIDS and the measured NIDS.

Definition 5. 

A measure ${\mu }_n$ on a measurable D-dimensional space $(W,\mathcal{S})$ is called σ-finite, if there exists the sequence of sets ${\Omega }_k\in \mathcal{S}$, such that

$$W=\bigcup _{k=1}^{\infty }{\Omega }_k,$$

(176)

and ${\mu }_n(D,{\Omega }_k)<\infty$, for all $k\in \mathbb{N}$.

An ordered triple $(W,\mathcal{S},{\mu }_n)$, consisting of the D-dimensional space W with $D\in (0,n]$, the σ-algebra $\mathcal{S}$ on W, and the σ-finite measure ${\mu }_n$ on $(W,\mathcal{S})$ is called the σ-finite measured NIDS.

Definition 6. 

Let $(W_1,{\mathcal{S}}_1)$ and $(W_2,{\mathcal{S}}_2)$ be measurable NIDS.

Then, the product ${\mathcal{S}}_1\otimes {\mathcal{S}}_2$ of σ-algebras ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ is the smallest σ-algebra on $W_1\times W_2$ that contains $\{{\Omega }_1\times {\Omega }_2:{\Omega }_k\in {\mathcal{S}}_k,k=1,2\}$.

Definition 7. 

Let $(W_k,{\mathcal{S}}_k,{\mu }_1({\alpha }_k,{\Omega }_k))$ with $k=1,\dots ,n$ be σ-finite measure NIDS, where $W_k$ are $D_k$-dimensional spaces with $D_k={\alpha }_k\in (0,1]$, and ${\Omega }_k\in {\mathcal{S}}_k$. For $\Omega \subset W:=W_1\times \dots \times W_n$, the products of measures ${\mu }_1({\alpha }_k,{\Omega }_k)$ is defined by

$${\mu }_n(D,\Omega ):={\mu }_1({\alpha }_1,{\Omega }_1)\times \dots \times {\mu }_1({\alpha }_n,{\Omega }_n)=$$

$${\int }_{W_1}\dots {\int }_{W_n}{\chi }_{\Omega }(x_1,\dots ,x_n)d{\mu }_1({\alpha }_1,x_1)\dots d{\mu }_1({\alpha }_n,x_n),$$

(177)

where ${\chi }_{\Omega }\left(\mathbf{x}\right)$ is the characteristic function of the set Ω, i.e., ${\chi }_{\Omega }\left(\mathbf{x}\right)=1$ if $\mathbf{x}\in \Omega$, and ${\chi }_{\Omega }\left(\mathbf{x}\right)=0$ if $\mathbf{x}\notin \Omega$.

A function $f:W\mapsto \mathbb{R}$ is called a measurable function on measurable NIDS, if $(W,\mathcal{S})$ is a measurable NIDS, where $f^{-1}\left(B\right)\in \mathcal{S}$ for all sets $B\subset \mathbb{R}$, where $f^{-1}\left(B\right)=\{\mathbf{x}\in W:f\left(\mathbf{x}\right)\in B\}$.

Let us give the formulation of the Fubini theorem for NIDS in the following form.

Theorem 5 

(Fubini theorem for NIDS). Let $(W_k,{\mathcal{S}}_k,{\mu }_1({\alpha }_k,{\Omega }_k))$ with $k=1,2$ be σ-finite measure NIDS, where each $W_k$ is the NIDS with dimension ${\alpha }_k\in (0,1]$, and let $f:W:=W_1\times W_2\mapsto \mathbb{R}$ be a measurable function on the measured NIDS $(W_1\times W_2,{\mathcal{S}}_1\otimes {\mathcal{S}}_2)$ with $D={\alpha }_1+{\alpha }_2\in (0,2]$, such that

$${\int }_W|f(x_1,x_2)|d{\mu }_2(D,\mathbf{x})<\infty ,$$

(178)

where

$$d{\mu }_2(D,\mathbf{x})=d{\mu }_1({\alpha }_1,x_1)d{\mu }_1({\alpha }_2,x_2).$$

(179)

Then, the functions

$$F_1\left(x_1\right):={\int }_{W_2}f(x_1,x_2)d{\mu }_1({\alpha }_2,x_2),$$

(180)

$$F_2\left(x_2\right):={\int }_{W_1}f(x_1,x_2)d{\mu }_1({\alpha }_1,x_1)$$

(181)

are measurable functions on the measurable ${\alpha }_k$-dimensional spaces $(W_k,{\mathcal{S}}_k)$, $k=1,2$.

The NIDS integration over $W=W_1\times W_2$ can be represented as

$${\int }_{W}f\left(\mathbf{x}\right)d{\mu }_2(D,\mathbf{x})={\int }_{W_1}d{\mu }_1({\alpha }_1,x_1){\int }_{W_2}d{\mu }_1({\alpha }_2,x_2)f(x_1,x_2)=$$

$${\int }_{W_2}d{\mu }_1({\alpha }_2,x_2){\int }_{W_1}d{\mu }_1({\alpha }_1,x_1)f(x_1,x_2).$$

(182)

Proof. 

The proof of Theorem 5 is similar to the proof of the standard Fubini theorem for spaces with integer dimensions, since the NIDS integration is represented as a standard integration with a special type of integration measure. □

Let us give the definition of the NIDS integration in the measured NIDS.

Definition 8 

(NIDS Integral in Measured NIDS). Let $(W_k,{\mathcal{S}}_k,{\mu }_1({\alpha }_k,{\Omega }_k))$ with $k=1,\dots ,n$ be σ-finite measured NIDSs, where $W_k$ are $D_k$-dimensional spaces with $D_k={\alpha }_k\in (0,1]$, and let $f:W:=W_1\times \dots \times W_n\mapsto \mathbb{R}$ be a measurable function on $(W,\mathcal{S})$ with $\mathcal{S}:={\mathcal{S}}_1\times \dots \times {\mathcal{S}}_n$ and

$${\int }_W|f\left(\mathbf{x}\right)|d{\mu }_n(D,\mathbf{x})<\infty .$$

(183)

Then, the multiple NIDS integration over the D-dimensional space W with the dimension

$$D=\sum _{k=1}^n{\alpha }_k\in (0,n]$$

(184)

is defined by the equation

$${\int }_{W}f\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x}):={\int }_{W_1}d{\mu }_1({\alpha }_1,x_1)\dots {\int }_{W_n}d{\mu }_1({\alpha }_n,x_n)f\left(\mathbf{x}\right),$$

(185)

where the single-variable NIDS integration in the $D_k$-dimensional space $W_k$ with $D_k={\alpha }_k\in (0,1]$ is defined by the equation

$${\int }_{W_k}f\left(x_k\right)d{\mu }_1({\alpha }_k,x_k):={\int }_{W_k}f\left(x_k\right)c_1({\alpha }_k,x_k)d{x}_k,$$

(186)

with the function

$$c_1({\alpha }_k,x_k):=c_1\left({\alpha }_k\right){|x^k|}^{{\alpha }_k-1}(k=1,\dots ,n).$$

(187)

The multiplier $c_1\left({\alpha }_k\right)$ is a function of the parameter ${\alpha }_k$, which is defined by the correspondence principles or the normalization conditions.

The multiple NIDS integral (185) can be also called the volume NIDS integral.

Remark 13. 

Note that the following notations will also be used in the volume NIDS integrals:

$$d{\mu }_n(D,\mathbf{x}):=c_n(D,\mathbf{x})d{\mu }_n(n,\mathbf{x}),$$

(188)

where

$$c_n(D,\mathbf{x}):=\prod _{k=1}^n{c}_1({\alpha }_k,x_k)=\prod _{k=1}^{n}c\left({\alpha }_k\right){|x_k|}^{{\alpha }_k-1},$$

(189)

$$d{\mu }_n(n,\mathbf{x}):=\prod _k^{n}d{x}_k,$$

(190)

and

$$d{\mu }_n(D,\mathbf{x})=\prod _{k=1}^{n}d{\mu }_1({\alpha }_k,x_k),$$

(191)

where

$$d{\mu }_1({\alpha }_k,x_k):=c_1({\alpha }_k,x^k)d{x}^k=c_1\left({\alpha }_k\right){|x^k|}^{{\alpha }_k-1}d{x}^k,(k=1,\dots ,n),$$

(192)

with all ${\alpha }_k\in (o,1]$.

Remark 14. 

For the D-dimensional space $W=W_1\times \dots \times W_n$, the parameter (184) is interpreted as a total dimension of space with $D\in (0,n]$. If all ${\alpha }_k=1$, we get $D=n$, i.e., the dimension of space is the standard integer dimension. If all ${\alpha }_k=\alpha$, where $0<\alpha <1$, we have an “isotropic” NIDS with $D=n\alpha$. In general, we have a NIDS, if at least one of the parameters ${\alpha }_k$ is not equal to 1.

Remark 15. 

As a result, Definition 8 defines the NIDS integration as represented in terms of the standard integration in integer-dimensional space with a special form of measure.

In the Wilson approach, the NIDS integration is also defined in terms of the standard integration in integer-dimensional space. The product measure approach generalizes the Wilson approach to the case where each orthogonal coordinate axis has own non-integer dimension ${\alpha }_k\in (0,1]$. As a result, we have integration for some anisotropic non-integer-dimensional space (anisotropic NIDS). The anisotropy of non-integer-dimensional space means that its properties, including dimensions, depend on directions.

The basic properties of the volume NIDS integrals are similar to the properties of the standard volume integral [141,142] of the integer-dimension space. Let us denote the volume (multiple) NIDS integrals of the function $f\left(\mathbf{x}\right)$ over the region $\Omega$ of the D-dimensional space $W=W_1\times \dots \times W_n$ as

$$I_{\Omega }^{D}f\left(\mathbf{x}\right)={\int }_W{\chi }_{\Omega }\left(\mathbf{x}\right)f\left(\mathbf{x}\right)c_n(D,\mathbf{x})\prod _{k=1}^{n}d{x}^k.$$

(193)

The volume NIDS integrals satisfy the following basic properties.

1. Linearity Property: If the functions $f_1\left(\mathbf{x}\right)$ and $f_2\left(\mathbf{x}\right)$ are integrable in the domain $\Omega \subset W$, then their linear combination is also integrable, and

$$I_{\Omega }^D\left(a{f}_1\left(\mathbf{x}\right)+b{f}_2\left(\mathbf{x}\right)\right)=a{I}_{\Omega }^D{f}_1\left(\mathbf{x}\right)+b{I}_{\Omega }^D{f}_2\left(\mathbf{x}\right),$$

(194)

where $a,b\in \mathbb{R}$.

2. Additivity Property: Let $f\left(\mathbf{x}\right)$ be a function that is integrable in the domains ${\Omega }_1\subset W$ and ${\Omega }_2\subset W$, such that

$${\Omega }_1\bigcap {\Omega }_2=\varnothing .$$

(195)

Then, the functions $f\left(\mathbf{x}\right)$ are integrable in the domain $\Omega :={\Omega }_1\bigcup {\Omega }_2$, and

$$I_{\Omega }^{D}f\left(\mathbf{x}\right)=I_{{\Omega }_1}^{D}f\left(\mathbf{x}\right)+I_{{\Omega }_2}^{D}f\left(\mathbf{x}\right).$$

(196)

3. Monotonicity Property: If the functions $f_1\left(\mathbf{x}\right)$ and $f_2\left(\mathbf{x}\right)$ are integrable in the domain $\Omega \subset W$, and $f_1\left(\mathbf{x}\right)⩽f_2\left(\mathbf{x}\right)$ for all $\mathbf{x}\in \Omega$, then the inequality

$$I_{\Omega }^D{f}_1\left(\mathbf{x}\right)⩽I_{\Omega }^D{f}_2\left(\mathbf{x}\right)$$

(197)

is satisfied.

4. Modulus Estimation Property: If the function $f\left(\mathbf{x}\right)$ is integrable in the domain $\Omega \subset W$, then $|f(\mathbf{x})|$ is integrable in this domain, and

$$\left|I_{\Omega }^{D}f\left(\mathbf{x}\right)\right|⩽I_{\Omega }^D|f\left(\mathbf{x}\right)|.$$

(198)

5. Mean Value Property: If the function $f\left(\mathbf{x}\right)$ is integrable in the domain $\Omega \subset W$, and the inequality $m⩽f\left(\mathbf{x}\right)⩽M$ holds for all $\mathbf{x}\in \Omega$, then

$$m{V}_D(\Omega )⩽I_{\Omega }^{D}f\left(\mathbf{x}\right)⩽M{V}_D(\Omega ),$$

(199)

where $V_D(\Omega )$ is the D-dimensional volume of the region $\Omega$.

The proofs of these properties are similar to the proofs for standard volume integrals in the integer-dimensional spaces (for example, see Budak and Fom [141,142], Kudryavtsev [143], Zorich [135,144], and Taylor and Mann [145].

Remark 16. 

Further, it will be proved that the parameter function $c_1\left({\alpha }_k\right)$ must have the form $c_1\left({\alpha }_k\right)={\pi }^{{\alpha }_k/2}/\Gamma ({\alpha }_k/2)$. This form of the function $c_1\left({\alpha }_k\right)$ was first proposed in the articles [18,19] and was subsequently used in the author’s papers. Note that in paper [15] (see Equations 2.6 and 3.8 in [15]) and other works [16,77,78,87], other forms of this function were used.

Let us consider the application of the second correspondence principle of NIDS integral calculus (Principle 2) for the product measure approach with $n=2$. Using this principle, we can derive the explicit form of the coefficient $c_1\left({\alpha }_k\right)$ from the requirement that the NIDS integration over the D-dimensional ball region must give the well-known equation of the volume of the D-dimensional ball.

Lemma 1. 

Let us assume that the proposed NIDS integration in the D-dimensional space $W=W_1\times W_2$ with $D={\alpha }_1+{\alpha }_2\in (0,2]$, over the D-dimensional ball region ${\Omega }_B=\{\mathbf{x}:|\mathbf{x}|⩽R,R>0\}$, gives the well-known expression:

$${\int }_W{\chi }_{{\Omega }_B}\left(\mathbf{x}\right)d{\mu }_2(D,\mathbf{x})=V_D\left({\Omega }_B\right):={\displaystyle \frac{2{\pi }^{D/2}}{D\Gamma (D/2)}}{R}^D,$$

(200)

for all ${\alpha }_1,{\alpha }_2\in (0,1]$ and all $R>0$, where

$$d{\mu }_2(D,\mathbf{x})=\prod _{k=1}^{2}d{\mu }_k({\alpha }_k,x^k)=\prod _{k=1}^2{c}_1({\alpha }_k,x^k)d{x}^k,$$

(201)

$$c_1({\alpha }_k,x_k):=c_1\left({\alpha }_k\right){|x_k|}^{{\alpha }_k-1},$$

(202)

and ${\chi }_{{\Omega }_B}\left(\mathbf{x}\right)$ is the characteristic function of the region ${\Omega }_B$.

Then, the coefficients $c_1\left({\alpha }_k\right)$ have the form

$$c_1\left({\alpha }_k\right)={\displaystyle \frac{{\pi }^{{\alpha }_k/2}}{\Gamma ({\alpha }_k/2)}},$$

(203)

where ${\alpha }_k\in (0,1]$.

Proof. 

For the D-dimensional ball ${\Omega }_B=\{\mathbf{x}:|\mathbf{x}|⩽RR>0\}$ with $D={\alpha }_1+{\alpha }_2$, the volume in NIDS is described by the NIDS integration as

$${\int }_W{\chi }_{{\Omega }_B}\left(\mathbf{x}\right)d{\mu }_2(D,\mathbf{x})={\int }_{{\Omega }_B}d{\mu }_2(D,\mathbf{x})={\int }_{{\Omega }_B}d{\mu }_1({\alpha }_1,x_1)d{\mu }_1({\alpha }_2,x_2)=$$

$$\prod _{k=1}^2{c}_1\left({\alpha }_k\right){\int }_{{\Omega }_B}|x_1{|}^{{\alpha }_1-1}{|x_2|}^{{\alpha }_2-1}d{x}_{1}d{x}_2.$$

(204)

Changing the coordinates

$$x_1=rcos\left(\varphi \right),x_2=rsin\left(\varphi \right),$$

(205)

and the Jacobian of the coordinate change $J_2(r,\varphi )=r$, we get

$${\int }_{{\Omega }_B}d{\mu }_2(D,\mathbf{x})=\prod _{k=1}^2{c}_1\left({\alpha }_k\right){\int }_0^{R}dr{\int }_0^{2\pi }d\varphi J_2(r,\varphi )r^{{\alpha }_1+{\alpha }_2-2}|cos\left(\varphi \right){|}^{{\alpha }_1-1}{|sin\left(\varphi \right)|}^{{\alpha }_2-1}.$$

(206)

Using the integrals

$${\int }_0^{R}dr{J}_2\left(r\right)r^{{\alpha }_1+{\alpha }_2-2}={\int }_0^{R}dr{r}^{{\alpha }_1+{\alpha }_2-1}={\displaystyle \frac{1}{{\alpha }_1+{\alpha }_2}}{R}^{{\alpha }_1+{\alpha }_2},$$

(207)

$${\int }_0^{2\pi }{d\varphi |cos\left(\varphi \right)|}^{{\alpha }_1-1}{|sin\left(\varphi \right)|}^{{\alpha }_2-1}=$$

$$4{\int }_0^{\pi /2}d\varphi {(cos\left(\varphi \right))}^{{\alpha }_1-1}{(sin\left(\varphi \right))}^{{\alpha }_2-1}={\displaystyle \frac{2\Gamma ({\alpha }_1/2)\Gamma ({\alpha }_2/2)}{\Gamma ({\alpha }_1/2+{\alpha }_2/2)}},$$

(208)

the substitution of Equations (207) and (208) into (206) gives

$${\int }_{{\Omega }_B}d{\mu }_2(D,\mathbf{x})=\prod _{k=1}^2{c}_1\left({\alpha }_k\right){\displaystyle \frac{2\Gamma ({\alpha }_1/2)\Gamma ({\alpha }_2/2)}{\Gamma ({\alpha }_1/2+{\alpha }_2/2)}}{\displaystyle \frac{1}{{\alpha }_1+{\alpha }_2}}{R}^{{\alpha }_1+{\alpha }_2}.$$

(209)

Then, requirement (200) and $D={\alpha }_1+{\alpha }_2$ gives

$$\prod _{k=1}^2{c}_1\left({\alpha }_k\right){\displaystyle \frac{2\Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)}{\Gamma ({\alpha }_1+{\alpha }_2)}}{\displaystyle \frac{1}{{\alpha }_1+{\alpha }_2}}{R}^{{\alpha }_1+{\alpha }_2}={\displaystyle \frac{2{\pi }^{({\alpha }_1+{\alpha }_2)/2}}{({\alpha }_1+{\alpha }_2)\Gamma ({\alpha }_1/2+{\alpha }_2/2)}}{R}^{{\alpha }_1+{\alpha }_2}.$$

(210)

Equation (210) gives the condition

$$c_1\left({\alpha }_1\right)c_1\left({\alpha }_2\right)={\displaystyle \frac{{\pi }^{{\alpha }_1/2}{\pi }^{{\alpha }_2/2}}{\Gamma ({\alpha }_1/2)\Gamma ({\alpha }_2/2)}}$$

(211)

that must be satisfied for all ${\alpha }_1,{\alpha }_2\in (0,1]$. For ${\alpha }_1=1$, Equation (211) gives

$$c_1\left({\alpha }_2\right)={\displaystyle \frac{{\pi }^{{\alpha }_2/2}}{\Gamma ({\alpha }_2/2)}}.$$

(212)

For ${\alpha }_2=1$, Equation (211) gives

$$c_1\left({\alpha }_1\right)={\displaystyle \frac{{\pi }^{{\alpha }_1/2}}{\Gamma ({\alpha }_1/2)}}.$$

(213)

As a result, we derived the explicit form of the coefficient

$$c_1\left({\alpha }_k\right)={\displaystyle \frac{{\pi }^{{\alpha }_k/2}}{\Gamma ({\alpha }_k/2)}},$$

(214)

for $k=1$ and $k=2$, where all ${\alpha }_k\in (0,1]$. □

Remark 17. 

It should be noted that Lemma 1 can be proved for an arbitrary value $n\in \mathbb{N}$ similarly to the proof of Lemma 1.

Let us consider the application of the second correspondence principle (Principle 2) for the case $n=1$. Principle 2 requires that the NIDS integration over the D-dimensional ball region ${\Omega }_B=\{x:|x|⩽R\}$, with $D=\alpha \in (0,1]$, must give the well-known equation of the volume of D-dimensional ball. Using this principle, we can derive the explicit form of the function $c_1(\alpha ,x)$, including the dependence of this function on the coordinate $x\in \mathbb{R}$ and the dimension $\alpha \in (0,1]$.

Lemma 2. 

Let us assume that the proposed NIDS integration in NIDS with $D\in (0,n]$ and $n=1$, over the D-dimensional ball ${\Omega }_B:=\{\mathbf{x}:|\mathbf{x}|⩽R\}$ with radius R, gives the standard value

$${\int }_W{\chi }_{{\Omega }_B}\left(\mathbf{x}\right)d{\mu }_1(D,\mathbf{x})=V_D\left({\Omega }_B\right):={\displaystyle \frac{2{\pi }^{D/2}}{D\Gamma (D/2)}}{R}^D,$$

(215)

for all $R>0$, where $D=\alpha \in (0,1]$.

If we assume that $c_1(\alpha ,x)=c_1(\alpha ,|x|)$, then this function has the form

$$c_1(\alpha ,x)={\displaystyle \frac{{\pi }^{\alpha /2}}{\Gamma (\alpha /2)}}{|x|}^{\alpha -1},$$

(216)

where $|x|=\sqrt{r(x,x)}=\sqrt{x^2}$.

Proof. 

Equation (215) has the form

$${\int }_{-R}^{R}d{\mu }_1(\alpha ,x)={\displaystyle \frac{2{\pi }^{D/2}}{D\Gamma (D/2)}}{R}^D,$$

(217)

since the ball region is defined by the equation $|\mathbf{x}|⩽R$ that gives $x\in [-R,R]$ for the case $n=1$. Let us asssume that $c_1(\alpha ,x)=c_1(\alpha ,|x|)$. Then, Equation (217) can be written as

$$2{\int }_0^R{c}_1(\alpha ,x)dx={\displaystyle \frac{2{\pi }^{\alpha /2}}{\alpha \Gamma (\alpha /2)}}{R}^{\alpha }.$$

(218)

Using the first fundamental theorem of the standard calculus in the form

$${\displaystyle \frac{d}{dR}}{\int }_0^{R}f\left(x\right)dx=f\left(R\right),$$

(219)

the application of the derivative $d/dR$ to Equation (218) gives

$$c_1(\alpha ,R)={\displaystyle \frac{{\pi }^{\alpha /2}}{\Gamma (\alpha /2)}}{R}^{\alpha -1},$$

(220)

for all $R>0$ and all $\alpha \in (0,1]$. □

The proved Lemmas 1 and 2 show that the second correspondence principle of the NIDS integral calculus (Principle 2) allows us to specify the normalization of the proposed NIDS integration.

2.6. Examples of NIDS Integrals in D-Dimensional Space

Let us consider, using examples, the differences between the NIDS integrals in spaces with non-integer dimensions $D=1.5$ and $D=2.7$ and standard integrals in spaces with integer dimensions $n\in \mathbb{N}$.

First, we note that the dimensions $D=1.5$ (and $D=2.7$) can be obtained from the equation

$$D:=\sum _{k=1}^n{\alpha }_k({\alpha }_k\in (0,1])$$

(221)

by using different values of n. Note that $D\in (0,n]$.

One can give the following examples for $D=1.5$.

(A1) For $n=2$, one can use, for example, the values

$$D={\alpha }_1+{\alpha }_2=1+0.5,$$

(222)

$$D={\alpha }_1+{\alpha }_2=0.7+0.8$$

(223)

to get $D=1.5$.

(A2) For $n=3$, one can use, for example, the values

$$D={\alpha }_1+{\alpha }_2+{\alpha }_3=1+0.3+0.2,$$

(224)

$$D={\alpha }_1+{\alpha }_2+{\alpha }_3=0.5+0.5+0.5,$$

(225)

$$D={\alpha }_1+{\alpha }_2+{\alpha }_3=0.3+0.5+0.7$$

(226)

to get $D=1.5$.

(A3) For $n=5$, one can use, for example, the values

$$D=\sum _{k=1}^5{\alpha }_k=0.3+0.3+0.3+0.3+0.3,$$

(227)

$$D=\sum _{k=1}^5{\alpha }_k=0.1+0.2+0.3+0.5+0.5$$

(228)

to get $D=1.5$.

One can give the following examples for $D=2.7$.

(B1) For $n=3$, one can use, for example, the values

$$D={\alpha }_1+{\alpha }_2+{\alpha }_3=1+1+0.7,$$

(229)

$$D={\alpha }_1+{\alpha }_2+{\alpha }_3=0.9+0.9+0.9$$

(230)

to get $D=2.7$.

(B2) For $n=5$, one can use, for example, the values

$$D=\sum _{k=1}^5{\alpha }_k=1+1+0.5+0.1+0.1,$$

(231)

$$D=\sum _{k=1}^5{\alpha }_k=0.1+0.3+0.6+0.8+0.9$$

(232)

to get $D=2.7$.

(B3) For $n=27$, one can use, for example, the values

$$D=\sum _{k=1}^{27}{\alpha }_k=0.1+\dots +0.1,$$

(233)

where all ${\alpha }_k-0.1$, to get $D=2.7$.

As a result, it can be stated that dimensions $D=1.5$ and $D=2.7$ can be obtained in an infinite number of ways. As a result, the corresponding NIDS integrals can also be different. Let us give several examples of such integrals.

For example, let us consider the region

$${\Omega }_{P,n}:=\{\mathbf{x}:a_k⩽x_k⩽b_k,k=1,2,\dots ,n\}.$$

(234)

The NIDS volume integral is represented as

$$I_{{\Omega }_{P,n}}^{D}f\left(\mathbf{x}\right)={\int }_{a_1}^{b_1}d{x}_1{\displaystyle \frac{{\pi }^{{\alpha }_1/2}{|x_1|}^{{\alpha }_1-1}}{\Gamma ({\alpha }_1/2)}}\dots {\int }_{a_n}^{b_n}d{x}_n{\displaystyle \frac{{\pi }^{{\alpha }_n/2}{|x_1|}^{{\alpha }_n-1}}{\Gamma ({\alpha }_n/2)}}f\left(\mathbf{x}\right),$$

(235)

and the standard integral in integer-dimensional space is

$$I_{{\Omega }_{P,n}}^{n}f\left(\mathbf{x}\right)={\int }_{a_1}^{b_1}d{x}_1\dots {\int }_{a_n}^{b_n}d{x}_{n}f\left(\mathbf{x}\right),$$

(236)

where $\mathbf{x}=(x_1,\dots x_n)$.

For the case (A1), with ${\alpha }_1=0.7$, ${\alpha }_2=0.8$, and $D=1.5$, we have

$$I_{{\Omega }_{P,2}}^{1.5}f(x,y)=$$

$${\int }_{a_1}^{b_1}dx{\displaystyle \frac{{\pi }^{0.35}{|x|}^{-0.3}}{\Gamma (0.35)}}{\int }_{a_2}^{b_2}dy{\displaystyle \frac{{\pi }^{0.4}{|y|}^{-0.2}}{\Gamma (0.4)}}f(x,y)=$$

$${\displaystyle \frac{{\pi }^{0.75}}{\Gamma (0.35)\Gamma (0.4)}}{\int }_{a_1}^{b_1}{\int }_{a_2}^{b_2}{f(x,y)|x|}^{-0.3}{|y|}^{-0.2}dxdy,$$

(237)

and the standard integral in integer-dimensional space is

$$I_{{\Omega }_{P,2}}^{2}f(x,y)={\int }_{a_1}^{b_1}dx{\int }_{a_2}^{b_2}dyf(x,y).$$

(238)

For the case (A2), with ${\alpha }_1=0.3$, ${\alpha }_2=0.5$, ${\alpha }_2=0.7$, and $D=1.5$, we have

$$I_{{\Omega }_{P,3}}^{1.5}f\left(\mathbf{x}\right)=$$

$${\int }_{a_1}^{b_1}dx{\displaystyle \frac{{\pi }^{0.15}{|x|}^{-0.7}}{\Gamma (0.15)}}{\int }_{a_2}^{b_2}dy{\displaystyle \frac{{\pi }^{0.25}{|y|}^{-0.5}}{\Gamma (0.25)}}{\int }_{a_3}^{b_3}dz{\displaystyle \frac{{\pi }^{0.35}{|z|}^{-0.3}}{\Gamma (0.35)}}f(x,y,z)=$$

$${\displaystyle \frac{{\pi }^{0.75}}{\Gamma (0.15)\Gamma (0.25)\Gamma (0.35)}}{\int }_{a_1}^{b_1}{\int }_{a_2}^{b_2}{\int }_{a_3}^{b_3}{f(x,y,z)|x|}^{-0.7}{y|}^{-0.5}{|z|}^{-0.3}dxdydz,$$

(239)

and the standard integral in integer-dimensional space is

$$I_{{\Omega }_{P,3}}^{3}f\left(\mathbf{x}\right)={\int }_{a_1}^{b_1}dx{\int }_{a_2}^{b_2}dy{\int }_{a_3}^{b_3}dzf(x,y,z).$$

(240)

For the case (B1) with ${\alpha }_1={\alpha }_2={\alpha }_3=0.9$, and $D=2.7$, we have

$$I_{{\Omega }_{P,3}}^{2.7}f\left(\mathbf{x}\right)=$$

$${\int }_{a_1}^{b_1}dx{\displaystyle \frac{{\pi }^{0.45}{|x|}^{-0.1}}{\Gamma (0.45)}}{\int }_{a_2}^{b_2}dy{\displaystyle \frac{{\pi }^{0.45}{|y|}^{-0.1}}{\Gamma (0.45)}}{\int }_{a_3}^{b_3}dz{\displaystyle \frac{{\pi }^{0.45}{|z|}^{-0.1}}{\Gamma (0.45)}}f(x,y,z)=$$

$${\displaystyle \frac{{\pi }^{1.35}}{{(\Gamma (0.45))}^3}}{\int }_{a_1}^{b_1}{\int }_{a_2}^{b_2}{\int }_{a_3}^{b_3}{f(x,y,z)|x|}^{-0.1}{y|}^{-0.1}{|z|}^{-0.1}dxdydz,$$

(241)

and the standard integral in integer-dimensional space is

$$I_{{\Omega }_{P,3}}^{3}f\left(\mathbf{x}\right)={\int }_{a_1}^{b_1}dx{\int }_{a_2}^{b_2}dy{\int }_{a_3}^{b_3}dzf(x,y,z).$$

(242)

The NIDS integrals with $D={\alpha }_1+\dots +{\alpha }_n$ and the standard integrals in spaces of integer dimension $n>\left[D\right]$ are written similarly.

Note that the Wilson approach represents the integration in the infinite-dimensional space with a special form of the standard integration in the integer-dimensional space and a special form of measure. Therefore, in application, we can use the NIDS integration in the form of the integration in the integer-dimensional space with a special form of measure. Note that this fact is also described in Collins [10].

2.7. NIDS Integration Satisfies Correspondence Principles

Let us consider the D-dimensional ball region with $D\in (0,n]$ for all values of $n\in \mathbb{N}$ by using the proposed volume NIDS integration in the (hyper)spherical coordinates.

In the following theorem, we prove that the proposed volume NIDS integration satisfies the second correspondence principle of NIDS integral calculus (Principle 2) for all values of the dimension $D>0$.

Theorem 6. 

Let $W=W_1\times \dots \times W_n$ be D-dimensional space with $D={\alpha }_1+\dots +{\alpha }_n$. Let ${\Omega }_B$ be the D-dimensional ball ${\Omega }_B:=\{\mathbf{x}:|\mathbf{x}|⩽R\}$ with radius $R>0$, and let function $f\left(\mathbf{x}\right)={\chi }_{{\Omega }_B}\left(\mathbf{x}\right)$ be the characteristic function of the region ${\Omega }_B$.

Then, the volume NIDS integration of function $f\left(\mathbf{x}\right)={\chi }_{{\Omega }_B}\left(\mathbf{x}\right)$ over the region ${\Omega }_B$ with the dimension $D\in (0,n]$ gives the well-known expression of the D-dimensional ball volume

$${\int }_W{\chi }_{{\Omega }_B}\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x})={\int }_{{\Omega }_B}\prod _{k=1}^n{c}_1({\alpha }_k,x_k)d{x}_k=V_D\left({\Omega }_B\right):={\displaystyle \frac{2{\pi }^{D/2}}{D\Gamma (D/2)}}{R}^D,$$

(243)

for all $n\in \mathbb{N}$ and $D\in (0,n]$, if the measures $d{\mu }_1({\alpha }_k,x_k)$ are defined as

$$d{\mu }_1({\alpha }_k,x_k)=c_1({\alpha }_k,x_k)d{x}_k=c_1\left({\alpha }_k\right){|x_k|}^{{\alpha }_k-1}d{x}_k,$$

(244)

where

$$c_1\left({\alpha }_k\right)={\displaystyle \frac{{\pi }^{{\alpha }_k/2}}{\Gamma ({\alpha }_k/2)}}.$$

(245)

Theorem 6 is proved in Appendix A.

Remark 18. 

Theorem 6 shows that the second correspondence principle of NIDS integral calculus (Principle 2) holds for the proposed definition of the NIDS integration for all $D>0$. Theorem 6 proves that Principle 2 holds, since the volume NIDS integration gives the well-known expression of the D-dimensional ball region ${\Omega }_B$ for all $D>0$ and all $R>0$.

As a result, in order for the volume NIDS integration

$${\int }_{W}f\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x})={\int }_{W}f(x_1,\dots ,x_n)\prod _{k=1}^{n}d{\mu }_1({\alpha }_k,x_k)=$$

$${\int }_{W}f(x_1,\dots ,x_n)\prod _{k=1}^n{c}_1({\alpha }_k,x_k)d{x}_k$$

(246)

to satisfy the second correspondence principle of NIDS integral calculus (principle 2), the functions $c_1({\alpha }_k,x_k)$ must be defined as

$$c_1({\alpha }_k,x_k)={\displaystyle \frac{{\pi }^{{\alpha }_k/2}}{\Gamma ({\alpha }_k/2)}}{|x_k|}^{{\alpha }_k-1},$$

(247)

with ${\alpha }_k\in (0,1]$.

This allows us to say that Principle 2 uniquely defines the type of integration measure and the form of the function (247).

Let us consider the Wilson normalization condition [9,10] for the volume NIDS integration over the entire D-dimensional space $W=W_1$ with $D=\alpha \in (0,1]$.

Theorem 7. 

The Wilson normalization condition for the D-dimensional space $W=W_1$ with $D=\alpha \in (0,1]$, in the form

$${\int }_{-\infty }^{+\infty }d{\mu }_1(\alpha ,x)exp(-x^2)={\pi }^{D/2}$$

(248)

is satisfied for all $D=\alpha \in (0,1]$, if

$$d{\mu }_1(\alpha ,x)=c_1(\alpha ,x)dx={\displaystyle \frac{{\pi }^{\alpha /2}}{\Gamma (\alpha /2)}}{|x|}^{\alpha -1}dx,$$

(249)

where ${|x|}^{\alpha -1}={\left(x^2\right)}^{\alpha /2-1/2}$, and $D=\alpha$.

Proof. 

Using measure (249), Equation (248) takes the form

$${\displaystyle \frac{{\pi }^{\alpha /2}}{\Gamma (\alpha /2)}}{\int }_{-\infty }^{+\infty }{|x|}^{\alpha -1}exp(-x^2)dx={\pi }^{D/2}.$$

(250)

Let us consider the integral

$${\int }_{-\infty }^{+\infty }{|x|}^{\alpha -1}exp(-x^2)dx=2{\int }_0^{+\infty }{\left(x^2\right)}^{\alpha /2-1/2}exp(-x^2)dx=$$

$$2{\int }_0^{+\infty }{\left(x^2\right)}^{\alpha /2-1}exp(-x^2)xdx={\int }_0^{+\infty }{z}^{\alpha /2-1}exp(-z)dz,$$

(251)

where we use the variable $z=x^2$. Then, using Equation (1) in Section 2.3.3 in [146], p. 322, in the form

$${\int }_0^{\infty }{z}^{\alpha /2-1}exp(-pz)dz=\Gamma (\alpha /2)p^{-\alpha /2},(\alpha >0,p>0)$$

(252)

or the definition of the gamma-function, Equation (251) takes the form

$${\int }_{-\infty }^{+\infty }{|x|}^{\alpha -1}exp(-x^2)dx=\Gamma (\alpha /2).$$

(253)

Substituting (253) into Equation (250) gives ${\pi }^{\alpha }={\pi }^D$, which holds if $D=\alpha$. □

Corollary 1. 

The Wilson normalization condition for the entire D-dimensional space $W=W_1\times \dots \times W_n$ in the form

$${\int }_{W}d{\mu }_n(D,\mathbf{x})exp(-{\mathbf{x}}^2)={\pi }^{D/2}$$

(254)

is satisfied for all $D\in (0,n]$ and all $n\in \mathbb{N}$, if

$$d{\mu }_n(D,\mathbf{x})=\prod _{k=1}^{n}d{\mu }_1({\alpha }_k,x^k),$$

(255)

where all $d{\mu }_1({\alpha }_k,x^k)$ have the form (249), and $D={\alpha }_1+\dots {\alpha }_n$, where all ${\alpha }_k\in (0,1]$.

Proof. 

Using Equation (255), the integral in Equation (254) gives

$${\int }_{W}d{\mu }_n(D,\mathbf{x})exp(-{\mathbf{x}}^2)=$$

$${\int }_{W_1}\dots {\int }_{W_n}exp(-x_1^2)\dots exp(-x_n^2)d{\mu }_1({\alpha }_1,x^1)\dots d{\mu }_1({\alpha }_n,x^n)=$$

$$\prod _{k-1}^n\left({\int }_{-\infty }^{+\infty }d{\mu }_1({\alpha }_k,x_k)exp(-x_k^2)\right)=$$

$$\prod _{k-1}^n{\pi }^{{\alpha }_k/2}={\pi }^{{\sum }_{k=1}^n{\alpha }_k/2}={\pi }^{D/2},$$

(256)

where $D={\alpha }_1+\dots +{\alpha }_n$. □

Remark 19. 

Note that the representation of $c_1({\alpha }_k,x_k)$ by Equation (247) was proposed in [18]. Note that the second correspondence principle of NIDS integral calculus (Principle 2) and Equation (243) cannot be satisfied for other forms of the functions $c_1({\alpha }_k,x_k)$, which were suggested in other articles on fractal physics (see references in [18,69]).

Note that the first correspondence principle of NIDS integral calculus (Principle 1) is satisfied, since

$${\int }_{W}f\left(\mathbf{x}\right)d{\mu }_n(n,\mathbf{x})={\int }_{W}f(x_1,\dots ,x_n)\prod _{k=1}^{n}d{\mu }_1(1,x_k)={\int }_{W}f(x_1,\dots ,x_n)\prod _{k=1}^{n}d{x}_k.$$

(257)

The proposed NIDS integration can be used to calculate the well-known expressions of D-dimensional volumes of the ball, the parallelepiped, and the cylinder regions in NIDS.

Remark 20. 

Let us consider the D-dimensional ellipsoid ${\Omega }_E$ in NIDS with $D={\sum }_{k=1}^n{\alpha }_k\in (0,n]$, which is defined by the equations

$$x_1=l_{1}rcos\left({\varphi }_1\right),$$

(258)

$$x_k=l_{k}rcos\left({\varphi }_k\right)\prod _{j=1}^{k-1}sin\left({\varphi }_j\right),(k=2,\dots ,n-1),$$

(259)

$$x_n=l_{n}r\prod _{j=1}^{n-1}sin\left({\varphi }_j\right),$$

(260)

where $r\in [0,1]$, ${\varphi }_k\in [0,\pi ]$ for $k=1,\dots ,n-2$, ${\varphi }_{n-1}\in [0,2\pi )$, and $l_k>0$ for $k=1,\dots ,n$.

Using similar transformations as in the proof of Theorem 6, the volume NIDS integration of the D-dimensional ellipsoid region ${\Omega }_E$ gives

$${\int }_W{\chi }_{{\Omega }_E}\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x})=V_D\left({\Omega }_E\right):={\displaystyle \frac{2{\pi }^{D/2}}{D\Gamma (D/2)}}\prod _{k=1}^n{l}_k^{{\alpha }_k},$$

(261)

for all $n\in \mathbb{N}$, all ${\alpha }_k\in (0,1]$, and all $D\in (0,n]$.

For $D=2$, Equation (261) gives the well-known area (two-dimensional volume) of the ellipsoid region

$$V_2\left({\Omega }_E\right):=\pi ab,$$

(262)

where $a=l_1$, $b=l_2$, since $\Gamma \left(1\right)=1$.

For $D=3$, Equation (261) gives the well-known volume of the ellipsoid region

$$V_3\left({\Omega }_E\right):={\displaystyle \frac{4\pi }{2}}abc,$$

(263)

where $a=l_1$, $b=l_2$, and $c=l_3$, since $\Gamma (3/2)={\pi }^{1/2}/2$.

Remark 21. 

The single-variable measures

$$d{\mu }_1({\alpha }_k,x_k)=c_1({\alpha }_k,x_k)d{x}_k={\displaystyle \frac{{\pi }^{{\alpha }_k/2}}{\Gamma ({\alpha }_k/2)}}{|x_k|}^{{\alpha }_k-1}d{x}_k$$

(264)

were proposed in paper [18] to describe anisotropic fractal media. From a physics point of view, the functions $c_1({\alpha }_k,x_k)$ can be interpreted as a density of states (DOS) along the $X_k$-axis [18,19,69,87]. The interpretation of the functions $c_1({\alpha }_k,x_k)$ as densities of states has been suggested in [18,87]. The density of states $c_n(D,\mathbf{x})$ describes the closely packed permitted states of matter particles (space points $\mathbf{x}$) in the space ${\mathbb{R}}^n$ that is defined as

$$c_n(D,\mathbf{x}):=\prod _{k=1}^n{c}_1({\alpha }_k,x_k),$$

(265)

where the parameter $D={\sum }_{k=1}^n{\alpha }_k$ is interpreted as a total dimension of NIDS with $D\in (0,n]$.

2.8. Product Measure and Induced Metric Approaches

2.8.1. Product Measure Approach and Correspondence Principle

In order to have correct definitions of the volume, surface, and line NIDS integrations in D-dimensional space, we should specify the correspondence principles for the NIDS integration. It should be noted that the following principles for surface and line NIDS integrations actually expand Principles 1 and 2 and allow us to propose adequate definitions of line and surface NIDS integrals. In the following subsections, we propose explicit equations that represent the line, surface, and volume NIDS integrals through the standard line, surface, and volume integrals in the integer-dimensional spaces.

The correspondence principle for volume, surface, and line NIDS integrals should be satisfied in the following form. Note that it is assumed that the line, surface, and volume NIDS integrals satisfy the axioms of the NIDS integration that are described in Section 2.3.

Principle 4 

(Correspondence Principle for Volume, Surface, and Line NIDS Integrals). For the D-dimensional space $W=W_1\times \dots \times W_n$ with $D\in (0,n]$, the volume, surface, and line NIDS integrals should give the usual volume, surface, and line integrals in the n-dimensional Euclidean space for all $D=n\in \mathbb{N}$.

In addition, the volume, surface, and line NIDS integrals should satisfy the following principles in the D-dimensional space $W=W_1\times \dots \times W_n$ for all $n\in \mathbb{N}$.

(I) Correspondence Principle for Volume NIDS Integrals: The volume NIDS integrals over the ball region

$${\Omega }_B:=\{\mathbf{x}:|\mathbf{x}|⩽R\}$$

(266)

in the D-dimensional space $W=W_1\times \dots \times W_n$ with $D\in (0,n]$ and the function $f\left(\mathbf{x}\right)={\chi }_{{\Omega }_B}\left(\mathbf{x}\right)$ should give the well-known expression of the volume

$$I_{{\Omega }_B}^D{\chi }_{{\Omega }_B}\left(\mathbf{x}\right)=V_D\left({\Omega }_B\right):={\displaystyle \frac{2{\pi }^{D/2}}{D\Gamma (D/2)}}{R}^D$$

(267)

that must hold for all $D\in (0,n]$, all $n\in \mathbb{N}$, and all $R>0$, where $D={\alpha }_1+\dots +{\alpha }_n$, and ${\chi }_{{\Omega }_B}\left(\mathbf{x}\right)$ is the characteristic function of the region ${\Omega }_B$.

(II) Correspondence Principle for Surface NIDS Integrals: The surface NIDS integrals over the D-dimensional semi-sphere

$${\Omega }_S:=\{\mathbf{x}:|\mathbf{x}|=R,x_n⩾0\}$$

(268)

in the D-dimensional space $W=W_1\times \dots \times W_n$ with $D\in (0,n]$ and $n⩾3$ ($n\in \mathbb{N}$) and the characteristic function $f\left(\mathbf{x}\right)={\chi }_{{\Omega }_S}\left(\mathbf{x}\right)$ must give the well-known expression of the surface area of the D-dimensional semi-sphere

$$I_{{\Omega }_S}^D{\chi }_{{\Omega }_S}\left(\mathbf{x}\right)=S_D\left({\Omega }_S\right):={\displaystyle \frac{{\pi }^{D/2}}{\Gamma (D/2)}}{R}^{D-1}$$

(269)

for all $D\in (0,n]$, all $n⩾3$ ($n\in \mathbb{N}$), and all $R>0$. For $n=2$, this principle gives the correspondence principle for line NIDS integrals.

(III) Correspondence Principle for Line NIDS Integrals: The line NIDS integrals over the length of the D-dimensional circle

$${\Omega }_C:=\{\mathbf{x}:|\mathbf{x}|=R\}$$

(270)

in the D-dimensional space $W=W_1\times W_2$ with dimension $D\in (0,2]$ and the characteristic function $f\left(\mathbf{x}\right)={\chi }_{{\Omega }_C}\left(\mathbf{x}\right)$ should give the well-known expression of the D-dimensional circle length

$$I_{{\Omega }_C}^D{\chi }_{{\Omega }_C}\left(\mathbf{x}\right)=L_D\left({\Omega }_C\right):={\displaystyle \frac{2{\pi }^{D/2}}{\Gamma (D/2)}}{R}^{D-1},$$

(271)

which must be satisfied for all $D\in (0,2]$ and all $R>0$, where $D={\alpha }_1+{\alpha }_2$.

(IV) The NIDS generalization of the standard Gauss–Ostrogradsky theorem, the Stoke theorem, and the gradient theorem must be satisfied to provide the consistency of the volume, surface, and line NIDS integrals and the NIDS vector differential operators.

Briefly, these correspondence principles require that the volume, surface, and line NIDS integrals of the ball volume, the area of a semi-sphere, and the circle length in D-dimensional space must give the well-known expressions for all $D>0$.

Remark 22. 

The correspondence principle for the volume NIDS integral is a special case of Principles 1 and 2. The fulfillment of these principles was proved in Theorem 6 of the previous subsection.

2.8.2. Metric Approach and Universal Metric Consistency Principle

Using the product of NIDS measures in the D-dimensional space $W=W_1\times \dots \times W_n$, one can define a metric tensor in this space. There is no unique way to define the metric of the space $W=W_1\times \dots \times W_n$ that can be used in the line, surface, and volume NIDS integrals.

For the correct definition of the metric tensor and the construction of the universal-induced metric for volumes, surfaces, and lines in D-dimensional space, we propose to use a principle that can be called the universal metric consistency principle. This principle requires the consistency of the metrics that are induced by the D-dimensional space on the surface and line. In fact, this principle defines the metric approach first proposed in [18].

Principle 5 

(Principle of Universal Metric Consistency). In D-dimensional space $W=W_1\times \dots \times W_n$ with $D\in (0,n]$, the metric ${\mathcal{G}}_{kl}$ must be consistent with the measure in the space W, and this metric must define the length of curves in the space W and generate induced metrics on the surfaces in the space W. This means that the following three conditions of metric self-consistency must be satisfied in the metric approach.

(I) In the D-dimensional space $W=W_1\times \dots \times W_n$ with $D={\alpha }_1+\dots +{\alpha }_n$, where all ${\alpha }_j\in (0,1]$, the product of measures must induce a positive-definite metric ${\mathcal{G}}_{kl}\left(\mathbf{x}\right)$, $k,l=1,\dots ,n$, such that the condition

$$\sqrt{|det({\mathcal{G}}_{kl}\left(\mathbf{x}\right))|}=c_n(D,\mathbf{x}):=\prod _{k=1}^n{c}_1({\alpha }_k,x^k)$$

(272)

is satisfied for all $n\in \mathbb{N}$, and the NIDS volume of the bounded region $\Omega \subset W$ is described as

$$V_D^{\ast }(\Omega )={\int }_{\Omega }\sqrt{|det({\mathcal{G}}_{kl}\left(\mathbf{x}\right))|}\prod _{k=1}^{n}d{x}^k,$$

(273)

for all $n\in \mathbb{N}$.

(II) The length $L_D^{\ast }[a,b]$ of the line L, which is described by equation $\mathbf{x}=\mathbf{x}\left(t\right)\in C^1[a,b]$ in the D-dimensional space $W=W_1\times \dots \times W_n$, should be defined by the metric ${\mathcal{G}}_{kl}\left(\mathbf{x}\right)$, such that

$$L_D^{\ast }[a,b]={\int }_a^b\sqrt{\sum _{k,l=1}^n{\mathcal{G}}_{kl}\left(\mathbf{x}\left(t\right)\right){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}{\displaystyle \frac{d{x}^l\left(t\right)}{dt}}}dt$$

(274)

is satisfied for all $n\in \mathbb{N}$.

(III) The metrics ${\mathcal{G}}_{kl}^S\left(\mathbf{x}\right)$ on the surface Σ, which is described by the equations $x^k=x^k(u^1,\dots ,u^m)$ with $\mathbf{u}\in U$ ($k=1,\dots ,n$, $m<n$ in the D-dimensional space $W=W_1\times \dots \times W_n$, should be induced by the metric ${\mathcal{G}}_{kl}\left(\mathbf{x}\right)$, such that

$${\mathcal{G}}_{ij}^S\left(\mathbf{u}\right):=\sum _{k,l=1}^n{\mathcal{G}}_{kl}\left(\mathbf{x}\left(\mathbf{u}\right)\right){\displaystyle \frac{\partial x^k\left(\mathbf{u}\right)}{\partial u^i}}{\displaystyle \frac{\partial x^l\left(\mathbf{u}\right)}{\partial u^j}}(i,j=1,\dots ,m)$$

(275)

is satisfied for all $n,m\in \mathbb{N}$, where $1<m<n$, and the NIDS area on the surface Σ is described by the equation

$$S_D^{\ast }(\Sigma )={\int }_U\sqrt{|det({\mathcal{G}}_{ij}^S\left(\mathbf{u}\right))|}\prod _{k=1}^{m}d{u}^k,$$

(276)

for all $n,m\in \mathbb{N}$, where $2⩽m⩽n-1$. Then, the metric ${\mathcal{G}}_{ij}^S\left(\mathbf{u}\right)$ can be called the universal metric induced on the surface by the NIDS metric ${\mathcal{G}}_{kl}\left(\mathbf{x}\right)$.

(IV) The NIDS generalization of the standard Gauss–Ostrogradsky theorem, the Stoke theorem, and the gradient theorem must be satisfied to provide the consistency of the volume, surface, and line NIDS integrals and the NIDS vector differential operators.

Remark 23. 

An obvious question arises: Does the correspondence Principle 4 hold for the volume, surface, and line NIDS integrals, if the principle of the universal metric consistency (Principle 5) is satisfied? It should be emphasized that the implementation of the universal metric consistency principle leads to the fact that the correspondence principles for the surface and line NIDS integrals are violated.

The approach to constructing the calculus in non-integer-dimensional spaces, which is based on the universal metric consistency principle (Principle 5), will be called the metric approach.

The approach to constructing the NIDS calculus, which is based on the correspondence principle for volume, surface, and line NIDS integrals (Principle 4), will be called the product measure approach.

Remark 24. 

It should be emphasized that the volume NIDS integrals in the metric approach and the product measure approach coincide:

$$I_{\Omega }^{D}f\left(\mathbf{x}\right)=I_{\Omega }^{D,\ast }f\left(\mathbf{x}\right).$$

(277)

In these approaches, the NIDS volumes also coincide:

$$V_D(\Omega )=V_D^{\ast }(\Omega ).$$

(278)

The difference between these two approaches is manifested in the definitions of the surface NIDS integrals and the line NIDS integrals. In general, the area of surface region Σ in the D-dimensional space may not coincide in these approaches:

$$S_D(\Sigma )\ne S_D^{\ast }(\Sigma ).$$

(279)

The lengths of the section Ω of the curved line in the D-dimensional space may also not coincide in these two approaches:

$$L_D(\Omega )\ne L_D^{\ast }\left({\Omega }_C\right).$$

(280)

As will be shown below, the inequalities are due to the following fact. The metric approach cannot give the well-known expression of the surface area of the D-dimensional semi-sphere

$$S_D^{\ast }\left({\Omega }_S\right)\ne {\displaystyle \frac{{\pi }^{D/2}}{\Gamma (D/2)}}{R}^{D-1},$$

(281)

for all $D\in (0,n]$, all $n⩾3$ ($n\in \mathbb{N}$), and all $R>0$ and cannot give the well-known expression of the D-dimensional circle length

$$L_D^{\ast }\left({\Omega }_C\right)\ne {\displaystyle \frac{2{\pi }^{D/2}}{\Gamma (D/2)}}{R}^{D-1},$$

(282)

for all $D\in (0,2]$ and all $R>0$, where $D={\alpha }_1+{\alpha }_2$.

In order to satisfy part I of the principle of induced metric consistency (principle 5), the volume NIDS integral of the function $f\left(\mathbf{x}\right)$ over the region $\Omega$ of the D-dimensional space $W=W_1\times \dots \times W_n$, in the form

$$I_{\Omega }^{D}f\left(\mathbf{x}\right)={\int }_W{\chi }_{\Omega }\left(\mathbf{x}\right)f\left(\mathbf{x}\right)c_n(D,\mathbf{x})\prod _{k=1}^{n}d{x}^k$$

(283)

must be satisfied for a wide class of bounded regions $\Omega \subset W$ and all $D\in (0,n]$. If Equation (283) holds for a wide class of bounded regions $\Omega \subset W$, then the equation

$$\sqrt{|det({\mathcal{G}}_{kl}\left(\mathbf{x}\right))|}=c_n(D,\mathbf{x}):=\prod _{k=1}^n{c}_1({\alpha }_k,x^k)$$

(284)

is satisfied for all $n\in \mathbb{N}$. In this case, function (284), which is used in the NIDS integral (283), can be interpreted as the root of the determinant $g\left(\mathbf{x}\right)$ of some metric tensor $g_{kl}\left(\mathbf{x}\right)$.

In paper [18], we proposed a metric tensor that satisfies part I of the principle of universal metric consistency (Principle 5) in the form

$$g_{kl}\left(\mathbf{x}\right)=c_1^2({\alpha }_k,x_k)(k=l),g_{kl}\left(\mathbf{x}\right)=0(k\ne l),$$

(285)

where $k,l=1,\dots ,n$. Note that $g_{kl}\left(\mathbf{x}\right)⩾0$ for all $\mathbf{x}\in W$, and this metric tensor is positive-definite $g(\mathbf{x},\mathbf{x})>0$ for every nonzero vector $\mathbf{x}$. The metric (285) can be called the natural NIDS metric. The determinant of metric (285) is

$$g\left(\mathbf{x}\right)=det\left(g_{kl}\left(\mathbf{x}\right)\right)=\prod _{k=1}^n{g}_{kk}\left(\mathbf{x}\right)=\prod _{k=1}^n{c}_1^2({\alpha }_k,x_k)=c_n^2(D,\mathbf{x}),$$

(286)

and

$$\sqrt{|g(\mathbf{x})|}=\prod _{k=1}^n{c}_1({\alpha }_k,x_k).$$

(287)

As a result, the metric tensor (285) can be used to define the volume NIDS integral and the NIDS volume of the regions in the NIDS.

The volume NIDS integral over the entire space W is defined by the equation

$$I_W^{D}f\left(\mathbf{x}\right):={\int }_{W}f\left(\mathbf{x}\right)\sqrt{|g(\mathbf{x})|}\prod _{k=1}^{n}d{x}^k={\int }_{W}f\left(\mathbf{x}\right)c_n(D,\mathbf{x})\prod _{k=1}^{n}d{x}^k,$$

(288)

and the volume NIDS integral over the region $\Omega \subset W$ is defined as

$$I_{\Omega }^{D}f\left(\mathbf{x}\right):={\int }_{W}f\left(\mathbf{x}\right){\chi }_{\Omega }\left(\mathbf{x}\right)\sqrt{|g(\mathbf{x})|}\prod _{k=1}^{n}d{x}^k={\int }_{\Omega }f\left(\mathbf{x}\right)c_n(D,\mathbf{x})\prod _{k=1}^{n}d{x}^k,$$

(289)

where ${\chi }_{\Omega }\left(\mathbf{x}\right)$ is the characteristic function of the region $\Omega$, i.e., ${\chi }_{\Omega }\left(\mathbf{x}\right)=1$ if $\mathbf{x}\in \Omega$ and ${\chi }_{\Omega }\left(\mathbf{x}\right)=0$ if $\mathbf{x}\notin \Omega$.

Using Equation (288), the NIDS volume of the bounded region $\Omega$ in the D-dimensional space $W=W_1\times \dots \times W_n$ is described by the volume NIDS integral in the form

$$V_D(\Omega )=I_W^D{\chi }_{\Omega }\left(\mathbf{x}\right)={\int }_{\Omega }\sqrt{|g(\mathbf{x})|}\prod _{k=1}^{n}d{x}^k={\int }_{\Omega }{c}_n(D,\mathbf{x})\prod _{k=1}^{n}d{x}^k.$$

(290)

In the previous subsection (see Theorem 6), we proved that the volume NIDS integrals (290) over the ball region ${\Omega }_B:=\{\mathbf{x}:|\mathbf{x}|⩽R\}$ give the well-known expression of the volume

$${\int }_{{\Omega }_B}d{\mu }_n(D,\mathbf{x})={\int }_{{\Omega }_B}\sqrt{|g(\mathbf{x})|}\prod _{k=1}^{n}d{x}^k={\displaystyle \frac{2{\pi }^{D/2}}{D\Gamma (D/2)}}{R}^D,$$

(291)

for all $D\in (0,n]$, all $n\in \mathbb{N}$, and all $R>0$.

Remark 25. 

Using the natural NIDS metric (285), the line and surface NIDS integrals can be defined by the equations of the principle of universal metric consistency (principle 5). One can state that the requirements of the principle of universal metric consistency are satisfied, if we use

$${\mathcal{G}}_{kl}\left(\mathbf{x}\right)=g_{kl}\left(\mathbf{x}\right),$$

(292)

where $g_{kl}\left(\mathbf{x}\right)$ is the natural metric tensor (285).

2.8.3. Two Approaches to Define Metric Tensors in NIDS

The Wilson axioms and the Wilson normalization condition (or correspondence principle for the volume NIDS integration) ensure the uniqueness of the NIDS integration [10], p. 65. They determine the special form of integration measure in an n-dimensional space [10], p. 65, which can be called the NIDS measure. The proposed representation of the NIDS integration gives explicit equations so that the D-dimensional integrals are written as a sequence of ordinary integrals. This representation allows us to prove standard properties of the integration and the scaling property reflecting the D-dimensionality. In addition, it ensures that there exists a self-consistent definition of the NIDS integrals [10].

The properties of the NIDS integration are related to (and are provided by) the properties of the NIDS integration measure, which is a special form of the usual integration measure in integer-dimensional space.

In the product measure approach, the volume NIDS integration $I_W^D$ in the D-dimensional space $W=W_1\times \dots \times W_n$ is defined as the sequence of the NIDS integration $I_{W_k}^{{\alpha }_k}$ ($k=1,\dots ,n$) in the $D_k={\alpha }_k$-dimensional spaces $W_k$ with $D_k={\alpha }_k\in (0,1]$ by the equation

$$I_W^{D}f\left(\mathbf{x}\right):=I_{W_1}^{{\alpha }_1}\dots I_{W_n}^{{\alpha }_n}f\left(\mathbf{x}\right).$$

(293)

The NIDS integration $I_{W_k}^{{\alpha }_k}$ ($k=1,\dots ,n$) in the ${\alpha }_k$-dimensional spaces $W_k$ with ${\alpha }_k\in (0,1]$ is represented as the ordinary integral in the space ${\mathbb{R}}^1$ with the special form of the integration measure

$$I_{W_k}^{{\alpha }_k}f\left(x_k\right):={\int }_{-\infty }^{+\infty }f\left(x_k\right)d{\mu }_1({\alpha }_k,x^k).$$

(294)

The measure ${\mu }_1({\alpha }_k,[a_k,b_k])$ is a special form of the ordinary measure in the space ${\mathbb{R}}^1$ that is defined as

$${\mu }_1({\alpha }_k,[a_k,b_k]):={\int }_{a_k}^{b_k}{c}_1({\alpha }_k,x^k)d{x}^k.$$

(295)

In the previous subsection, using the product measure approach and the correspondence principles, we derived and proved the exact expression of the function

$$c_1({\alpha }_k,x_k)={\displaystyle \frac{{\pi }^{{\alpha }_k/2}}{\Gamma ({\alpha }_k/2)}}{|x_k|}^{{\alpha }_k-1},{\alpha }_k\in (0,1].$$

(296)

Therefore in the product measure approach, the volume NIDS integral $I_W^D$ in the D-dimensional space $W=W_1\times \dots \times W_n$ with $D={\alpha }_l+\dots +{\alpha }_n\in (0,n]$ can be represented as the ordinary integral in the space ${\mathbb{R}}^n$ with the special form of the integration measure

$$I_W^{D}f\left(\mathbf{x}\right):={\int }_{W}f\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x})={\int }_{W}f\left(\mathbf{x}\right)c_n(D,\mathbf{x})\prod _{k=1}^{n}d{x}^k,$$

(297)

where

$$c_n(D,\mathbf{x})=\prod _{k=1}^n{c}_1({\alpha }_k,x^k)=\prod _{k=1}^n{\displaystyle \frac{{\pi }^{{\alpha }_k/2}}{\Gamma ({\alpha }_k/2)}}{|x^k|}^{{\alpha }_k-1}({\alpha }_k\in (0,1]).$$

(298)

The measure ${\mu }_n(D,\Omega )$ with $\Omega \subset {\mathbb{R}}^n$ is a special form of the ordinary measure in the space ${\mathbb{R}}^n$ that is defined as the product of the measure ${\mu }_1({\alpha }_k,[a_k,b_k])$ in the product measure approach

$${\mu }_n(D,\Omega ):={\int }_{\Omega }\prod _{k=1}^{n}d{\mu }_1({\alpha }_k,x^k)={\int }_{\Omega }{c}_n(D,\mathbf{x})\prod _{k=1}^{n}d{x}^k.$$

(299)

The volume NIDS integral (297) can also be written as

$$I_W^{D}f\left(\mathbf{x}\right)={\int }_{W}f\left(\mathbf{x}\right)\sqrt{|det({\mathcal{G}}_{kl}\left(\mathbf{x}\right))|}\prod _{k=1}^{n}d{x}^k,$$

(300)

if ${\mathcal{G}}_{kl}\left(\mathbf{x}\right)$ is a metric tensor in the space ${\mathbb{R}}^n$ that is induced by the NIDS measure, such that the condition

$$\sqrt{|det({\mathcal{G}}_{kl}\left(\mathbf{x}\right))|}=c_n(D,\mathbf{x})$$

(301)

holds for all $n\in \mathbb{N}$ and all $D\in (0,n]$.

The product of measures of the $D_k$-dimensional spaces $W_k$ with $D_k={\alpha }_k\in (0,1]$, $k=1,\dots ,n$ forms the measure of the D-dimensional space $W=W_1\times \dots \times W_n$ with $D={\alpha }_1+\dots {\alpha }_n\in (0,n]$. The product of measures $d{\mu }_1({\alpha }_k,x^k)$ of the spaces $W_k$ induces the measure of the D-dimensional space W as

$$d{\mu }_n(D,\mathbf{x})=\prod _{k=1}^{n}d{\mu }_1({\alpha }_k,x^k)=c_n(D,\mathbf{x})\prod _{k=1}^{n}d{x}^k$$

(302)

and induces the NIDS metric tensor ${\mathcal{G}}_{kl}\left(\mathbf{x}\right)$ as a positive-definite metric tensor of the D-dimensional space W.

It should be emphasized that the metric ${\mathcal{G}}_{kl}\left(\mathbf{x}\right)$ is induced by the NIDS measures. Therefore, the NIDS metric (285) can be called the metric induced by the product of NIDS measures $d\mu ({\alpha }_k,x^k)$.

Note that we can define different types of metrics ${\mathcal{G}}_{kl}\left(\mathbf{x}\right)$ in the D-dimensional space $W=W_1\times \dots \times W_n$ by using the NIDS measure. For example, for the volume NIDS integration, we can consider the following two types of metric tensors:

$${\mathcal{G}}_{kl}\left(\mathbf{x}\right)=g_{kl}\left(\mathbf{x}\right):=c_1^2({\alpha }_k,x^k){\delta }_{kl},$$

(303)

$${\mathcal{G}}_{kl}\left(\mathbf{x}\right)={\widehat{G}}_{kl}^{n,n}\left(\mathbf{x}\right):=c_n^{2/n}(D,\mathbf{x}){\delta }_{kl}.$$

(304)

For these two metric tensors, the determinants have the form

$$det\left({\mathcal{G}}_{kl}\left(\mathbf{x}\right)\right)=c_n^2(D,\mathbf{x}).$$

(305)

As a result, part I of Principle 5 of induced metric consistency is satisfied for these two type of metric tensors.

Note that the metric of types (303) and (304) are used for the volume NIDS integrals in the metric and product measures approached, respectively. The metric of type (303) is also used for the line and surface NIDS integrals in the metric approach. However, the metric of type (304) cannot be used for the line and surface NIDS integrals in the product measure approach. The reason for this fact is that the correspondence principle for the volume, surface, and line NIDS integrals (Principle 4) is violated for the surface and line NIDS integrals in the product measure approach.

In the D-dimensional space, the natural NIDS metric $g_{kl}\left(\mathbf{x}\right)$, which allows us calculate volumes in the NIDS, cannot give the NIDS lengths and the NIDS areas that satisfy correspondence Principle 4 for the surface and line NIDS integrals. The NIDS length of the circle and the NIDS area of the semi-sphere in NIDS with the natural NIDS metric $g_{kl}\left(\mathbf{x}\right)$ cannot give the well-known D-dimensional expressions for non-integer values. Therefore, one can state that NIDS does not have a “universal” NIDS metric that satisfies correspondence Principle 4. Therefore, the NIDS cannot be considered a classical Riemannian manifold, if correspondence Principle 4 is satisfied. The Riemannian manifolds with the natural NIDS metric $g_{kl}\left(\mathbf{x}\right)$ violate correspondence Principle 4. However, there are equations of the NIDS integrals that satisfy the correspondence principles and allow us to calculate the lengths, areas, and volumes in NIDS to give well-known expressions. These equations will be presented in the next subsection in the framework of the product measure approach. In order to satisfy correspondence Principle 4, the concept of the Riemannian manifold should be generalized.

Let us describe the main differences between the metric and product measure approaches.

1.

In the metric approach, the NIDS can be considered as a classical Riemannian manifold with the natural NIDS metric $g_{kl}\left(\mathbf{x}\right)$, and the well-known equations calculated by the surface and line NIDS integrals with the natural NIDS metric (285) can be used. However, correspondence Principle 4 of the NIDS integration is satisfied only for the volume NIDS integral. Correspondence Principle 4 for the surface and line NIDS integrals is not satisfied. The surface and line NIDS integrations of the area of m-parametric surface and length of curved lines give non-standard expressions that violate correspondence Principle 4 for the surface and line NIDS integrals.

As a result, we can state the following. In the metric approach, correspondence Principle 4 is violated for the surface and line NIDS integrals. Therefore, the metric approach cannot give the well-known expression of the surface area of the D-dimensional semi-sphere

$$S_D^{\ast }\left({\Omega }_S\right)\ne {\displaystyle \frac{{\pi }^{D/2}}{\Gamma (D/2)}}{R}^{D-1},$$

(306)

for all $D\in (0,n]$, all $n⩾3$ ($n\in \mathbb{N}$), and all $R>0$ and cannot give the well-known expression of the D-dimensional circle length

$$L_D^{\ast }\left({\Omega }_C\right)\ne {\displaystyle \frac{2{\pi }^{D/2}}{\Gamma (D/2)}}{R}^{D-1},$$

(307)

for all $D\in (0,2]$ and all $R>0$, where $D={\alpha }_1+{\alpha }_2$.

2.

In the product measure approach, the NIDS and surfaces in the NIDS cannot be considered as a classical Riemannian manifold with a/universal NIDS metric. The condition that correspondence Principle 4 for the surface and line NIDS integrals must be satisfied leads to the absence of the “universal” metric, i.e., the metric ${\mathcal{G}}_{kl}\left(\mathbf{x}\right)$ that satisfies the principle of universal metric consistency (Principle 5). As a result, there is no metric that is universal for the line, surface, and volume NIDS integrals similar to the integer-dimensional space. This means that the principle of universal metric consistency (Principle 5) is violated in the product measure approach. In this case, the concept of the Riemannian manifolds should be generalized for the D-dimensional case. Such generalized (non-integer-dimensional) Riemannian manifolds can be described by the following set of NIDS metrics, which will be considered in the next subsections (see Section 2.9 and Section 2.10).

Definition 9 

(Metrics in the Product Measure Approach). Let $W=W_1\times \dots \times W_n$ be the D-dimensional space with $D\in (0,n]$ and $n\in \mathbb{N}$.

• In the product measure approach, the set of NIDS metrics

$$\{{\widehat{G}}_{ij}^{n,m}\left(\mathbf{u}\right),n,m\in \mathbb{N},1⩽m⩽n\},$$

(308)

where $\mathbf{x}=(x^1,\dots ,x^n)$, $\mathbf{u}=(u^1,\dots ,u^m)$, and the NIDS metrics ${\widehat{G}}_{ij}^{n,m}\left(\mathbf{u}\right)$ ($i,j=1,\dots ,m$) are defined by the equation

$${\widehat{G}}_{ij}^{n,m}\left(\mathbf{u}\right):=\sum _{k=1}^n{\widehat{c}}_{n,(k1,\dots ,km)}^{2/m}(D,\mathbf{x}\left(\mathbf{u}\right)){\displaystyle \frac{\partial x^k\left(\mathbf{u}\right)}{\partial u^i}}{\displaystyle \frac{\partial x^k\left(\mathbf{u}\right)}{\partial u^j}},$$

(309)

$$n,m\in \mathbb{N},1⩽m⩽n,kj\ne ki\mathit{if}j\ne i,$$

(310)

$${\widehat{c}}_{n,(k1,\dots ,km)}(D,\mathbf{x}\left(\mathbf{u}\right))=\left(\prod _{j=1}^m{c}_1({\alpha }_{kj},x^{kj})\left(\mathbf{u}\right)\right)\left(\prod _{j=k+1}^m{\widehat{c}}_1({\alpha }_{kj},x^{kj}\left(\mathbf{u}\right))\right)$$

(311)

$${\widehat{c}}_1({\alpha }_j,x^j\left(\mathbf{u}\right)):=\left\{\begin{array}{cc}1\hfill & {\displaystyle \mathit{if}\frac{\partial x^j\left(\mathbf{u}\right)}{\partial u^k}=0\mathit{for}\mathit{all}k=1,\dots ,m,}\hfill \\ c_1({\alpha }_j,x^j\left(\mathbf{u}\right))\hfill & {\displaystyle \mathit{if}\frac{\partial x^j\left(\mathbf{u}\right)}{\partial u^k}\ne 0\mathit{at}\mathit{least}\mathit{for}\mathit{one}k=1,\dots ,m,}\hfill \end{array}\right.$$

(312)

$$c_1({\alpha }_j,x^j\left(\mathbf{u}\right))={\displaystyle \frac{{\pi }^{{\alpha }_j/2}}{\Gamma ({\alpha }_j/2)}}|x^j\left(\mathbf{u}\right)|,$$

(313)

where $\mathbf{x}\left(\mathbf{u}\right)\in C^1\left(U\right)$ for the region U of the m-parametric space, and $\mathbf{u}=(u^1,\dots ,u^m)$. For $m=1$, Equation (309) gives the metric on the line in the NIDS. For $m=2,\dots ,n-1$, Equation (309) gives the metric on the m-parametric surface in the NIDS. For $m=n$, Equation (309) gives the metric in the n-parametric region in the NIDS.

• The justification for the type of Equation (309) will be given in the following subsections about the line and surface NIDS integrals of scalar fields. Metrics (309) can be used to calculate the NIDS lengths of curved lines and the NIDS areas of surfaces in D-dimensional spaces, which satisfy correspondence Principle 4 of the line and surface NIDS integrations. For the integer-dimensional spaces, $D=n$, ${\alpha }_k=1$, and $c_1({\alpha }_k,x^k)=1$, for all $k=1,\dots ,n$. In this case, we have ${\widehat{G}}_{ij}^{n,m}\left(\mathbf{u}\right)=G_{ij}\left(\mathbf{u}\right)$ with $m=2,\dots ,n-1$, where $G_{ij}\left(\mathbf{u}\right)$ is a standard metric induced on an m-parametric surface by the n-dimensional space.

Remark 26. 

As a result, the following conclusion can be made.

When constructing a definition of a metric satisfying Principle 4 (the correspondence principle for volume, surface, and line NIDS integrals) and Principle 5 (the principle of universal metric consistency), it becomes necessary to select only one of these principles.

In fact, it is possible to define either a metric satisfying Principle 4 and violating Principle 5 or a metric satisfying Principle 5 and violating Principle 4. As a result, we have two approaches that are called the metric approach and the product measure approach, respectively. Therefore, both approaches are described in this article.

Remark 27. 

The proposed term “the NIDS-induced metric” is caused by the fact that such metric tensors can be interpreted as metrics induced by D-dimensional space W in n-dimension space on an m-parametric surface and curved lines in W.

For $m=n$, the metric (309) should be used in the volume NIDS integrals in the space W. For $m\in [2,n-1]$, the metric (309) should be used in the surface NIDS integrals over an m-parametric surface in the space W. For $m=1$, the metric (309) should be used in the line NIDS integrals along the lines in the space W.

As will be shown below, in the product measure approach, the proposed NIDS-induced metrics define equations of the volume, surface, and line NIDS integrals that satisfy the correspondence principles for these types of the NIDS integrals.

The definition of the surface NIDS integral and the line NIDS integral that satisfy correspondence Principle 4 is proposed in the next subsection. In these subsections, we prove that the volume, surface, and line NIDS integrals satisfy the following principle.

Principle 6 

(Principle of Induced Metric Consistency in the Product Measure Approach). In the D-dimensional space, $W=W_1\times \dots \times W_n$ with $D\in (0,n]$, the NIDS-induced metrics $G_{ij}^{n,m}$ with $1⩽m⩽n$ and $n,m\in \mathbb{N}$ must be consistent with the NIDS measures.

• The metrics ${\widehat{G}}_{kl}^{n,1}\left(\mathbf{x}\left(t\right)\right)$ must define the length of curves in the region of the space W.
• The metrics ${\widehat{G}}_{ij}^{n,m}\left(\mathbf{u}\right)$ with integer values $m\in [2,n-1]$ must define the induced metrics on the m-parametric surfaces in the region of the space W.
• The metrics ${\widehat{G}}_{kl}^{n,n}\left(\mathbf{x}\right)$ must define the induced metrics in the region of the space W.

This means that the following three conditions must be satisfied, such that the correspondence principle for volume, surface, and line NIDS integrals (Principle 4) holds in the product measure approach.

(I) The length $L_D[a,b]$ of the line L, described by the equation $\mathbf{x}=\mathbf{x}\left(t\right)\in C^1[a,b]$ in the D-dimensional space $W=W_1\times \dots \times W_n$, should be defined by the metric ${\widehat{G}}_{kl}^{n,1}\left(\mathbf{x}\right)$, such that

$$L_D[a,b]={\int }_a^b\sqrt{\sum _{k,l=1}^n{\widehat{G}}_{kl}^{n,1}\left(\mathbf{x}\left(t\right)\right){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}{\displaystyle \frac{d{x}^l\left(t\right)}{dt}}}dt$$

(314)

is satisfied for all $n\in \mathbb{N}$.

(II) The area of the m-parametric surface Σ with integer values $m\in [2,n-1]$, which is described by equations $x^k=x^k(u^1,\dots ,u^m)$ ($k=1,\dots ,n$) for $\mathbf{u}\in U$, $2⩽m⩽n-1$ in the D-dimensional space $W=W_1\times \dots \times W_n$, should be defined by the metric ${\widehat{G}}_{kl}^{n,m}\left(\mathbf{x}\right)$, such that

$$S_D(\Sigma )={\int }_U\sqrt{det\left({\widehat{G}}_{ij}^{n,m}\left(\mathbf{u}\right)\right)}\prod _{j=1}^{n}d{u}^j(i,j=1,\dots ,m)$$

(315)

is satisfied for all $n,m\in \mathbb{N}$, where $2⩽m⩽n-1$.

(III) The volume $V_D(\Omega )$ of the bounded region Ω in the D-dimensional space $W=W_1\times \dots \times W_n$ should be defined by the metric ${\widehat{G}}_{kl}^{n,n}\left(\mathbf{x}\right)$, such that

$$V_D(\Omega )={\int }_{\Omega }\sqrt{det\left({\widehat{G}}_{kl}^{n,n}\left(\mathbf{x}\right)\right)}\prod _{k=1}^{n}d{x}^k$$

(316)

is satisfied for all $n\in \mathbb{N}$.

If the conditions of Principle 6 of induced metric consistency are satisfied, then there are the set of metrics

$$\{{\widehat{G}}_{kl}^{n,m}\left(\mathbf{x}\right),1⩽m⩽n,n,m\in \mathbb{N}\},$$

(317)

for which correspondence Principle 4 is satisfied for the line, surface, and volume NIDS integrals, and these integrals are described by the equations

$$I_L^{D}f\left(\mathbf{x}\right)={\int }_a^{b}f\left(\mathbf{x}\left(t\right)\right)\sqrt{\sum _{k,l=1}^n{\widehat{G}}_{kl}^{n,1}\left(\mathbf{x}\left(t\right)\right){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}{\displaystyle \frac{d{x}^l\left(t\right)}{dt}}}dt,$$

(318)

$$I_{\Sigma }^{D}f\left(\mathbf{x}\right)={\int }_{U}f\left(\mathbf{x}\left(\mathbf{u}\right)\right)\sqrt{det\left({\widehat{G}}_{ij}^{n,m}\left(\mathbf{u}\right)\right)}\prod _{j=1}^{n}d{u}^j(i,j=1,\dots ,m),$$

(319)

$$I_{\Omega }^{D}f\left(\mathbf{x}\right)={\int }_{\Omega }f\left(\mathbf{x}\right)\sqrt{det\left({\widehat{G}}_{kl}^{n,n}\left(\mathbf{x}\right)\right)}\prod _{k=1}^{n}d{x}^k,$$

(320)

for a wide class of functions.

If the metric is defined by Equation (309) from Definition 9, then the principle of induced metric consistency in the product measure approach (Principle 6) is satisfied. This fact will be proven in the following subsections on the line and surface NIDS integrals of scalar fields.

Remark 28. 

In the product measure approach, the NIDS cannot be considered as a classical Riemannian manifold with the natural NIDS metric, if the correspondence Principle 4 for the surface and line NIDS integrals are satisfied. In this case, the concept of the Riemannian manifolds should be generalized for the D-dimensional case. This NIDS generalization of the Riemannian manifolds can be described by the set of the metrics

$$\{{\widehat{G}}_{ij}^{n,m}\left(\mathbf{u}\right),n,m\in \mathbb{N},1⩽m⩽n\},$$

(321)

which satisfy Principle 6. The metrics ${\widehat{G}}_{ij}^{n,m}\left(\mathbf{u}\right)$ can be interpreted as the metric induced on the curved lines or on the m-parametric surfaces by the D-dimensional space. For the set of metrics (308), the principle of universal metric consistency (Principle 5) is violated,

Metrics ${\widehat{G}}_{ij}^{n,m}\left(\mathbf{u}\right)$ can be used to calculate the NIDS lengths of curved lines and the NIDS areas of surfaces in D-dimensional spaces, which satisfy correspondence Principle 6. For the case integer-dimensional spaces, when $D=n$ (if ${\alpha }_k=1$ for all $k=1,\dots ,n$), we have ${\widehat{G}}_{ij}^{n,m}\left(\mathbf{u}\right)=G_{ij}\left(\mathbf{u}\right)$.

2.9. Line NIDS Integrals of Scalar Function and NIDS Length

2.9.1. Line NIDS Integration in the Metric Approach

Let us consider the standard integer-dimensional (n-dimensional) case. It is known that a smooth manifold together with a positive-definite metric tensor is called the Riemannian manifold [147]. In classical n-dimensional differential geometry [147], the metric tensor $G_{kl}\left(\mathbf{x}\right)$ specifies the infinitesimal distance on the n-dimensional Riemannian manifold by the equation

$${\left(d{L}_n^{\ast }\right)}^2=\sum _{k,l=1}^n{G}_{kl}\left(\mathbf{x}\right)d{x}^{k}d{x}^l,$$

(322)

where $G_{kl}\left(\mathbf{x}\right)$ is positive-definite. It is known that, on an n-dimensional Riemannian manifold, the length $L_n^{\ast }[a,b]$ of a smooth curve $\mathbf{x}=\mathbf{x}\left(t\right)$ with $t\in [a,b]$ between two points $\mathbf{x}\left(a\right)$ and $\mathbf{x}\left(b\right)$ is defined by the line integral

$$L_n^{\ast }[a,b]={\int }_a^b\sqrt{\sum _{k,l=1}^n{G}_{kl}\left(\mathbf{x}\left(t\right)\right){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}{\displaystyle \frac{d{x}^k\left(t\right)}{dt}}}.$$

(323)

The Riemannian manifold is a metric space, if the distance between the points $\mathbf{x}\left(a\right)$ and $\mathbf{x}\left(b\right)$ is defined as the infimum of the lengths of all such curves.

Let us assume that the infinitesimal distance on the D-dimensional space $W=W_1\times \dots \times W_n$ with $D={\alpha }_1+\dots +{\alpha }_n\in (0,n]$ is defined by the natural NIDS metric $g_{kl}\left(\mathbf{x}\right)$ in the form

$${\left(d{L}_D^{\ast }\right)}^2=\sum _{k=1}^n\sum _{l=1}^n{g}_{kl}\left(\mathbf{x}\right)d{x}^{k}d{x}^l=\sum _{k=1}^n{g}_{kk}\left(\mathbf{x}\right){\left(d{x}^k\right)}^2=\sum _{k=1}^n{c}_1^2({\alpha }_k,x^k){\left(d{x}^k\right)}^2,$$

(324)

where

$$g_{kl}\left(\mathbf{x}\right)=c_1^2({\alpha }_k,x_k){\delta }_{kl}={\left({\displaystyle \frac{{\pi }^{{\alpha }_k/2}}{\Gamma ({\alpha }_k/2)}}{|x^k|}^{{\alpha }_k-1}\right)}^2{\delta }_{kl}.$$

(325)

Then, it is possible to define the line NIDS integral in the metric approach.

Definition 10. 

Let L be a continuously differentiable curve

$$L:=\{\mathbf{x}:\mathbf{x}=\mathbf{x}\left(t\right)\in C^1[a,b],t\in [a,b]\}$$

(326)

in the D-dimensional space $W=W_1\times \dots \times W_n$ with $D={\alpha }_1+\dots +{\alpha }_n$, where all ${\alpha }_k\in (0,1]$.

Then, in the metric approach, the line NIDS integral of the function $f\left(\mathbf{x}\right(t\left)\right)\in C[a,b]$ is defined as

$$I_L^{D,\ast }f\left(\mathbf{x}\right):={\int }_a^{b}f\left(\mathbf{x}\left(t\right)\right)\sqrt{\sum _{k=1}^n{g}_{kk}\left(\mathbf{x}\left(t\right)\right){\left({\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt,$$

(327)

for all $n\in \mathbb{N}$, where $g_{kl}\left(\mathbf{x}\right)$ is the natural NIDS metric defined by Equation (325).

In this case, the NIDS length $L_D^{\ast }[A,B]$ of a continuously differentiable curve (326) is defined as

$$L_D^{\ast }[A,B]:={\int }_a^b\sqrt{\sum _{k=1}^n{g}_{kk}\left(\mathbf{x}\left(t\right)\right){\left({\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt,$$

(328)

for all $n\in \mathbb{N}$, where A and B are the starting and ending points of the line with coordinates ${\mathbf{x}}_A=\mathbf{x}\left(a\right)$ and ${\mathbf{x}}_B=\mathbf{x}\left(b\right)$, respectively.

Since the metric must specify the length of the curve, an obvious question arises. Can the natural NIDS metric $g_{kl}\left(\mathbf{x}\right)$, which is defined by Equation (325), be used to calculate the lengths of curves by Equation (328) and satisfy the correspondence principle for the line NIDS integration?

Let us first consider the case $n=1$ and the length of the line in the D-dimensional space $W=W_1$ with $D=\alpha \in (0,1]$.

Example 4. 

Let A and B be points of the D-dimensional space $W=W_1$ with $D=\alpha \in (0,1]$, where coordinates of points A and B will be designated as $x_A=x\left(a\right)$ and $x_B=x\left(b\right)$, respectively. Then, the length of the segment $[A,B]$ on the coordinate axis X is given by the equation

$$L_D^{\ast }[A,B]:={\int }_a^b\sqrt{{\left(c_1(\alpha ,x\left(t\right)){\displaystyle \frac{dx\left(t\right)}{dt}}\right)}^2}dt=c\left(\alpha \right){\int }_a^b{|x\left(t\right)|}^{\alpha -1}|x_t^{\left(1\right)}\left(t\right)|dt,$$

(329)

where $c\left(\alpha \right)={\pi }^{\alpha /2}/\Gamma (\alpha /2)$. The line segment $[A,B]$ in the space W can be described as

$$x\left(t\right)=(x_B-x_A)t+x_A(0⩽t⩽1),$$

(330)

where $x\left(0\right)=x_A$, $x\left(1\right)=x_B$, and $a=0$, $b=1$. Then, $|x_t^{\left(1\right)}\left(t\right)=|x_B-x_A|$, and Equation (329) has the form

$$L_D^{\ast }[A,B]:=c\left(\alpha \right)|x_B-x_A|{\int }_0^1{|(x_B-x_A)t+x_A|}^{\alpha -1}dt.$$

(331)

Changing the variable $x=(x_B-x_A)t+x_A$, Equation (331) gives

$$L_D^{\ast }[A,B]:=c\left(\alpha \right){\displaystyle \frac{|x_B-x_A|}{x_B-x_A}}{\int }_{x_A}^{x_B}{|x|}^{\alpha -1}dx=c\left(\alpha \right)sgn(x_B-x_A){\int }_{x_A}^{x_B}{|x|}^{\alpha -1}dx,$$

(332)

where $sgn\left(x\right)$ is the sign function ($sgn\left(x\right)=1$ if $x>0$, $sgn\left(x\right)=-1$ if $x<0$, and $sgn\left(x\right)=0$ if $x=0$). The integral of Equation (332) can be written as

$${\int }_{x_A}^{x_B}{|x|}^{\alpha -1}dx={\int }_{sgn\left(x_A\right)|x_A|}^{sgn\left(x_B\right)|x_B|}{|x|}^{\alpha -1}dx=$$

$${\int }_{sgn\left(x_A\right)|x_A|}^{sgn\left(x_B\right)|x_B|}{|x|}^{\alpha -1}dx={\int }_0^{sgn\left(x_B\right)|x_B|}{|x|}^{\alpha -1}dx+{\int }_{sgn\left(x_A\right)|x_A|}^0{|x|}^{\alpha -1}dx=$$

$$sgn\left(x_B\right){\int }_0^{sgn\left(x_B\right)|x_B|}{|x|}^{\alpha -1}d|x|+sgn\left(x_A\right){\int }_{sgn\left(x_A\right)|x_A|}^0{|x|}^{\alpha -1}d|x|=$$

$$sgn\left(x_B\right){\int }_0^{sgn\left(x_B\right)|x_B|}{\xi }^{\alpha -1}d\xi -sgn\left(x_A\right){\int }^{sgn{\left(x_A\right)}_0|x_A|}{\xi }^{\alpha -1}d\xi =$$

$$sgn\left(x_B\right){\displaystyle \frac{|x_B{|}^{\alpha }}{\alpha }}-sgn\left(x_A\right){\displaystyle \frac{|x_A{|}^{\alpha }}{\alpha }}.$$

(333)

Substitution of Equation (333) into Equation (332) gives the length of the segment $AB$ in the D-dimensional space with $D=\alpha \in (0,1]$ in the form

$$L_D^{\ast }[A,B]:={\displaystyle \frac{{\pi }^{\alpha /2}}{\alpha \Gamma (\alpha /2)}}sgn(x_B-x_A)\left(sgn\left(x_B\right)|x_B{|}^{\alpha }-sgn\left(x_A\right){|x_A|}^{\alpha }\right).$$

(334)

Let us give some special cases of Equation (334).

• If $x_B>x_A>0$, Equation (334) gives

$$L_D^{\ast }[A,B]:={\displaystyle \frac{{\pi }^{\alpha /2}}{\alpha \Gamma (\alpha /2)}}\left(x_B^{\alpha }-x_A^{\alpha }\right).$$

(335)

If $x_B>0>x_A$, Equation (334) is

$$L_D^{\ast }[A,B]:={\displaystyle \frac{{\pi }^{\alpha /2}}{\alpha \Gamma (\alpha /2)}}\left(x_B^{\alpha }+{|x_A|}^{\alpha }\right).$$

(336)

If $0>x_B>x_A$, Equation (334) has the form

$$L_D^{\ast }[A,B]:={\displaystyle \frac{{\pi }^{\alpha /2}}{\alpha \Gamma (\alpha /2)}}\left(|x_A{|}^{\alpha }-{|x_B|}^{\alpha }\right).$$

(337)

If $x_A=-R$, and $x_B=R$, Equation (334) gives

$$L_{\alpha }^{\ast }[A,B]={\displaystyle \frac{2{\pi }^{\alpha /2}}{\alpha \Gamma (\alpha /2)}}{R}^{\alpha },$$

(338)

which is interpreted as the volume of the D-dimensional ball for $D=\alpha \in (0,1]$.

Note that the D-dimensional volume of the parallelepiped region

$${\Omega }_P:=\{\mathbf{x}:0⩽x_k⩽l_k,k=1,\dots ,n\}$$

(339)

in the D-dimensional space $W:=W_1\times \dots \times W_n$ is described by the equation

$$V_D\left({\Omega }_P\right)={\int }_{{\Omega }_P}\prod _{k=1}^n{c}_1({\alpha }_k,x_k)d{x}^k=\prod _{k=1}^n{L}_{{\alpha }_k}[A_k,B_k]=P_n({\alpha }_1,\dots ,{\alpha }_n)\prod _{k=1}^n{\left(l_k\right)}^{{\alpha }_k},$$

(340)

where $[A_k,B_k]$ is a segment along the axis $X_k$, when the point $A_k$ has all coordinates equal to zero $x_k\left(a\right)=0$, and the point $B_k$ has coordinates $x_k\left(b\right)=l_k$ and $x_j\left(b\right)=0$ for all $j\ne k$. The function $P_n({\alpha }_1,\dots ,{\alpha }_n)$ is

$$P_n({\alpha }_1,\dots ,{\alpha }_n)=\prod _{k=1}^n{\displaystyle \frac{{\pi }^{{\alpha }_k/2}}{{\alpha }_k\Gamma ({\alpha }_k/2)}},$$

(341)

and all ${\alpha }_k\in (0,1]$. If all ${\alpha }_k=1$, then $D=n$, $P_n({\alpha }_1,\dots ,{\alpha }_n)=1$, and $V_D\left({\Omega }_P\right)=V_n\left({\Omega }_P\right)={\prod }_{k=1}^n{l}_k$.

Let us next consider the case $n=2$, for the D-dimensional space $W=W_1\times W_2$ with dimension $D={\alpha }_1+{\alpha }_2\in (0,2]$. We are interested in the following question. Does the natural NIDS metric (285) give the well-known expression for the length of a circle in D-dimensional space $W=W_1\times W_2$ with $D={\alpha }_1+{\alpha }_2$ for all $D\in (0,2)$? Unfortunately, in D-dimensional space, the metric (285) cannot give the well-known expression for the NIDS length of a circle for all $D\in (0,2]$.

Theorem 8. 

Let $W=W_1\times W_2$ be a D-dimensional space with $D={\alpha }_1+{\alpha }_2\in (0,2]$.

The NIDS length, which is defined by Equation (328) with $n=2$, does not satisfy the second correspondence principle for line NIDS integrals, since Equation (328) of the length $L_D^{\ast }\left({\Omega }_C\right)$ of the circle

$${\Omega }_C:=\{\mathbf{x}:|\mathbf{x}|=R\}$$

(342)

in the D-dimensional space $W=W_1\times W_2$ gives the inequality

$$L_D^{\ast }\left({\Omega }_C\right)\ne {\displaystyle \frac{2{\pi }^{D/2}}{\Gamma (D/2)}}{R}^{D-1},$$

(343)

if $0<D<2$, and $R>0$.

Proof. 

Let us consider the circle (342) that is described by the equation

$$\sum _{k=1}^2{\left(x^k\right)}^2=R^2.$$

(344)

For $D\in (0,n]$ with $n=2$, the length of the continuously differentiable curve $\mathbf{x}=\mathbf{x}\left(t\right)$ is calculated by the equation

$$L_D^{\ast }\left({\Omega }_C\right):={\int }_a^b\sqrt{\sum _{k=1}^2{g}_{kk}\left(\mathbf{x}\left(t\right)\right){\left({\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt.$$

(345)

Equation (345) with metric (285) has the form

$$L_D^{\ast }\left({\Omega }_C\right):={\int }_a^b\sqrt{\sum _{k=1}^2{c}_1^2({\alpha }_k,x_k\left(t\right)){\left({\displaystyle \frac{d{x}_k\left(t\right)}{dt}}\right)}^2}dt=$$

$${\int }_a^b\sqrt{\sum _{k=1}^2{\displaystyle \frac{{\pi }^{{\alpha }_k}}{{\Gamma }^2({\alpha }_k/2)}}{|x_k|}^{2{\alpha }_k-2}{\left({\displaystyle \frac{d{x}_k\left(t\right)}{dt}}\right)}^2}dt.$$

(346)

The equation $\mathbf{x}=\mathbf{x}\left(t\right)$ of the circle (344) can be written in the form

$$x_1\left(t\right)=Rcos\left(t\right),x_2\left(t\right)=Rsin\left(t\right),$$

(347)

where $t\in [0,2\pi )$, and $R=const$, $R>0$. Then, Equation (346) of the length $L_D^{\ast }\left({\Omega }_C\right)$ of the NIDS circle with $D={\alpha }_1+{\alpha }_2$, where ${\alpha }_1,{\alpha }_2\in (0,1]$, gives

$$L_D^{\ast }\left({\Omega }_C\right):=4{\int }_0^{\pi /2}\sqrt{{\displaystyle \frac{{\pi }^{{\alpha }_1}{R}^{2{\alpha }_1}}{{\Gamma }^2({\alpha }_1/2)}}{sin}^2\left(t\right){cos}^{2{\alpha }_1-2}\left(t\right)+{\displaystyle \frac{{\pi }^{{\alpha }_2}{R}^{2{\alpha }_2}}{{\Gamma }^2({\alpha }_2/2)}}{cos}^2\left(t\right){sin}^{2{\alpha }_2-2}\left(t\right)}dt.$$

(348)

For the case ${\alpha }_1={\alpha }_2=\alpha \in (0,1]$, we have $D=2\alpha$. Then, Equation (348) gives

$$L_D^{\ast }\left({\Omega }_C\right):=K\left(\alpha \right)R^{\alpha },$$

(349)

where

$$K\left(\alpha \right):={\displaystyle \frac{4{\pi }^{\alpha /2}}{\Gamma (\alpha /2)}}{\int }_0^{\pi /2}\sqrt{{sin}^2\left(t\right){cos}^{2\alpha -2}\left(t\right)+{cos}^2\left(t\right){sin}^{2\alpha -2}\left(t\right)}dt.$$

(350)

Using the well-known equation

$$L_D\left({\Omega }_C\right)={\displaystyle \frac{2{\pi }^{D/2}}{\Gamma (D/2)}}{R}^{D-1}={\displaystyle \frac{2{\pi }^{\alpha }}{\Gamma \left(\alpha \right)}}{R}^{2\alpha -1}$$

(351)

for $D=2\alpha$ and $\alpha \in (0,1]$ and Equation (349), we can get

$$L_D^{\ast }\left({\Omega }_C\right)\ne L_D\left({\Omega }_C\right),$$

(352)

for $\alpha \in (0,1)$ if $R>0$, since $R^{\alpha }=R^{2\alpha -1}$ only if $\alpha =1$.

As a result, Equation (345) of the length $L_D^{\ast }(\Omega )$ of the circle ${\Omega }_C:=\{\mathbf{x}:|\mathbf{x}|=R\}$ in the D-dimensional space $W=W_1\times W_2$ gives an expression that differs from the well-known expression for the length of the D-dimensional circle

$$L_D^{\ast }\left({\Omega }_C\right)\ne {\displaystyle \frac{2{\pi }^{D/2}}{\Gamma (D/2)}}{R}^{D-1}$$

(353)

in the metric approach, if $0<D<2$, and $R>0$. □

Theorem 8 states that the correspondence principle for line NIDS integration is violated for the equations of the length $L_D^{\ast }\left({\Omega }_C\right)$, with the natural NIDS metric $g_{kl}\left(\mathbf{x}\right)$. As a result, Equation (328) with metric (285) satisfies the principle of universal metric consistency, but the correspondence principle for line NIDS integrals is violated. Note that the correspondence principle for line NIDS integrals is satisfied in the product measure approach. This fact will be proved in the next subsection.

2.9.2. Line NIDS Integration in the Product Measure Approach

To define the line NIDS integral and the NIDS length of line in the framework of the product measure approach, simplifying notations should be defined.

Definition 11. 

Let L be a continuously differentiable curve that is defined as

$$L:=\{\mathbf{x}:\mathbf{x}=\mathbf{x}\left(t\right)\in c^1[a,b],a⩽t⩽b\},$$

(354)

in the D-dimensional space $W=W_1\times \dots \times W_n$ with $D={\alpha }_1+\dots +{\alpha }_n\in (0,n]$, and $n\in \mathbb{N}$, and $\mathbf{x}\left(t\right)=(x_1\left(t\right),\dots ,x_n\left(t\right))$ are points of W for all $t\in [a,b]$.

Then, the functions ${\widehat{c}}_n(D,\mathbf{x}\left(t\right))$ and ${\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right))$ are defined as

$${\widehat{c}}_n(D,\mathbf{x}\left(t\right)):=\prod _{j=1}^n{\widehat{c}}_1({\alpha }_j,x^j\left(t\right)),$$

(355)

$${\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)):=c_1({\alpha }_k,x^k\left(t\right))\prod _{j=1,j\ne k}^n{\widehat{c}}_1({\alpha }_j,x^j\left(t\right)),$$

(356)

where

$${\widehat{c}}_1({\alpha }_j,x^j\left(t\right)):=\left\{\begin{array}{cc}1\hfill & {\displaystyle \mathit{if}d{x}^j\left(t\right)/dt=0,}\hfill \\ c_1({\alpha }_j,x^j\left(t\right))\hfill & {\displaystyle \mathit{if}d{x}^j\left(t\right)/dt\ne 0,}\hfill \end{array}\right.$$

(357)

and

$$c_1({\alpha }_j,x^j\left(t\right))={\displaystyle \frac{{\pi }^{{\alpha }_j/2}}{\Gamma ({\alpha }_j/2)}}{|x^j\left(t\right)|}^{{\alpha }_j-1},{\alpha }_j\in (0,1].$$

(358)

This notation means that, if a finite segment of the line L is parallel to the coordinate axis $X_k$, then the functions $c_1({\alpha }_j,x^j\left(t\right))$ with $j\ne k$ are equal to one on this region. Similarly, if the finite segment of the line L is parallel to the coordinate plane $X_k{X}_l$, then the functions $c_1({\alpha }_j,x^j\left(t\right))$ with $j\ne k,l$ are equal to one, and so on. In other words, if the function ${\widehat{c}}_n(D,\mathbf{x}\left(t\right))$ does not change along any axis in some line segment, then this function does not depend on the corresponding coordinates.

Remark 29. 

In fact, the functions ${\widehat{c}}_n(D,\mathbf{x}\left(t\right))$ and ${\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right))$ have the natural property of the line NIDS integral and the NIDS length. If, on a segment of the line L, the coordinate $x^j$ does not change when the parameter $t\in [a,b]$ changes, then the value of the dimension ${\alpha }_j$ should not affect the length of this segment. For example, if the line L is parallel to the coordinate axis $X_k$, then dimensions ${\alpha }_j$ with $j\ne k$ of the D-dimensional space do not affect the value of the NIDS integral and the NIDS length of this segment of the line L. Similarly, if a segment of the line L is parallel to the coordinate plane $X_k{X}_l$, then the dimensions ${\alpha }_j$ with $j\ne k,l$ do not affect the NIDS integral and the NIDS length of this segment of L.

Let us define the line NIDS integral and the length of lines in NIDS in the framework of the product measure approach.

Definition 12. 

Let L be a continuously differentiable curve that is defined as

$$L:=\{\mathbf{x}:\mathbf{x}=\mathbf{x}(t),a⩽t⩽b\}$$

(359)

in the D-dimensional space $W=W_1\times \dots \times W_n$ with $D\in (0,n]$, $n\in \mathbb{N}$, and $\mathbf{x}\left(t\right)=(x_1\left(t\right),\dots ,x_n\left(t\right))$ are points of the NIDS for all $t\in [a,b]$.

Then, the line NIDS integral of function $f\left(\mathbf{x}\right(t\left)\right)\in C[a,b]$ along the curve L is given as

$$I_L^{D}f\left(\mathbf{x}\right):={\int }_a^{b}f\left(\mathbf{x}\left(t\right)\right){\widehat{c}}_n(D,\mathbf{x}\left(t\right))\sqrt{\sum _{k=1}^n{\left({\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt,$$

(360)

or

$$I_L^{D}f\left(\mathbf{x}\right):={\int }_a^{b}f\left(\mathbf{x}\left(t\right)\right)\sqrt{\sum _{k=1}^n{\left({\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt,$$

(361)

where ${\widehat{c}}_n$ and ${\widehat{c}}_{n,l}$ are defined by Equations (355) and (356), and the points $\mathbf{x}\left(a\right)$ and $\mathbf{x}\left(b\right)$ are the endpoints of the line L in the NIDS.

As a special case of Equations (360) and (361), the NIDS length $L_D[a,b]$ of the line L is given as the NIDS integral

$$L_D[a,b]:={\int }_a^b{\widehat{c}}_n(D,\mathbf{x}\left(t\right))\sqrt{\sum _{k=1}^n{\left({\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt,$$

(362)

or

$$L_D[a,b]:={\int }_a^b\sqrt{\sum _{k=1}^n{\left({\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt,$$

(363)

for all $D\in (0,n]$, and $n\in \mathbb{N}$ in the D-dimensional space $W=W_1\times \dots \times W_n$.

Remark 30. 

Let us note that Equations (360) and (361) are equivalent equations by Definition 11. Let us prove this fact. Since the equation

$${\widehat{c}}_1({\alpha }_k,x^k\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}=c_1({\alpha }_k,x^k\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}$$

(364)

holds by definition of the function ${\widehat{c}}_1({\alpha }_k,x^k\left(t\right))$ in Equation (357), we get

$${\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}={\widehat{c}}_n(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}.$$

(365)

As a result,

$$I_L^{D}f\left(\mathbf{x}\right):={\int }_a^{b}f\left(\mathbf{x}\left(t\right)\right){\widehat{c}}_n(D,\mathbf{x}\left(t\right))\sqrt{\sum _{k=1}^n{\left({\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt=$$

$$\sqrt{\sum _{k=1}^n{\left({\widehat{c}}_n(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt=$$

$${\int }_a^{b}f\left(\mathbf{x}\left(t\right)\right)\sqrt{\sum _{k=1}^n{\left({\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt,$$

(366)

for all $D\in (0,n]$.

Equations (362) and (363) describe equations for calculating the NIDS length of curved lines in the product measure approach.

For the case $n=1$, Equation (362) gives the same result as the equation with the natural metric in Example 4. Therefore, the length of the segment $AB$ in the D-dimensional space with $D=\alpha \in (0,1]$ in the metric and product measure approaches is

$$L_D[A,B]:={\displaystyle \frac{{\pi }^{\alpha /2}}{\alpha \Gamma (\alpha /2)}}sgn(x_B-x_A)\left(sgn\left(x_B\right)|x_B{|}^{\alpha }-sgn\left(x_A\right){|x_A|}^{\alpha }\right),$$

(367)

where $x_A=x\left(a\right)$, and $x_B=x\left(b\right)$. Let us emphasize that Equations (329) and (334) are true both for the metric approach and for the product approach, i.e., $L_{\alpha }^{\ast }[A,B]=L_{\alpha }[A,B]$.

For the case $n=2$, Equation (362) gives a different result than the equation with the natural metric in the metric approach. Let us prove that Definition 12 gives the equations of the NIDS length of the curve in the D-dimensional space, for which the correspondence principle of the line NIDS integrals is satisfied.

Theorem 9. 

Let $W=W_1\times W_2$ be the D-dimensional space with $D={\alpha }_1+{\alpha }_2$, where all ${\alpha }_1,{\alpha }_2\in (0,1]$, and let ${\Omega }_C=\{\mathbf{x}:|\mathbf{x}|=R\}$ be the circle with radius $R>0$ in W.

Then, the NIDS length of the circle ${\Omega }_C$ can be calculated by the equation

$$L_D\left({\Omega }_C\right):={\int }_a^b{\widehat{c}}_2(D,\mathbf{x}\left(t\right))\sqrt{\sum _{k=1}^2{\left({\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt,$$

(368)

where ${\widehat{c}}_2(D,\mathbf{x}\left(t\right))=c_2(D,\mathbf{x}\left(t\right))$, and Equation (368) gives

$$L_D\left({\Omega }_C\right)={\displaystyle \frac{2{\pi }^{D/2}}{\Gamma (D/2)}}{R}^{D-1},$$

(369)

for all $R>0$ and all ${\alpha }_1,{\alpha }_2\in (0,1]$, where $D={\alpha }_1+{\alpha }_2\in (0,2]$.

Proof. 

Using the equation $\mathbf{x}=\mathbf{x}\left(t\right)$ of the circle in the form

$$x_1=Rcost,x_2=Rsint,$$

(370)

where $t\in [0,2\pi )$, we get

$$c_2(D,\mathbf{x}\left(t\right))=c_1({\alpha }_1,x_1\left(t\right))c_1({\alpha }_2,x_2\left(t\right))=$$

$${\displaystyle \frac{{\pi }^{({\alpha }_1+{\alpha }_2)/2}}{\Gamma ({\alpha }_1/2)\Gamma ({\alpha }_2/2)}}|x_1{\left(t\right)|}^{{\alpha }_1-1}{|x_2\left(t\right)|}^{{\alpha }_2-1}=$$

$${\displaystyle \frac{{\pi }^{({\alpha }_1+{\alpha }_2)/2}}{\Gamma ({\alpha }_1/2)\Gamma ({\alpha }_2/2)}}{R}^{{\alpha }_1+{\alpha }_2-2}{|cost|}^{{\alpha }_1-1}{|sint|}^{{\alpha }_2-1},$$

(371)

and

$$\sqrt{\sum _{k=1}^2{\left({\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}=\sqrt{{\left(Rsint\right)}^2+{\left(Rcost\right)}^2}=R.$$

(372)

Then, the NIDS length of the circle is

$$L_D[0,2\pi ]:={\int }_0^{2\pi }{c}_2(D,\mathbf{x}\left(t\right))\sqrt{\sum _{k=1}^2{\left({\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt=$$

$${\displaystyle \frac{{\pi }^{({\alpha }_1+{\alpha }_2)/2}}{\Gamma ({\alpha }_1/2)\Gamma ({\alpha }_2/2)}}{R}^{{\alpha }_1+{\alpha }_2-1}{\int }_0^{2\pi }{|cost|}^{{\alpha }_1-1}{|sint|}^{{\alpha }_2-1}dt.$$

(373)

Using the equation

$${\int }_0^{2\pi }{|cost|}^{{\alpha }_1-1}{|sint|}^{{\alpha }_2-1}dt=4{\int }_0^{\pi /2}{(cost)}^{{\alpha }_1-1}{(sint)}^{{\alpha }_2-1}dt={\displaystyle \frac{2\Gamma ({\alpha }_1/2)\Gamma ({\alpha }_2/2)}{\Gamma ({\alpha }_1/2+{\alpha }_2/2)}},$$

(374)

we get the NIDS circle length

$$L_D[0,2\pi ]:={\displaystyle \frac{2{\pi }^{({\alpha }_1+{\alpha }_2)/2}}{\Gamma ({\alpha }_1/2+{\alpha }_2/2)}}{R}^{{\alpha }_1+{\alpha }_2-1}={\displaystyle \frac{2{\pi }^{D/2}}{\Gamma (D/2)}}{R}^{D-1},$$

(375)

for all $R>0$ and all ${\alpha }_1,{\alpha }_2\in (0,1]$, where $D={\alpha }_1+{\alpha }_2\in (0,2]$. □

As a result, Theorem 9 proves that, for the line NIDS integral defined by Equation (368), the the NIDS length in the product measure approach gives the classical expression for the D-dimensional length of the curved line.

Remark 31. 

Since ${\alpha }_1,{\alpha }_2\in (0,1]$, the dimension $D={\alpha }_1+{\alpha }_2\in (0,2]$, then, in the equation

$$L_D[0,2\pi ]:={\displaystyle \frac{2{\pi }^{D/2}}{\Gamma (D/2)}}{R}^{D-1},$$

(376)

we have $d=D-1<0$, if $D\in (0,1)$. In the case $D\in (0,1)$, the NIDS circle length exhibits an unusual behavior in which the circle length decreases as the radius increases.

Note also that in quantum field theory, the negative value of the dimension of NIDS is also considered (for example, see Section 4.2 in [10], pp. 68–71).

In a general case [19], we can consider the NIDS with the dimension D of the ball region $B_D$ and the dimension d of its boundary $S_d=\partial B_D$ that are not related by the equation $d=D-1$, i.e.,

$$dim(\partial B_D)\ne dim\left(B_D\right)-1,$$

(377)

where $dim\left(B_D\right)=D$. In such a generalized NIDS, one can assume that the dimension of the boundary $S_d=\partial B_D$ is

$$dim\left(S_d\right)=d,$$

(378)

and the parameter

$${\alpha }_r=D-d$$

(379)

is interpreted as a dimension along the radial direction [19]. Such generalizations of the NIDS are not considered in the following sections.

Remark 32. 

Equation (362) can be represented as

$$L_D[a,b]={\int }_a^b\sqrt{\sum _{k,l=1}^n{\widehat{G}}_{kl}^{n,1}\left(\mathbf{x}\left(t\right)\right){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}{\displaystyle \frac{d{x}^l\left(t\right)}{dt}}}dt,$$

(380)

where ${\widehat{G}}_{kl}^{n,1}\left(\mathbf{x}\right)$ is a special case of the metric ${\widehat{G}}_{kl}^{n,m}\left(\mathbf{x}\right)$ with $m=1$, such that

$${\widehat{G}}_{kl}^{n,1}\left(\mathbf{x}\left(t\right)\right)={\widehat{c}}_n^2(D,\mathbf{x}\left(t\right)){\delta }_{kl}.$$

(381)

The function (381) can be interpreted as the metric induced by the measure product on the line L in the space $W=W_1\times \dots \times W_n$.

It should be emphasized that the metric (381) is not applicable to the volume NIDS integrals, since the Wilson normalization condition and the correspondence principle for volume NIDS integrals are not satisfied for it. For the volume NIDS integrals, we must use the metric ${\widehat{G}}_{kl}^{n,n}\left(\mathbf{x}\right)$ that is defined by ${\widehat{c}}_n^{2/n}(D,\mathbf{x})$ instead of ${\widehat{c}}_n^2(D,\mathbf{x})$. The correspondence principle for surface NIDS integrals is also not satisfied for the metric (381). For the surface NIDS integrals over an m-parametric surface, we must use the metric ${\widehat{G}}_{kl}^{n,m}\left(\mathbf{x}\right)$ that is defined by ${\widehat{c}}_n^{2/m}(D,\mathbf{x})$.

As shown in this subsection, the proposed NIDS-induced metric (381) defines equations of the line NIDS integrals that satisfy the correspondence principles for the line NIDS integrals.

In the product measure approach, the NIDS-induced metrics ${\widehat{G}}_{kl}^{n,m}$ with $m=1,\dots ,n$ for line surface and volume regions in the D-dimensional space $W=W_1\times \dots \times W_n$ with $D\in (0,n]$ must be consistent with the NIDS measures. Principle 6 states that the conditions for the line, surface, and volume NIDS integrals must be satisfied in the product measure approach. The first condition of principle 6 states that the metrics ${\widehat{G}}_{kl}^{n,1}\left(\mathbf{x}\left(t\right)\right)$, given by Equation (381), must define the length of curves in the space W. The length $L_D[a,b]$ of the line L, which is described by equation $\mathbf{x}=\mathbf{x}\left(t\right)\in C^1[a,b]$ in D-dimensional space $W=W_1\times \dots \times W_n$, should be defined by the metric (381), such that Equation (380) is satisfied for all $n\in \mathbb{N}$.

2.9.3. Physical Interpretation of the NIDS Length

Let us give a physical interpretation of the NIDS length $L_D^{\ast }$ and $L_D$ via the notion of the optical length in the metric and product measure approaches. In physics, the optical length (or optical path length) is the measure of the path that light (an electromagnetic wave) takes through a medium [136,137]. It is not a geometric path length, since this value takes into account how the medium affects the speed at which light takes this path.

The concept of the optical path length is based on the refractive index (see Sections 3.1.2 and 3.3 in [136], pp. 122, 135–141). In a linear anisotropic dielectric medium, the component of the electric flux density $\mathbf{D}\left(\mathbf{x}\right)$ is a linear combination of the components of the electric field $\mathbf{E}\left(\mathbf{x}\right)$, as $D_k\left(\mathbf{x}\right)={\epsilon }_{kl}\left(\mathbf{x}\right)E_l\left(\mathbf{x}\right)$. The permittivity tensor has the form ${\epsilon }_{kl}\left(\mathbf{x}\right)={\epsilon }_{kk}\left(\mathbf{x}\right){\delta }_{kl}$ corresponding to the refractive indexes $n_k\left(\mathbf{x}\right)=\sqrt{{\epsilon }_{kk}\left(\mathbf{x}\right)/{\epsilon }_0}$, which are called the principal refractive indexes, where ${\epsilon }_0$ is the permittivity of free space. In such optically anisotropic medium, the optical path length is described by the equation

$$L_{OPL}[a,b]={\int }_a^b\sqrt{\sum _{k=1}^n{n}_k^2\left(\mathbf{x}\left(t\right)\right){\left({\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt,$$

(382)

if the continuously differentiable path is defined as $\mathbf{x}=\mathbf{x}\left(t\right)$ with $a⩽t⩽b$. Note that Equation (382) can be written as

$$L_{OPL}[a,b]={\int }_a^b\sqrt{{\displaystyle \frac{1}{{\epsilon }_0}}\sum _{k=1}^n{\epsilon }_{kk}\left(\mathbf{x}\left(t\right)\right){\left({\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt.$$

(383)

As a result, the NIDS lengths $L_D^{\ast }$ and $L_D$ in the D-dimensional space can be interpreted as the optical path length $L_{OPL}[a,b]$ in optically anisotropic media with the principal refractive indexes

$$n_k\left(\mathbf{x}\right)={\lambda }_k{c}_1({\alpha }_k,x_k)$$

(384)

or

$$n_k\left(\mathbf{x}\right)={\lambda }_{n,k}{\widehat{c}}_{n,k}(D,\mathbf{x})$$

(385)

for the metric and product measure approaches, respectively. Here ${\lambda }_k$ and ${\lambda }_{n,k}$ are constants with the physical dimension inverse to the physical dimension of the functions $c_1({\alpha }_k,x_k)$ and ${\widehat{c}}_{n,k}(D,\mathbf{x})$, respectively. These constants are necessary for the quantity $n\left(\mathbf{x}\right)$ to be dimensionless ($\left[n_k\right]={\left[x\right]}^0$).

Remark 33. 

Let us note some properties of the NIDS length and the physical analogy of these properties with the properties of the optical length.

Note that the optical path lengths [136,137] of the two geometrically identical segments (identical from the point of view of standard geometry of ${\mathbb{R}}^3$) are not equal (even along the axis), in the general case. For example, using Equation (362), we have

$${\int }_0^{x_b}{c}_1(\alpha ,x)dx\ne {\int }_{x_b}^{2x_b}{c}_1(\alpha ,x)dx,$$

(386)

but the property of the additivity of length is preserved:

$${\int }_0^{2x_b}{c}_1(\alpha ,x)dx={\int }_0^{x_b}{c}_1(\alpha ,x)dx+{\int }_{x_b}^{2x_b}{c}_1(\alpha ,x)dx.$$

(387)

In other words, let us consider a segment $AB$ lying along the axis $X_1$ with coordinates $x_A$, $x_B$. If the point C has coordinate $x_C=(x_A+x_B)/2$, and $x_A<x_B<x_C$, then the NIDS lengths of the segments $AC$ and $CB$ are not equal:

$$L_{\alpha }^{\ast }[x_A,x_C]\ne L_{\alpha }^{\ast }[x_C,x_B],$$

(388)

although $x_C-x_A=x_B-x_C$, for the case $\alpha \in (0,1)$. However, the additivity property of the NIDS length holds

$$L_{\alpha }^{\ast }[x_A,x_C]+L_{\alpha }^{\ast }[x_C,x_B]=L_{\alpha }^{\ast }[x_A,x_B],$$

(389)

if $x_A<x_C<x_B$. The optical path (382) has the same properties in the general case. Note that the additivity property is important for generalizations of standard calculus and its application.

2.9.4. Kronecker Delta in D-Dimensional Space

In n-dimensional space, the Kronecker delta ${\delta }_{kl}$ by definition is equal to $+1$, if $k=l$, and is equal to zero otherwise. Therefore, the trace of the Kronecker delta

$$Tr\left({\delta }_{kl}\right)=\sum _{k=1}^n{\delta }_{kk}=n$$

(390)

is equal to the integer dimension $n\in \mathbb{N}$, i.e., $Tr\left({\delta }_{kl}\right)=n$.

In D-dimensional space, the Kronecker delta ${\delta }_{kl}^D$ in NIDS by definition is equal to ${\alpha }_k$, if $k=l$, and is equal to zero otherwise. Therefore in the NIDS, the trace of the Kronecker delta is equal to $D\in (0,n]$, and we have the equation

$$Tr\left({\delta }_{kl}^D\right)=\sum _{k=1}^n{\delta }_{kk}^D=\sum _{k=1}^n{\alpha }_k=D.$$

(391)

The property (391) is used in the representation of the tensors in NIDS. For example, tensors $E_{kl}\left(\mathbf{x}\right)$ can be represented as

$$E_{kl}\left(\mathbf{x}\right)=A_{kl}\left(\mathbf{x}\right)+B_{kl}\left(\mathbf{x}\right),$$

(392)

$$A_{kl}\left(\mathbf{x}\right)=E_{kl}\left(\mathbf{x}\right)-{\displaystyle \frac{1}{D}}{\delta }_{kl}^D\sum _{j=1}^n{E}_{jj}\left(\mathbf{x}\right),$$

(393)

$$B_{kl}\left(\mathbf{x}\right)={\displaystyle \frac{1}{D}}{\delta }_{kl}^D\sum _{j=1}^n{E}_{jj}\left(\mathbf{x}\right).$$

(394)

In the first tensor $A_{kl}\left(\mathbf{x}\right)$, the sum of diagonal terms is equal to zero. In the second tensor $B_{kl}\left(\mathbf{x}\right)$, the non-diagonal terms are equal to zero ($Tr\left(A_{kl}\right)=0$). Here, we use Equation (391) for a non-integer-dimensional space

$$Tr\left(B_{kl}\right)=\sum _{j=1}^n{E}_{jj}\left(\mathbf{x}\right).$$

(395)

For details about the Kronecker delta and the proof of Equation (391), one can see property 4 in Section 4.3 of [10]). Note that property (391) and Equation (392) are important to describe the elasticity of fractal materials by using the NIDS approach [113].

2.10. Surface NIDS Integral of Scalar Function and NIDS Area

Let us consider the surface NIDS integrals and the NIDS areas of a surface in the framework of the metric and product measure approaches.

2.10.1. Surface NIDS Integrals in $W=W_1\times W_2$

The surface NIDS integrals in the D-dimensional space $W=W_1\times W_2$ with $D={\alpha }_1+{\alpha }_2\in (0,2]$ are a special case of the volume NIDS integral for $n=2$. The surface NIDS integrals for $n⩾3$ will be considered in the next sub-subsections.

The surface NIDS integral of function $f\left(\mathbf{x}\right)\in C(\Omega )$ over the bounded region $\Omega \subset W$ with the coordinates $x_1$ and $x_2$ (where $(x_1,x_2)\in \Omega$) is

$$I_{\Omega }^{D}f\left(\mathbf{x}\right)={\int }_{\Omega }f\left(\mathbf{x}\right)d{\mu }_2(D,\mathbf{x}):={\int }_{W}f\left(\mathbf{x}\right){\chi }_{\Omega }\left(\mathbf{x}\right)d{\mu }_2(D,\mathbf{x})=$$

$${\int }_{W_1}{\int }_{W_2}f\left(\mathbf{x}\right){\chi }_{\Omega }\left(\mathbf{x}\right)d{\mu }_1({\alpha }_1,x_1)d{\mu }_1({\alpha }_2,x_2)=$$

$${\int }_{W_1}{\int }_{W_2}f\left(\mathbf{x}\right){\chi }_{\Omega }\left(\mathbf{x}\right)c_1({\alpha }_1,x_1)c_1({\alpha }_2,x_2)d{x}_{1}d{x}_2,$$

(396)

if this integral exists (if the boundary of the region is a piecewise smooth Jordan plane curve), where ${\chi }_{\Omega }\left(\mathbf{x}\right)$ is a characteristic function of $\Omega$.

For example, if the region $\Omega$ is bounded from below and above by the piecewise smooth curves

$$\Omega :=\{(x,y):a⩽x⩽b,y_1\left(x\right)⩽y⩽y_2\left(x\right)\},$$

(397)

then the surface NIDS integral is given by the double NIDS integral

$$I_{\Omega }^{D}f\left(\mathbf{x}\right)={\int }_{\Omega }f\left(\mathbf{x}\right)d{\mu }_2(D,\mathbf{x}):={\int }_a^{b}d{\mu }_1({\alpha }_1,x){\int }_{y_1\left(x\right)}^{y_1\left(x\right)}f(x,y)d{\mu }_1({\alpha }_2,y),$$

(398)

where

$$d{\mu }_1({\alpha }_k,x_k)=c_1({\alpha }_k,x_k)d{x}_k,$$

(399)

with $x_1=x$, $x_2=y$.

Therefore, we have the following definition of the surface NIDS integral over the region of the D-dimensional space $W=W_1\times W_2$.

Definition 13. 

Let $W=W_1\times W_2$ be the D-dimensional space with $D={\alpha }_1+{\alpha }_2$, and let $\Omega \subset W$ be a bounded region with the coordinates $x_1$ and $x_2$.

Then, the surface NIDS integral of the function $f\left(\mathbf{x}\right)\in C(\Omega )$ with $\mathbf{x}=(x_1,x_2)\in \Omega$ is defined by the equation

$$I_{\Omega }^{D}f\left(\mathbf{x}\right):=I_W^D{\chi }_{\Omega }\left(\mathbf{x}\right)f\left(\mathbf{x}\right)=$$

$${\int }_{\Omega }f\left(\mathbf{x}\right)c_2(D,\mathbf{x})d{x}_{1}d{x}_2={\int }_W{\chi }_{\Omega }\left(\mathbf{x}\right)f\left(\mathbf{x}\right)c_2(D,\mathbf{x})d{x}_{1}d{x}_2,$$

(400)

where $c_2(D,\mathbf{x})=c_1({\alpha }_1,x_1)c_1({\alpha }_2,x_2)$.

The area $S_D$ of a bounded region $\Omega \subset W$ is

$$S_D(\Omega )=I_W^D{\chi }_{\Omega }\left(\mathbf{x}\right)={\int }_{\Omega }{c}_2(D,\mathbf{x})d{x}_{1}d{x}_2,$$

(401)

where $D={\alpha }_1+{\alpha }_2\in (0,n]$ with $n=2$.

Remark 34. 

Definition 13 is a special case of the volume NIDS integral for $n=2$. Due to this, Definition 13 is true in the metric and product measure approaches.

The fulfillment of the correspondence principle for this NIDS integral was proved in the previous section (see Theorem 6). The area $S_D$ of a circle ${\Omega }_c=\{\mathbf{x}:|\mathbf{x}|⩽R\}$, which is calculated by the surface NIDS integral (401), is equal to the well-known equation

$$S_D\left({\Omega }_c\right)={\displaystyle \frac{2{\pi }^{D/2}}{D\Gamma (D/2)}}{R}^D,$$

(402)

where $D={\alpha }_1+{\alpha }_2$.

The change in the coordinates in the surface NIDS integral is implemented in the standard way. Using a one-to-one change in the coordinates

$$x_1=x_1(u,v),x_2=x_2(u,v),$$

(403)

where $\mathbf{x}=(x_1,x_2)\in \Omega$, Equation (400) for coordinates u and v takes the form

$$I_{\Omega }^{D}f\left(\mathbf{x}\right)={\int }_{\Omega }f\left(\mathbf{x}\right)d{\mu }_2(D,\mathbf{x})=$$

$${\int }_{U}d{\mu }_2(D,\mathbf{x}(u,v))={\int }_{U}f\left(\mathbf{x}(u,v)\right)c_2(D,\mathbf{x}(u,v))|J(u,v)|dudv=$$

$${\int }_{U}f\left(\mathbf{x}(u,v)\right)c_1({\alpha }_1,x_1(u,v))c_1({\alpha }_2,x_2(u,v))|J(u,v)|dudv,$$

(404)

where U is the region of the $(u,v)$-plane corresponding to the region $\Omega \subset W$, such that

$$U:=\left\{\right(u,v):\mathbf{x}(u,v)\in \Omega \}.$$

(405)

Here, $J(u,v)$ is the Jacobian of the coordinate transformation

$$J(u,v):={\displaystyle \frac{\partial (x_1,x_2)}{\partial (u,v)}}:={\displaystyle \frac{\partial x_1(u,v)}{\partial u}}{\displaystyle \frac{\partial x_2(u,v)}{\partial v}}-{\displaystyle \frac{\partial x_2(u,v)}{\partial u}}{\displaystyle \frac{\partial x_1(u,v)}{\partial v}},$$

(406)

where it is assumed that $x_k(u,v)\in C^1\left(U\right)$ for $k=1,2$.

2.10.2. Surface NIDS Integrals in $W=W_1\times W_2\times W_3$ in the Metric Approach

Let us define the surface NIDS integral $I_{\Omega }^{D,\ast }$ in the metric approach for the bounded region $\Omega$ of the surface in the D-dimensional space $W=W_1\times W_2\times W_3$. Let us assume that the natural NIDS metric $g_{kl}\left(x\right)$ can be used in the definitions of the volume, surface, and line NIDS integrations in the metric approach.

Definition 14. 

Let Ω be the surface region

$$\Omega :=\{\mathbf{x}:\mathbf{x}=\mathbf{x}(u,v),(u,v)\in U\}$$

(407)

in the D-dimensional space $W=W_1\times W_2\times W_3$ with $D={\alpha }_1+{\alpha }_2+{\alpha }_3$.

Then, in the metric approach, the surface NIDS integral $I_{\Omega }^{D,\ast }$ of the function $f\left(\mathbf{x}(u,v)\right)\in C^1\left(U\right)$ is given by the equation

$$I_{\Omega }^{D,\ast }f\left(\mathbf{x}\right):={\int }_{U}f\left(\mathbf{x}(u,v)\right)\sqrt{|det(G_{ij}^g(u,v))|}dudv,$$

(408)

and the NIDS area $S_D^{\ast }$ of the surface region Ω in the D-dimensional space W is

$$S_D^{\ast }(\Omega )={\int }_U\sqrt{|det(G_{ij}^g(u,v))|}dudv,$$

(409)

where the integral is taken over the region U in the $(u,v)$-plane, and

$$det\left(G_{ij}^g(u,v)\right)=G_{11}^g(u,v)G_{22}^g(u,v)-{\left(G_{12}^g(u,v)\right)}^2$$

(410)

is the determinant of the metric

$$G_{ij}^g(u,v):=\sum _{k,l=1}^3{g}_{kl}\left(\mathbf{x}(u,v)\right){\displaystyle \frac{\partial x^k(u,v)}{\partial u^i}}{\displaystyle \frac{\partial x^l(u,v)}{\partial u^j}}(u^1=u,u^2=v)$$

(411)

induced on the surface $\mathbf{x}=\mathbf{x}(u,v)$ by the natural NIDS metric

$$g_{kl}\left(\mathbf{x}\right)=c_1^2({\alpha }_k,x^k){\delta }_{kl},$$

(412)

where

$$c_1({\alpha }_k,x^k)={\displaystyle \frac{{\pi }^{{\alpha }_k/2}}{\Gamma ({\alpha }_k/2)}}{|x^k|}^{{\alpha }_k-1},$$

(413)

which is induced by the product of the NIDS measures of the D-dimensional space W.

Remark 35. 

Note that Equation (409) does not satisfy the second correspondence principle for surface NIDS integration. The violation of this correspondence principle for Equation (409) is manifested in the fact that, for the NIDS semi-sphere

$${\Omega }_S:=\{\mathbf{x}:|\mathbf{x}|=R,x_3⩾0\},$$

(414)

Equation (409) does not give the well-known expression, and we have the inequality

$$S_D^{\ast }\left({\Omega }_S\right)\ne S_D\left({\Omega }_S\right)={\displaystyle \frac{{\pi }^{D/2}}{\Gamma (D/2)}}{R}^{D-1},$$

(415)

for all $R>0$ and all ${\alpha }_1,{\alpha }_2,{\alpha }_3\in (0,1]$, where $D={\alpha }_1+{\alpha }_2+{\alpha }_3$.

Using Definition 14, one can give a theorem that describes some special cases of the surface NIDS integrals in the metric approach.

Theorem 10. 

Let Ω be a bounded region on a surface in the D-dimensional space $W=W_1\times W_2\times W_3$ with $D={\alpha }_1+{\alpha }_2+{\alpha }_3$.

Then, in the metric approach, the surface NIDS integrals over the region Ω of the surface in the space $W=W_1\times W_2\times W_3$ with the coordinates $x_1=x$, $x_2=y$, $x_3=z$ can be given by the following equations.

(A) If a surface in $W=W_1\times W_2\times W_3$ is given in the form

$$x_1(x,y)=x,x_2(x,y)=y,x_3=z(x,y),$$

(416)

and if Ω is a bounded region of this surface with projection U onto the $(x,y)$-plane $W_1\times W_2$, then the equation for surface NIDS integral can be represented as

$$I_{\Omega }^{D,\ast }f\left(\mathbf{x}\right):={\int }_{U}f\left(\mathbf{x}\left(\mathbf{u}\right)\right)c_3(D,\mathbf{x}(x,y))\sqrt{{\left(D_x^{{\alpha }_1}z(x,y)\right)}^2+{\left(D_y^{{\alpha }_2}z(x,y)\right)}^2+{\left(D_z^{{\alpha }_3}z\right)}^2}dxdy,$$

(417)

where $(x,y)\in U\subset W_1\times W_2$, the function

$$c_3(D,\mathbf{x}(x,y))=c_1({\alpha }_1,x)c_1({\alpha }_2,y)c_1({\alpha }_3,z(x,y)),$$

(418)

and the operator

$$D_{x^k}^{{\alpha }_k}F\left(\mathbf{x}\right)={\displaystyle \frac{1}{c_1({\alpha }_k,x^k)}}{\displaystyle \frac{\partial F(x_1,x_2,x_3)}{\partial x^k}}={\pi }^{-{\alpha }_k/2}\Gamma ({\alpha }_k/2){|x^k|}^{1-{\alpha }_k}{F}_{x^k}^{\left(1\right)}\left(\mathbf{x}\right),$$

(419)

which will then be defined as the partial NIDS derivative of the function $f\left(\mathbf{x}\right)$.

(B) If a surface in $W=W_1\times W_2\times W_3$ is given by the equation $F(x,y,z)=0$, and if Ω is a region on this surface, which projects in a one-to-one onto a region U of the $(x,y)$-plane, then the surface NIDS integral can be written as

$$I_{\Omega }^{D,\ast }f\left(\mathbf{x}\right):={\int }_{U}f\left(\mathbf{x}\left(\mathbf{u}\right)\right){\left({\displaystyle \frac{|{Grad}_{\left(\alpha \right)}\left(F(x,y,z)\right)|}{|D_z^{{\alpha }_3}F(x,y,z)|}}\right)}_{F=0}{c}_1({\alpha }_1,x)c_1({\alpha }_2,y)dxdy,$$

(420)

where $U\subset W_1\times W_2$, and $D_z^{{\alpha }_3}F(x,y,z)\ne 0$ for all $(x,y,z)\in \Omega$, and

$$|{Grad}_{\left(\alpha \right)}\left(F(x,y,z)\right)|=\sqrt{\sum _{k=1}^3{\left(D_{x^k}^{{\alpha }_k}F\left(\mathbf{x}\right)\right)}^2},$$

(421)

which will be defined as the NIDS gradient of the function $F(x,y,z)$ in the metric approach.

Proof. 

The proof is based on Definition 14 of the surface NIDS integral

$$I_{\Omega }^{D,\ast }f\left(\mathbf{x}\right):={\int }_{U}f\left(\mathbf{x}\left(\mathbf{u}\right)\right)\sqrt{|G^g(x,y)|}dxdy,$$

(422)

where we use $u=x$, $v=y$, and $G^g(x,y)$ is the induced metric $G_{ij}^g(x,y)$, $i,j=1,2$, such that

$$G^g(x,y)=det\left(G_{ij}^g(x,y)\right)=G_{11}^g(x,y)G_{22}^g(x,y)-{\left(G_{12}^g(x,y)\right)}^2.$$

(423)

(A) For statement A, using $x^1(x,y)=x$, $x^2(x,y)=y$, $x^3(x,y)=z(x,y)$, we get

$$G_{11}^g(x,y)=\sum _{k=1}^3{g}_{kk}\left(x^k\right){\left({\displaystyle \frac{\partial x^k(x,y)}{\partial x}}\right)}^2=c_1^2({\alpha }_1,x)+c_1^2({\alpha }_3,z(x,y)){\left({\displaystyle \frac{\partial z(x,y)}{\partial x}}\right)}^2,$$

(424)

$$G_{22}^g(x,y)=\sum _{k=1}^3{g}_{kk}\left(x^k\right){\left({\displaystyle \frac{\partial x^k(x,y)}{\partial y}}\right)}^2=c_1^2({\alpha }_2,y)+c_1^2({\alpha }_3,z(x,y)){\left({\displaystyle \frac{\partial z(x,y)}{\partial y}}\right)}^2,$$

(425)

$$G_{12}^g(x,y)=\sum _{k=1}^3{g}_{kk}\left(x^k\right){\displaystyle \frac{\partial x^k(x,y)}{\partial x}}{\displaystyle \frac{\partial x^k(x,y)}{\partial y}}=c_1^2({\alpha }_3,z(x,y)){\displaystyle \frac{\partial z(x,y)}{\partial x}}{\displaystyle \frac{\partial z(x,y)}{\partial y}}.$$

(426)

Substituting Equations (424)–(426) into Equation (423) gives the determinant of this metric

$$G^g(x,y)=c_1^2({\alpha }_1,x)c_1^2({\alpha }_2,y)+$$

$$c_1^2({\alpha }_1,x)c_1^2({\alpha }_3,z(x,y)){\left(z_y^{\left(1\right)}(x,y)\right)}^2+c_1^2({\alpha }_2,y)c_1^2({\alpha }_3,z(x,y)){\left(z_x^{\left(1\right)}(x,y)\right)}^2=$$

$$c_1^2({\alpha }_1,x)c_1^2({\alpha }_2,y)c_1^2({\alpha }_3,z(x,y))({\displaystyle \frac{1}{c_1^2({\alpha }_3,z(x,y))}}+$$

$${\displaystyle \frac{1}{c_1^2({\alpha }_2,y)}}{\left(z_y^{\left(1\right)}(x,y)\right)}^2+{\displaystyle \frac{1}{c_1^2({\alpha }_1,x)}}{\left(z_x^{\left(1\right)}(x,y)\right)}^2)=$$

$$c_3^2(D,\mathbf{x}(x,y))\sum _{k=1}^3{\left(D_{x^k}^{{\alpha }_k}z(x,y)\right)}^2,$$

(427)

where we use the operator (419), and

$$c_3(D,\mathbf{x}(x,y))=c_1^2({\alpha }_1,x)c_1^2({\alpha }_2,y)c_1^2({\alpha }_3,z(x,y)).$$

(428)

Substitution of Equations (427) and (428) into Equation (422) gives Equation (417).

(B) For Statement B, equation $F(x,y,z)=0$ can be represented as $z=(x,y)$, if the condition $\partial F(x,y,z)/\partial z\ne 0$ holds. Differentiating the identity $F(x,y,z(x,y\left)\right)=0$ with respect to x gives

$${\displaystyle \frac{\partial F(x,y,z(x,y\left)\right)}{\partial x}}+{\left({\displaystyle \frac{\partial F(x,y,z)}{\partial z}}\right)}_{z=z(x,y)}{\displaystyle \frac{\partial z(x,y)}{\partial x}}=0.$$

(429)

Using Equation (429), we get

$${\displaystyle \frac{\partial z(x,y)}{\partial x}}=-{\displaystyle \frac{\partial F(x,y,z(x,y\left)\right)/\partial x}{{\partial F(x,y,z)/\partial z)}_{F=0}}}.$$

(430)

Similarly, we obtain

$${\displaystyle \frac{\partial z(x,y)}{\partial y}}=-{\displaystyle \frac{\partial F(x,y,z(x,y\left)\right)/\partial y}{{\partial F(x,y,z)/\partial z)}_{F=0}}}.$$

(431)

Therefore, the determinant of the metric (427) is

$$G^g(x,y)=c_1^2({\alpha }_2,y)c_1^2({\alpha }_3,z(x,y)){\left({\displaystyle \frac{\partial F(x,y,z(x,y\left)\right)/\partial x}{{\partial F(x,y,z)/\partial z)}_{F=0}}}\right)}^2+$$

$$c_1^2({\alpha }_1,x)c_1^2({\alpha }_3,z(x,y)){\left({\displaystyle \frac{\partial F(x,y,z(x,y\left)\right)/\partial y}{{\partial F(x,y,z)/\partial z)}_{F=0}}}\right)}^2+c_1^2({\alpha }_1,x)c_1^2({\alpha }_2,y).$$

Using the operator (419), the determinant $G^g(x,y)$ can be written as

$$G^g(x,y)=c_1^2({\alpha }_1,x)c_1^2({\alpha }_2,y){\left({\displaystyle \frac{c_1^{-1}({\alpha }_1,x)\partial F(x,y,z(x,y))/\partial x}{c_1^{-1}({\alpha }_3,z(x,y)){\partial F(x,y,z)/\partial z)}_{F=0}}}\right)}^2+$$

$$c_1^2({\alpha }_1,x)c_1^2({\alpha }_2,y){\left({\displaystyle \frac{c_1^{-1}({\alpha }_2,y)\partial F(x,y,z(x,y))/\partial y}{c_1^{-1}({\alpha }_3,z(x,y)){\partial F(x,y,z)/\partial z)}_{F=0}}}\right)}^2+c_1^2({\alpha }_1,x)c_1^2({\alpha }_2,y)=$$

$$c_1^2({\alpha }_1,x)c_1^2({\alpha }_2,y){\left({\displaystyle \frac{D_x^{{\alpha }_1}F(x,y,z(x,y))}{{\left(D_z^{{\alpha }_3}F\right)}_{F=0}}}\right)}^2+$$

$$c_1^2({\alpha }_1,x)c_1^2({\alpha }_2,y){\left({\displaystyle \frac{D_y^{{\alpha }_2}F(x,y,z(x,y))}{{\left(D_z^{{\alpha }_3}F\right)}_{F=0}}}\right)}^2+c_1^2({\alpha }_1,x)c_1^2({\alpha }_2,y).$$

(432)

As a result, we get

$$G^g(x,y)={\displaystyle \frac{c_1^2({\alpha }_1,x)c_1^2({\alpha }_2,y)}{{\left(D_{x^3}^{{\alpha }_3}F\left(\mathbf{x}\right)\right)}_{F=0}^2}}\left(\sum _{k=1}^3{\left(D_{x^k}^{{\alpha }_2}F\left(\mathbf{x}\right)\right)}_{F=0}^2\right).$$

(433)

Substitution of Equation (433) into Equation (422) gives Equation (420). □

Remark 36. 

Let us note that metric $G_{ij}^g(u,v)$, which is induced on the surface $\mathbf{x}=\mathbf{x}(u,v)$ in the D-dimensional space $W=W_1\times W_2\times W_3$, is represented by Theorem 10 in terms of the partial NIDS derivatives and the NIDS gradient that is defined in the next section.

2.10.3. Surface NIDS Integrals in $W=W_1\times W_2\times W_3$ in the Product Measure Approach

To define the surface NIDS integral and the NIDS area within the product measure approach, simplifying notations must be defined. Let us give the definition of the notations for a two-parameter surface in the D-dimensional space $W=W_1\times \dots \times W_n$ with $n=3,4,5,\dots$.

Definition 15. 

Let Ω be a continuously differentiable two-parameter surface that is defined as

$$\Omega :=\{\mathbf{x}:\mathbf{x}=\mathbf{x}(u,v)\in C^1\left(U\right),(u,v)\in U\},$$

(434)

in the D-dimensional space $W=W_1\times \dots \times W_n$ with $D={\alpha }_1+\dots +{\alpha }_n\in (0,n]$, $n⩾3$, $n\in \mathbb{N}$.

Then, the functions ${\widehat{c}}_n(D,\mathbf{x}(u,v))$ and ${\widehat{c}}_{n,(k1,k2)}(D,\mathbf{x}(u,v))$ are defined as

$${\widehat{c}}_n(D,\mathbf{x}(u,v)):=\prod _{k=1}^n{\widehat{c}}_1({\alpha }_k,x^k(u,v)),$$

(435)

$${\widehat{c}}_{n,(q,p)}(D,\mathbf{x}(u,v)):=c_1({\alpha }_q,x^q(u,v))c_1({\alpha }_p,x^p(u,v))\prod _{j=1,j\ne q,p}^n{\widehat{c}}_1({\alpha }_j,x^j(u,v)),$$

(436)

where

$${\widehat{c}}_1({\alpha }_j,x^j(u,v)):=\left\{\begin{array}{cc}1\hfill & {\displaystyle \mathit{if}\partial x^j(u,v)/\partial u=0\mathrm{and}\partial x^j(u,v)/\partial v=0,}\hfill \\ c_1({\alpha }_j,x^j(u,v))\hfill & {\displaystyle \mathit{if}\partial x^j(u,v)/\partial u\ne 0\mathrm{or}\partial x^j(u,v)/\partial v\ne 0,}\hfill \end{array}\right.$$

(437)

and

$$c_1({\alpha }_j,x^j(u,v))={\displaystyle \frac{{\pi }^{{\alpha }_j/2}}{\Gamma ({\alpha }_j/2)}}{|x^j(u,v)|}^{{\alpha }_j-1},{\alpha }_j\in (0,1].$$

(438)

Remark 37. 

The notation of functions, proposed in Definition 15, means that if part of the region Ω is parallel to the coordinate plane $X_k{X}_l$, then the functions $c_1({\alpha }_j,x^j(u,v))$ with $k\ne j,i$ are equal to one. If Ω is a surface, where there are no regions in which the functions $x^k=x^k(u,v)$ are constant, then ${\widehat{c}}_n(D,\mathbf{x}(u,v))=c_n(D,\mathbf{x}(u,v))$.

In fact, this is a natural property of the surface NIDS integral and the NIDS area. If, on a part of the surface Ω, the coordinate $x^j$ does not change when the parameters $(u,v)$ change, then the value of the dimension ${\alpha }_j$ should not affect the area of that part of the surface. For example, if a part of the surface Ω is parallel to the coordinate plane $X_k{X}_l$, then the dimensions ${\alpha }_j$ with $k\ne j,i$ do not affect the NIDS integral and the NIDS area of this part of Ω.

Example 5. 

Let Ω be a surface region in the D-dimensional space $W=W_1\times W_2\times W_3$, such that

$$\Omega =\{\mathbf{x}:x^1=x,x^2=y,x^3=z(x,y)(x,y)\in U\}.$$

(439)

Then,

$${\widehat{c}}_{3,(q,p)}(D,\mathbf{x})={\widehat{c}}_{3,xy}(D,\mathbf{x})=c_1({\alpha }_1,x)c_1({\alpha }_2,y){\widehat{c}}_1({\alpha }_3,z(x,y)).$$

(440)

If $z(x,y)=const$ for all $(x,y)\in U_0\subset U$, then

$${\widehat{c}}_1({\alpha }_3,z(x,y))=1{\widehat{c}}_{3,xy}(D,\mathbf{x}=c_1({\alpha }_1,x)c_1({\alpha }_2,y)((x,y)\in U_0\subset U).$$

(441)

If $z(x,y)\ne const$ for all $(x,y)\in U_0\subset U$, then

$${\widehat{c}}_{3,xy}(D,\mathbf{x}=c_3(D,\mathbf{x})=c_1({\alpha }_1,x)c_1({\alpha }_2,y)c_1({\alpha }_3,z(x,y))((x,y)\in U_0\subset U),$$

(442)

where all ${\alpha }_k\in (0,1]$.

Let us define the surface NIDS integration in the D-dimensional space $W=W_1\times W_2\times W_3$.

Definition 16 

(Surface NIDS integral of Scalar Function). Let Ω be the two-parameter surface region in the D-dimensional space $W=W_1\times W_2\times W_3$, such that

$$\Omega =\{\mathbf{x}:x^k=x^k(u,v),k=1,2,3,(u,v)\in U\}.$$

(443)

Then, the surface NIDS integration of the function $f\left(\mathbf{x}\right(u,v\left)\right)\in C\left(U\right)$ over the surface region Ω is given by the equation

$$I_{\Omega }^{D}f\left(\mathbf{x}\right):={\int }_{U}f\left(\mathbf{x}(u,v)\right)\sqrt{{\widehat{G}}^{3,2}(u,v)}dudv,$$

(444)

where the integral is taken over the region U in the $(u,v)$-plane, and

$$\widehat{G}(u,v):=det\left({\widehat{G}}_{ij}^{3,2}(u,v)\right)={\widehat{G}}_{11}^{3,2}(u,v){\widehat{G}}_{22}^{3,2}(u,v)-{\left({\widehat{G}}_{12}^{3,2}(u,v)\right)}^2$$

(445)

is a determinant of the metric

$${\widehat{G}}_{ij}^{3,2}(u,v):=\sum _{k3=1}^3{\widehat{c}}_{3,(k1,k2)}(D,\mathbf{x}(u,v)){\displaystyle \frac{\partial x^{k3}(u,v)}{\partial u^i}}{\displaystyle \frac{\partial x^{k3}(u,v)}{\partial u^j}}$$

(446)

induced on the surface $\mathbf{x}=\mathbf{x}(u,v)$ of the D-dimensional space W, where

$$k3\ne k1,k2,k2\ne k1,u^1=u,u^2=v,$$

(447)

and ${\widehat{c}}_{3,(k1,k2)}(D,\mathbf{x}(u,v))$ is defined by Equation (436).

As a special case, the NIDS area $S_D$ of the surface region Ω is given by the equation

$$S_D(\Omega )={\int }_U\sqrt{{\widehat{G}}^{3,2}(u,v)}dudv,$$

(448)

where the integral is taken over the region U.

Remark 38. 

In this remark, an extension of Definition 16 for a two-parameter surface in the D-dimensional space $W=W_1\times \dots \times W_n$ for arbitrary integer values $n=3,4,5,\dots$ is given.

Let Ω be the two-parameter surface region in the D-dimensional space $W=W_1\times \dots \times W_n$, such that

$$\Omega =\{\mathbf{x}:x^k=x^k(u,v),k=1,\dots ,n,(u,v)\in U\}.$$

(449)

Then, the surface NIDS integration of the function $f\left(\mathbf{x}\right(u,v\left)\right)\in C\left(U\right)$ over the surface region Ω is given by the equation

$$I_{\Omega }^{D}f\left(\mathbf{x}\right):={\int }_{U}f\left(\mathbf{x}(u,v)\right)\sqrt{{\widehat{G}}^{n,2}(u,v)}dudv$$

(450)

with $n=3,4,5\dots$, where the integral is taken over the region U in the $(u,v)$-plane, and

$$\widehat{G}(u,v):=det\left({\widehat{G}}_{ij}^{n,2}(u,v)\right)={\widehat{G}}_{11}^{n,2}(u,v){\widehat{G}}_{22}^{n,2}(u,v)-{\left({\widehat{G}}_{12}^{n,2}(u,v)\right)}^2$$

(451)

is a determinant of the metric

$${\widehat{G}}_{ij}^{n,2}(u,v):=\sum _{k3=1}^n{\widehat{c}}_{n,(q,p)}(D,\mathbf{x}(u,v)){\displaystyle \frac{\partial x^k(u,v)}{\partial u^i}}{\displaystyle \frac{\partial x^k(u,v)}{\partial u^j}}$$

(452)

induced on the surface $\mathbf{x}=\mathbf{x}(u,v)$ in the D-dimensional space W, where

$$k\ne q,p,p\ne q,u^1=u,u^2=v,$$

(453)

and the function ${\widehat{c}}_{n,(q,p)}(D,\mathbf{x}(u,v))$ is defined by Equation (436).

As a special case, the NIDS area $S_D$ of the surface region Ω is given by the equation

$$S_D(\Omega )={\int }_U\sqrt{{\widehat{G}}^{n,2}(u,v)}dudv,$$

(454)

where the integral is taken over the region U.

Remark 39. 

Note that the notation in Equation (446) means that

$${\widehat{G}}_{ij}(u,v):={\widehat{c}}_{3,(2,3)}(D,\mathbf{x}(u,v)){\displaystyle \frac{\partial x^1(u,v)}{\partial u^i}}{\displaystyle \frac{\partial x^1(u,v)}{\partial u^j}}+$$

$${\widehat{c}}_{3,(1,3)}(D,\mathbf{x}(u,v)){\displaystyle \frac{\partial x^2(u,v)}{\partial u^i}}{\displaystyle \frac{\partial x^2(u,v)}{\partial u^j}}+{\widehat{c}}_{3,(1,2)}(D,\mathbf{x}(u,v)){\displaystyle \frac{\partial x^3(u,v)}{\partial u^i}}{\displaystyle \frac{\partial x^3(u,v)}{\partial u^j}},$$

(455)

where we can use $x^1=x$, $x^2=y$, $x^3=z$, and

$${\widehat{c}}_{3,(2,3)}(D,\mathbf{x}(u,v))={\widehat{c}}_{3,yz}(D,\mathbf{x}(u,v)),$$

(456)

$${\widehat{c}}_{3,(1,3)}(D,\mathbf{x}(u,v))={\widehat{c}}_{3,xz}(D,\mathbf{x}(u,v)),$$

(457)

$${\widehat{c}}_{3,(1,2)}(D,\mathbf{x}(u,v))={\widehat{c}}_{3,xy}(D,\mathbf{x}(u,v))$$

(458)

as an additional notation.

Let us consider the surface NIDS integral for the special case when there are no regions parallel to the coordinate planes.

Theorem 11 

(First Special Case of the Surface NIDS Integral of Scalar Function). Let Ω be a surface region in the D-dimensional space $W=W_1\times W_2\times W_3$, such that

$$\Omega =\{\mathbf{x}:x^k=x^k(u,v),k=1,2,3.(u,v)\in U\},$$

(459)

where there are no regions in which the functions $x^k=x^k(u,v)$ are constant.

Then, the surface NIDS integration of the function $f\left(\mathbf{x}\right(u,v\left)\right)\in C\left(U\right)$ over the surface region Ω is given by the equation

$$I_{\Omega }^{D}f\left(\mathbf{x}\right)={\int }_{U}f\left(\mathbf{x}(u,v)\right)c_3(D,\mathbf{x}(u,v))\sqrt{G(u,v)}dudv,$$

(460)

where the integral is taken over the region U in the $(u,v)$-plane, and the function

$$G(u,v):=det\left(G_{ij}(u,v)\right)=G_{11}(u,v)G_{22}(u,v)-{\left(G_{12}(u,v)\right)}^2$$

(461)

is a determinant of the standard metric

$$G_{ij}(u,v):=\sum _{k=1}^3{\displaystyle \frac{\partial x^k(u,v)}{\partial u^i}}{\displaystyle \frac{\partial x^k(u,v)}{\partial u^j}},(u^1=u,u^2=v)$$

(462)

induced on the surface $\mathbf{x}=\mathbf{x}(u,v)$.

As a special case, the NIDS area $S_D$ of the surface region Ω is given by the equation

$$S_D(\Omega )={\int }_U{c}_3(D,\mathbf{x}(u,v))\sqrt{G(u,v)}dudv,$$

(463)

where the integral is taken over the region U.

Proof. 

If there are no regions in which the functions $x^k=x^k(u,v)$ are constant, then

$${\widehat{c}}_1({\alpha }_k,x^k(u,v))=c_1({\alpha }_k,x^k(u,v)),$$

(464)

for all $k=1,2,3$ and all $(u,v)\in U$. In this case,

$${\widehat{c}}_{3,(k1,k2)}(D,\mathbf{x}(u,v))=c_3(D,\mathbf{x}(u,v)),$$

(465)

for all $k1,k2=1,2,3$ and all $(u,v)\in U$.

Then,

$${\widehat{G}}_{ij}(u,v):=\sum _{k3=1}^3{\widehat{c}}_{3,(k1,k2)}(D,\mathbf{x}(u,v)){\displaystyle \frac{\partial x^{k3}(u,v)}{\partial u^i}}{\displaystyle \frac{\partial x^{k3}(u,v)}{\partial u^j}}=$$

$$c_3(D,\mathbf{x}(u,v))\sum _{k3=1}^3{\displaystyle \frac{\partial x^{k3}(u,v)}{\partial u^i}}{\displaystyle \frac{\partial x^{k3}(u,v)}{\partial u^j}}=c_3(D,\mathbf{x}(u,v))G_{ij}(u,v),$$

(466)

and

$$\widehat{G}(u,v):=det\left({\widehat{G}}_{ij}(u,v)\right)=c_3^2(D,\mathbf{x}(u,v))det\left(G_{ij}(u,v)\right)=c_3^2(D,\mathbf{x}(u,v))G(u,v).$$

(467)

As a result, we get

$$\sqrt{\widehat{G}(u,v)}=c_3(D,\mathbf{x}(u,v))\sqrt{G(u,v)},$$

(468)

and Equation (444) gives Equation (460). □

Let us prove the corollary of Theorem 11 that describes a special case of the surface NIDS integrals for the surface described by the equation $F(x,y,z)=0$.

Corollary 2. 

Let Ω be a bounded region on a surface in the D-dimensional space $W=W_1\times W_2\times W_3$ with $D={\alpha }_1+{\alpha }_2+{\alpha }_3$. Let $\Omega \subset W$ be the surface region, where there are no regions in which the functions $x^k=x^k(u,v)$ are constant.

Let the surface be given by the equation $F(x,y,z)=0$ with the coordinates $x_1=x$, $x_2=y$, $x_3=z$, and $F_z^{\left(1\right)}(x,y,z)\ne 0$ for all $(x,y,z)\in \Omega$, and let Ω be projected in a one-to-one ratio onto a region U of the plane $(x,y)$.

Then, the surface NIDS integral of the function $f\left(\mathbf{x}\right)\in C(\Omega )$ can be represented as

$$I_{\Omega }^{D}f\left(\mathbf{x}\right)={\int }_U{\left(f\left(\mathbf{x}\right)\right)}_{F(x,y,z)=0}{\left(c_3(D,\mathbf{x})\right)}_{F(x,y,z)=0}{\left({\displaystyle \frac{|grad(F(x,y,z))|}{|F_z^{\left(1\right)}(x,y,z)|}}\right)}_{F(x,y,z)=0}dxdy,$$

(469)

and the equation for the NIDS area of Ω becomes

$$S_D(\Omega )={\int }_U{\left(c_3(D,\mathbf{x})\right)}_{F(x,y,z)=0}{\left({\displaystyle \frac{|grad(F(x,y,z))|}{|F_z^{\left(1\right)}(x,y,z)|}}\right)}_{F(x,y,z)=0}dxdy,$$

(470)

where $U\subset W_1\times W_2$, and $F_z^{\left(1\right)}(x,y,z)\ne 0$ for all $(x,y,z)\in \Omega$.

Proof. 

The statements of Corollary 2 are proved similarly to the proof of Theorem 7.3.5 in [147], pp. 75–76. The proof is based on Equation (460) in the form

$$I_{\Omega }^{D}f\left(\mathbf{x}\right):={\int }_{U}f\left(\mathbf{x}(u,v)\right)c_3(D,\mathbf{x}(u,v))\sqrt{G(u,v)}dudv$$

(471)

and Equation (463) of the NIDS area $S_D(\Omega )$ of a bounded region $\Omega$ of the surface in the form

$$S_D(\Omega ):={\int }_U{c}_3(D,\mathbf{x}(u,v))\sqrt{G(u,v)}dudv$$

(472)

by using the induced metric $G_{ij}(u,v)$, $i,j=1,2$ with the determinant

$$G(x,y)=G_{11}(x,y)G_{22}(x,y)-{\left(G_{12}(x,y)\right)}^2,$$

(473)

where there are no regions in which the functions $x^k=x^k(u,v)$ are constant.

For the surface $F(x,y,z)=0$, the induced metric is

$$G_{11}(x,y)={\displaystyle \frac{{\left(F_x^{\left(1\right)}(x,y,z)\right)}^2}{{\left(F_z^{\left(1\right)}(x,y,z)\right)}^2}}+1,$$

(474)

$$G_{22}(x,y)={\displaystyle \frac{{\left(F_y^{\left(1\right)}(x,y,z)\right)}^2}{{\left(F_z^{\left(1\right)}(x,y,z)\right)}^2}}+1,$$

(475)

$$G_{12}(x,y)={\displaystyle \frac{F_x^{\left(1\right)}(x,y,z)F_y^{\left(1\right)}(x,y,z)}{{\left(F_z^{\left(1\right)}(x,y,z)\right)}^2}}.$$

(476)

Substitution of Equations (474)–(476) into Equation (473) gives the determinant of this metric in the form

$$G(x,y)={\displaystyle \frac{{\left(F_x^{\left(1\right)}(x,y,z)\right)}^2}{{\left(F_z^{\left(1\right)}(x,y,z)\right)}^2}}+{\displaystyle \frac{{\left(F_y^{\left(1\right)}(x,y,z)\right)}^2}{{\left(F_z^{\left(1\right)}(x,y,z)\right)}^2}}+1=$$

$${\displaystyle \frac{{\left(F_x^{\left(1\right)}(x,y,z)\right)}^2+{\left(F_y^{\left(1\right)}(x,y,z)\right)}^2+{\left(F_z^{\left(1\right)}(x,y,z)\right)}^2}{{\left(F_z^{\left(1\right)}(x,y,z)\right)}^2}}={\displaystyle \frac{{|grad\left(F(x,y,z)\right)|}^2}{|F_z^{\left(1\right)}{(x,y,z)|}^2}},$$

(477)

where $(u,v)\in U$.

Substitution of Equation (477) into Equations (471) and (472) gives (469) and (470). □

Remark 40. 

Note that Corollary 2 can be easily extended to the case of a two-parameter surface in the D-dimensional space $W=W_1\times \dots \times W_n$ for arbitrary integer values $n=3,4,5,\dots$ by using Remark 38.

Let us consider the surface NIDS integral for the special case when all areas are parallel to one or another coordinate plane.

Theorem 12 

(Second Special Case of the Surface NIDS Integral of Scalar Function). Let Ω be a surface region in the D-dimensional space $W=W_1\times W_2\times W_3$ with $D={\sum }_{k=1}^3{\alpha }_k$, such that

$$\Omega :={\Omega }_{23}\bigcup {\Omega }_{13}\bigcup {\Omega }_{12},$$

(478)

where ${\Omega }_{ij}$ are surface regions that are parallel to the coordinate planes $X_{ij}=X_i{X}_j$, where $i\ne j$.

Then, the surface NIDS integral of a continuously differentiable scalar field $f\left(\mathbf{x}\right)$ over the surface region Ω is represented by the equation

$$I_{\Omega }^{D}f\left(\mathbf{x}\right)=I_{{\Omega }_{23}}^{D}f\left(\mathbf{x}\right)+I_{{\Omega }_{13}}^{D}f\left(\mathbf{x}\right)+I_{{\Omega }_{12}}^{D}f\left(\mathbf{x}\right),$$

(479)

where $I_{{\Omega }_{ij}}^D$ ($i\ne j$) are the surface NIDS integrals in the $d_{ij}$-dimensional space $W=W_i\times W_j$ with the dimension $d_{ij}={\alpha }_i+{\alpha }_j$, as defined in Definition 13, and are special cases of the volume NIDS integral for $n=2$, such that

$$I_{{\Omega }_{ij}}^{D}f\left(\mathbf{x}\right)={\int }_{{\Omega }_{ij}}f\left(\mathbf{x}\right)d{\mu }_2(d_{ij},\mathbf{x}),$$

(480)

where

$$d{\mu }_2(d_{ij},\mathbf{x})=c_2(d_{ij},\mathbf{x})d{x}^{i}d{x}^j,(i\ne j),$$

(481)

and

$$c_2(d_{ij},\mathbf{x}):=c_1({\alpha }_i,x^i)c_1({\alpha }_j,x^j),(i\ne j)$$

(482)

with all ${\alpha }_k\in (0,1]$ and $d_{ij}\in (0,2]$.

Proof. 

If regions ${\Omega }_{ij}$ of the surface are parallel to the coordinate planes $X_i{X}_j$, then

$${\widehat{c}}_{3,(i,j)}(D,\mathbf{x}(u,v))=c_1({\alpha }_i,x^i(u,v))c_1({\alpha }_j,x^j(u,v)),(i\ne j),$$

(483)

and ${\widehat{c}}_1({\alpha }_k,x^k(u,v))=1$ for $k\ne i,j$. For $(u,v)\in {\Omega }_{ij}$, we can use $u=x^j$ and $v=x^j$. Then, for $(u,v)\in {\Omega }_{ij}$, we get

$$\sqrt{{\widehat{G}}^{3,2}(u,v)}dudv=c_2(d_{ij},\mathbf{x}(u,v))\sqrt{G(u,v)}dudv=$$

$$c_2(d_{ij},\mathbf{x}(u,v)){\displaystyle \frac{\partial (x^i,x^j)}{\partial (u,v)}}dudv=c_2(d_{ij},\mathbf{x})d{x}^{i}d{x}^j,$$

(484)

where

$$c_2(d_{ij},\mathbf{x})=c_1({\alpha }_i,x^i)c_1({\alpha }_j,x^j),(i\ne j).$$

(485)

As a result, the surface NIDS integral

$$I_{{\Omega }_{ij}}^{D}f\left(\mathbf{x}\right)={\int }_{{\Omega }_{ij}}f\left(\mathbf{x}\right)c_2(d_{ij},\mathbf{x})d{x}^{i}d{x}^j$$

(486)

is the volume NIDS integral in the $d_{ij}$-dimensional space $W=W_i\times W_j$ with the dimension $d_{ij}={\alpha }_i+{\alpha }_j$, as defined in Definition 13. □

Remark 41. 

The surface NIDS integral (479), which is described in Theorem 12 for the case when all areas are parallel to one or another coordinate plane, can be written as

$${\int }_{\Omega }f\left(\mathbf{x}\right)d{S}_d:={\int }_{{\Omega }_{23}}f\left(\mathbf{x}\right)d{S}_{d_{23}}+{\int }_{{\Omega }_{13}}f\left(\mathbf{x}\right)d{S}_{d_{13}}+{\int }_{{\Omega }_{12}}f\left(\mathbf{x}\right)d{S}_{d_{12}}=$$

$${\int }_{{\Omega }_{23}}f\left(\mathbf{x}\right)c_2(d_{23},\mathbf{x})d{S}_{23}+{\int }_{{\Omega }_{13}}f\left(\mathbf{x}\right)c_2(d_{13},\mathbf{x})d{S}_{13}+{\int }_{{\Omega }_{12}}f\left(\mathbf{x}\right)c_2(d_{12},\mathbf{x})d{S}_{12},$$

(487)

where $d{S}_{d_{ij}}$, $d{S}_{ij}$ and $c_2(d_{ij},\mathbf{x})$ are defined by the equation

$$d{S}_{d_{ij}}:=d{\mu }_2(d_{ij},\mathbf{x})=c_2(d_{ij},\mathbf{x})d{S}_{ij},(i\ne j),$$

(488)

$$d{S}_{ij}:=d{\mu }_2(2,\mathbf{x})=d{x}^{i}d{x}^j,(i\ne j),$$

(489)

$$c_2(d_{ij},\mathbf{x}):=c_1({\alpha }_i,x^i)c_1({\alpha }_j,x^j),(i\ne j),$$

(490)

with $(x^i,x^j)\in {\Omega }_{ij}$ and $d_{ij}={\alpha }_i+{\alpha }_j$.

Remark 42. 

Let us explain why the functions ${\widehat{c}}_n(D,\mathbf{x})$ are needed in the definition of the surface NIDS integrals. The surface $\Omega =\{(x^1,x^2,x^3):x^k=x^k(u,v)(u,v)\in U,k=1,2,3\}$ in the D-dimensional space $W=W_1\times W_2\times W_3$ can be considered as a surface in $(D+{\alpha }_4)$-dimensional space $W_1\times W_2\times W_3\times W_4$ (and as a surface in the $(D+d)$-dimensional space $W_1\times \dots \times W_{3+q}$ with $q\in \mathbb{N}$). In this case, the functions

$$c_4(D+{\alpha }_4,\mathbf{x})=\left(\prod _{k=1}^3{c}_1({\alpha }_k,x^k)\right)c_1({\alpha }_4,x^4).$$

(491)

For example, one can consider the semi-sphere D-dimensional space $W=W_1\times W_2\times W_3$ that is described by the equation

$$\sum _{k=1}^3{\left(x^k\right)}^2=R^2,x^3⩾0.$$

(492)

This semi-sphere can be considered in the $(D+{\alpha }_4)$-dimensional space $W_1\times W_2\times W_3\times W_4$ by the equations

$$\sum _{k=1}^3{\left(x^k\right)}^2=R^2,x^3⩾0x^4=x_0^4=const.$$

(493)

If the surface NIDS integration uses the function $c_1({\alpha }_j,x^j)$ instead of the function ${\widehat{c}}_1({\alpha }_j,x^j)$, then the surface NIDS integrals in the spaces $W_1\times W_2\times W_3\times W_4$ and $W=W_1\times W_2\times W_3$ give expressions differing in the constant factor

$$I_{\Omega }^{D+{\alpha }_4}f\left(\mathbf{x}\right)=c_1({\alpha }_4,x_0^4)I_{\Omega }^{D}f\left(\mathbf{x}\right),$$

(494)

where $c_1({\alpha }_4,x_0^4)=const$. Therefore, the integral $I_{\Omega }^{D+{\alpha }_4}f\left(\mathbf{x}\right)$ with $c_4(D+{\alpha }_4,\mathbf{x})$ gives the incorrect result. This means that the Wilson normalization condition and Principle 4, which is the correspondence principle for volume, surface, and line NIDS integrals, are not satisfied for the surface NIDS integration.

If we use ${\widehat{c}}_1({\alpha }_4,x^4)$ in the definition of the surface NIDS integrals, then ${\widehat{c}}_1({\alpha }_4,x_0^4)=1$, since $x_0^k=const$, and

$$I_{\Omega }^{D+{\alpha }_4}f\left(\mathbf{x}\right)=I_{\Omega }^{D}f\left(\mathbf{x}\right),$$

(495)

if $x^4=x_0^4=const$.

As a result, the use of the function ${\widehat{c}}_1({\alpha }_4,x^4)$ in the definition of the surface NIDS integrals leads to the fact that Principle 4, which is the correspondence principle for volume, surface, and line NIDS integrals, is satisfied for the surface NIDS integration.

Remark 43. 

Let us emphasize that Equation (460) contains the function ${\widehat{c}}_3(D,\mathbf{x})$ and not the function $c_2(D,\mathbf{x})$. If Equation (460) contained the function $c_2(D,\mathbf{x})$, then the second correspondence principle for the surface NIDS integrals could not be satisfied.

Let us consider a surface in $W=W_1\times W_2\times W_3$ that is described by $z=z(x,y)$, and let Ω be a region of this surface with projection U onto the $(x,y)$-plane $X_1{X}_2$.

Note that the equation with $c_1({\alpha }_1,x)$ and $c_1({\alpha }_2,y)$ in the form

$$S_D^{\ast }(\Omega )={\int }_U\sqrt{{\left(z_x^{\left(1\right)}(x,y)\right)}^2+{\left(z_y^{\left(1\right)}(x,y)\right)}^2+1}d{\mu }_1({\alpha }_x,x)d{\mu }_1({\alpha }_y,y)=$$

$${\int }_U{c}_1({\alpha }_x,x)c_1({\alpha }_y,y)\sqrt{{\left(z_x^{\left(1\right)}(x,y)\right)}^2+{\left(z_y^{\left(1\right)}(x,y)\right)}^2+1}dxdy,$$

(496)

where $U\subset W_1\times W_2$, cannot describe the NIDS surface area, since Equation (496) does not satisfy the second correspondence principle of surface NIDS integration. The violation of this correspondence principle is manifested in the fact that, for the NIDS semi-sphere ${\Omega }_S:=\{\mathbf{x}:|\mathbf{x}|=R,x_3⩾0\}$, Equation (496) does not give the well-known expression, and we have the inequality

$$S_D^{\ast }\left({\Omega }_S\right)\ne S_D\left({\Omega }_S\right)={\displaystyle \frac{{\pi }^{D/2}}{\Gamma (D/2)}}{R}^{D-1}$$

(497)

for $D={\alpha }_1+{\alpha }_2+{\alpha }_3$, if ${\alpha }_3\ne 1$. Equation (496) gives the correct expression, only if ${\alpha }_3=1$.

The fact that Equation (460) satisfies the second correspondence principle of surface NIDS integration will be proved below in this subsection.

2.10.4. Surface NIDS Integrals in $W=W_1\times \dots \times W_n$ in the Product Measure Approach

Let us give the equations of the surface NIDS integral and surface area in the D-dimensional space $W=W_1\times \dots \times W_n$ with dimension $D\in (0,n]$.

Let us first consider the standard case of the m-dimensional surface lying in n-dimensional Euclidean space. Let us assume that in a neighborhood of a non-singular point of such a surface, $m\in \mathbb{N}$ coordinates (parameters) may be defined on it, and this surface is given in parametric form $\mathbf{x}=\mathbf{x}\left(\mathbf{u}\right)$, such as $x^k=x^k(u^1,\dots ,u^m)$, where $k=1,\dots ,n$. The metric $G_{ij}\left(\mathbf{x}\right)$ that induced on the m-parametric surface by the metric $G_{kl}\left(\mathbf{x}\right)$ ($k,l=1,\dots ,n$) of the n-dimensional space is defined by the equation

$$G_{ij}\left(\mathbf{u}\right)=\sum _{k,l=1}^n{G}_{kl}\left(\mathbf{x}\left(\mathbf{u}\right)\right){\displaystyle \frac{\partial x^k\left(\mathbf{u}\right)}{\partial u^i}}{\displaystyle \frac{\partial x^l\left(\mathbf{u}\right)}{\partial u^j}}.$$

(498)

To simplify the description, we consider here n-dimensional Euclidean space with the metric $G_{kl}\left(\mathbf{x}\right)={\delta }_{kl}$ and the induced metric

$$G_{ij}\left(\mathbf{u}\right)=\sum _{k=1}^n{\displaystyle \frac{\partial x^k\left(\mathbf{u}\right)}{\partial u^i}}{\displaystyle \frac{\partial x^k\left(\mathbf{u}\right)}{\partial u^j}}.$$

(499)

The surface NIDS integral of a function $f\left(\mathbf{x}\right(\mathbf{u}\left)\right)\in C\left(U\right)$ over the bounded region $\Omega$ is defined as

$$I_{\Omega }^{D}f\left(\mathbf{x}\left(\mathbf{u}\right)\right):={\int }_{U}f\left(\mathbf{x}\left(\mathbf{u}\right)\right)\sqrt{|det(G_{ij}\left(\mathbf{u}\right)|}d{u}^1\dots d{u}^m,$$

(500)

and the area $S_D$ of the bounded region $\Omega$ of the m-parametric surface in the n-dimensional space is defined by the equation

$$S_D(\Omega )={\int }_U\sqrt{|det(G_{ij}\left(\mathbf{u}\right))|}d{u}^1\dots d{u}^m,$$

(501)

where $U:=\{\mathbf{u}:\mathbf{x}(\mathbf{u})\in \Omega \}$.

To define the surface NIDS integral and the NIDS area in the framework of the product measure approach, simplifying notations should be defined.

Definition 17. 

Let $W=W_1\times \dots \times W_n$ be D-dimensional space with $D={\alpha }_1+\dots +{\alpha }_n$, and let Ω be a continuously differentiable m-parameter surface that is defined as

$$\Omega :=\{\mathbf{x}:\mathbf{x}=\mathbf{x}\left(\mathbf{u}\right)\in C^1\left(U\right),\mathbf{u}=(u^1,\dots ,u^m)\in U\}$$

(502)

in the D-dimensional space $W=W_1\times \dots \times W_n$, with $D={\alpha }_1+\dots +{\alpha }_n\in (0,n]$, $n\in \mathbb{N}$, and $2⩽m⩽n-1$.

Then, the functions ${\widehat{c}}_n(D,\mathbf{x}\left(\mathbf{u}\right))$ and ${\widehat{c}}_{n,(k1...km)}(D,\mathbf{x}\left(\mathbf{u}\right))$ are defined as

$${\widehat{c}}_n(D,\mathbf{x}\left(\mathbf{u}\right)):=\prod _{k=1}^n{\widehat{c}}_1({\alpha }_k,x^k\left(\mathbf{u}\right)),$$

(503)

$${\widehat{c}}_{n,(k1,...,km)}(D,\mathbf{x}\left(\mathbf{u}\right)):=\left(\prod _{i=1}^m{c}_1({\alpha }_{ki},x^{ki}\left(\mathbf{u}\right))\right)\prod _{j=1,j\ne k1,...,km}^n{\widehat{c}}_1({\alpha }_j,x^j\left(\mathbf{u}\right)),$$

(504)

where

$${\widehat{c}}_1({\alpha }_j,x^j\left(\mathbf{u}\right)):=\left\{\begin{array}{cc}1\hfill & {\displaystyle \mathit{if}\partial x^j\left(\mathbf{u}\right)/\partial u^i=0\mathrm{for}\mathrm{all}i=1,\dots ,m,}\hfill \\ c_1({\alpha }_j,x^j\left(\mathbf{u}\right))\hfill & {\displaystyle \mathit{if}\mathrm{at}\mathrm{least}\mathrm{one}\partial x^j(u,v)/\partial u^i\ne 0,}\hfill \end{array}\right.$$

(505)

and

$$c_1({\alpha }_j,x^j\left(\mathbf{u}\right))={\displaystyle \frac{{\pi }^{{\alpha }_j/2}}{\Gamma ({\alpha }_j/2)}}{|x^j\left(\mathbf{u}\right)|}^{{\alpha }_j-1},$$

(506)

where all ${\alpha }_j\in (0,1]$.

This notation is a natural property of the surface NIDS integral and the NIDS area. If, on a part of the surface $\Omega$, the coordinate $x^j$ does not change when the parameters $\mathbf{u}=(u^1,\dots ,u^m)$ change, then the value of the dimension ${\alpha }_j$ should not affect the area of that part of the surface and the surface NIDS integral.

Example 6. 

Let us consider an example for the case $n=5$ and $m=3$. The function ${\widehat{c}}_{5,(1,3,5)}(D,\mathbf{x})$ has the form

$${\widehat{c}}_{5,(1,3,5)}(D,\mathbf{x})=\prod _{j=2,4}{\widehat{c}}_1({\alpha }_j,x^j)\prod _{k=1,3,5}{c}_1({\alpha }_k,x^k),$$

(507)

where $k1=1$, $k2=3$, $k3=5$.

In the product measure approach, one can define a metric induced by the measure of the D-dimensional space $W=W_1\times \dots \times W_n$.

Definition 18. 

In the product measure approach, the metric ${\widehat{G}}_{ij}^{n,m}\left(\mathbf{u}\right)$ ($i,j=1,\dots ,m$), which is induced on the m-parametric surfaces $\mathbf{x}\left(\mathbf{u}\right)$ (the lines and space regions) by the measured product of the D-dimensional space $W=W_1\times \dots \times W_n$ with $D\in (0,n]$, is defined by the equation

$${\widehat{G}}_{ij}^{n,m}\left(\mathbf{u}\right):=\sum _{k=1}^n{\widehat{c}}_{n,(k1,\dots ,km)}^{2/m}(D,\mathbf{x}\left(\mathbf{u}\right)){\displaystyle \frac{\partial x^k\left(\mathbf{u}\right)}{\partial u^i}}{\displaystyle \frac{\partial x^k\left(\mathbf{u}\right)}{\partial u^j}},$$

(508)

$$n,m\in \mathbb{N},1⩽m⩽n,kj\ne kj\mathit{if}j\ne i,$$

(509)

where $\mathbf{x}\left(\mathbf{u}\right)\in C^1\left(U\right)$ for the region U of the m-parametric space. For $m\in [2,n-1]$, $m\in \mathbb{N}$, Equation (508) gives the metric on the surface. For $m=1$, Equation (508) gives the metric on the line. For $m=n$, Equation (508) gives the metric on the n-parametric region in the space W.

The metrics in (508) will be called the NIDS-induced metrics.

The proposed term “the NIDS-induced metric” is caused by the fact that such metric tensors can be interpreted as metrics induced by D-dimensional space W in the n-dimensional space ($n=m$), on the m-parametric surface ($m\in [2,n-1]$), and on the curved line ($m=1$).

Example 7. 

For $n=3$ and $m=2$, the metrics in (508) have the form

$${\widehat{G}}_{ij}^{3,2}\left(\mathbf{u}\right):=\sum _{k=1}^3{\widehat{c}}_{3,(k1,k2)}(D,\mathbf{x}\left(\mathbf{u}\right)){\displaystyle \frac{\partial x^k\left(\mathbf{u}\right)}{\partial u^i}}{\displaystyle \frac{\partial x^k\left(\mathbf{u}\right)}{\partial u^j}}=$$

$$\sum _{k=1}^3{c}_1({\alpha }_{k1},x^{k1}\left(\mathbf{u}\right))c_1({\alpha }_{k2},x^{k2}\left(\mathbf{u}\right)){\widehat{c}}_1({\alpha }_k,x^k\left(\mathbf{u}\right)){\displaystyle \frac{\partial x^k\left(\mathbf{u}\right)}{\partial u^i}}{\displaystyle \frac{\partial x^k\left(\mathbf{u}\right)}{\partial u^j}},$$

(510)

where $k\ne k1,k2$ and $k1\ne k2$.

Let us now give the definition of the surface NIDS integral for the D-dimensional space with $D\in (0,n]$ with arbitrary $n\in \mathbb{N}$.

Definition 19. 

Let W be the D-dimensional space $W=W_1\times \dots \times W_n$ with $D={\alpha }_1+\dots ,{\alpha }_n$, and let Ω be a bounded region of the surface in the space W.

Then, in the product measure approach, the surface NIDS integration of a function $f\left(\mathbf{x}\right(\mathbf{u}\left)\right)\in C\left(U\right)$ over the region Ω of the m-parametric surface in the space W is defined as

$$I_{\Omega }^{D}f\left(\mathbf{x}\left(\mathbf{u}\right)\right):={\int }_{U}f\left(\mathbf{x}\left(\mathbf{u}\right)\right)\sqrt{|det({\widehat{G}}_{ij}^{n,m}\left(\mathbf{u}\right)|}d{u}^1\dots d{u}^m,$$

(511)

where

$$U:=\{\mathbf{u}:\mathbf{x}(\mathbf{u})\in \Omega \}.$$

(512)

The NIDS area $S_D$ of the bounded region Ω of the surface in the D-dimensional space W is defined as

$$S_D(\Omega )={\int }_U\sqrt{|det({\widehat{G}}_{ij}^{n,m}\left(\mathbf{u}\right))|}d{u}^1\dots d{u}^m,$$

(513)

where $2⩽m⩽n-1$.

Let us give a theorem for the surfaces that do not contain regions parallel to the coordinate planes $X_i{X}_j$, $i,j=1\dots ,n$, for which $x^k\left(\mathbf{u}\right)\ne const$ for all $k=1,\dots ,n$, and all $\mathbf{u}\in U$.

Theorem 13 

(Special case of the Surface NIDS Integral). Let W be the D-dimensional space $W=W_1\times \dots \times W_n$ with $D={\alpha }_1+\dots ,{\alpha }_n$, and let Ω be a bounded region Ω of the surface in the D-dimensional space W such that there are no regions of Ω, in which the functions $x^k=x^k\left(\mathbf{u}\right)$$k=1,\dots ,n$, are constant.

Then, in the product measure approach, the surface NIDS integration of the function $f\left(\mathbf{x}\right(\mathbf{u}\left)\right)\in C\left(U\right)$ over the region Ω of the m-parametric surface in the space W can be written as

$$I_{\Omega }^{D}f\left(\mathbf{x}\right):={\int }_{U}f\left(\mathbf{x}\left(\mathbf{u}\right)\right)c_n(D,\mathbf{x}\left(\mathbf{u}\right))\sqrt{|det(G_{ij}\left(\mathbf{u}\right)|}d{u}^1\dots d{u}^m,$$

(514)

where $U:=\{\mathbf{u}:\mathbf{x}(\mathbf{u})\in \Omega \}$, and

$$c_n(D,\mathbf{x}\left(\mathbf{u}\right))=\prod _{k=1}^n{c}_1({\alpha }_k,x^k\left(\mathbf{u}\right)).$$

(515)

In this case, the NIDS area $S_D$ of the bounded region Ω of the surface in the D-dimensional space W is defined as

$$S_D(\Omega )={\int }_U{c}_n(D,\mathbf{x}\left(\mathbf{u}\right))\sqrt{|det(G_{ij}\left(\mathbf{u}\right))|}d{u}^1\dots d{u}^m,$$

(516)

where $2⩽m⩽n-1$.

Proof. 

If there are no regions on the surface $\Omega$, in which the functions $x^k=x^k\left(\mathbf{u}\right)$ are constant, then

$${\widehat{c}}_{n,(k1,\dots ,km})(D,\mathbf{x}\left(\mathbf{u}\right))=c_n(D,\mathbf{x}\left(\mathbf{u}\right)).$$

(517)

If the surface $\Omega$ does not contain regions in which the functions $x^k=x^k\left(\mathbf{u}\right)$, $k=1,\dots ,n$, are constant, then

$${\widehat{G}}_{ij}^{n,m}\left(\mathbf{u}\right)=c_n^{2/m}(D,\mathbf{x}\left(\mathbf{u}\right))G_{ij}\left(\mathbf{u}\right),(i,j=1,\dots m),$$

(518)

and the determinant of the metrics in (508) is

$$det\left({\widehat{G}}_{ij}^{n,m}\left(\mathbf{u}\right)\right)=c_n^2(D,\mathbf{x}\left(\mathbf{u}\right))det\left(G_{ij}\left(\mathbf{u}\right)\right),$$

(519)

where $1⩽m⩽n$.

In this case,

$$\sqrt{|det({\widehat{G}}_{ij}^{n,m}\left(\mathbf{u}\right))|}=c_n(D,\mathbf{x}\left(\mathbf{u}\right))\sqrt{|det(G_{ij}\left(\mathbf{u}\right))|}.$$

(520)

Therefore, Equation (511) gives Equation (514). □

Remark 44. 

Note that for metrics in (508) with $m=n$, Equation (514) describes the coordinates’ changes in the volume NIDS integral in the form

$${\int }_{\Omega }f\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x})={\int }_{\Omega }f\left(\mathbf{x}\right)\sqrt{|det({\widehat{G}}_{kl}^{n,n}\left(\mathbf{x}\right)|}d{\mu }_n(n,\mathbf{x})=$$

$${\int }_{{\Omega }^{\prime }}f\left(\mathbf{x}\left({\mathbf{x}}^{\prime }\right)\right)\sqrt{|det({\widehat{G}}_{kl}^{n,n}\left({\mathbf{x}}^{\prime }\right))|}d{\mu }_n(n,{\mathbf{x}}^{\prime }),$$

(521)

where ${\mathbf{x}}^{\prime }=(x^1,\dots ,x^n)$.

Remark 45. 

Note that when representing NIDS integration in the D-dimensional spaces through integration in n-dimensional spaces with the special form of the measured product, the metrics in (508) are induced by the product of NIDS measures. Unlike the standard case (see Section 88 in [148]), the NIDS-induced metrics (508) depend on both the number n of coordinates $x^k$ ($k=1,\dots ,n$) of the D-dimensional space $W=W_1\times \dots \times W_n$ and the number of parameters $m\in \mathbb{N}$ ($1⩽m⩽n$) of the m-parametric surface in this space W. In addition, the metrics in (508) depend on the dimensions ${\alpha }_k\in (0,1]$ of the spaces $W_k$.

Let us emphasize that the volume, surface, and line NIDS integrals satisfy Principle 6, which is the principle of induced metric consistency in the product measure approach.

Remark 46. 

As will be shown below in the following subsection, the proposed NIDS-induced metrics define equations of the surface NIDS integrals that satisfy Principle 4, which is the correspondence principle for volume, surface, and line NIDS integrals for the product measure approach.

In the product measure approach, the NIDS-induced metrics ${\widehat{G}}_{ij}^{n,m}$ of D-dimensional space $W=W_1\times \dots \times W_n$ with $D\in (0,n]$ are consistent with the NIDS measure. Principle 6 means that the following condition is satisfied in the product measure approach.

The metrics ${\widehat{G}}_{ij}^{n,m}\left(\mathbf{u}\right)$ are the induced metrics on the m-parametric surfaces in the space $W=W_1\times \dots \times W_n$. The NIDS area of the m-parametric surface Σ, which is described by equations $x^k=x^k(u^1,\dots ,u^m)$ ($k=1,\dots ,n$) for $\mathbf{u}\in U$, $1<m<n$ in the D-dimensional space $W=W_1\times \dots \times W_n$, should be defined by the metric ${\widehat{G}}_{ij}^{n,m}\left(\mathbf{u}\right)$, such that

$$S_D(\Sigma )={\int }_U\sqrt{det\left({\widehat{G}}_{ij}^{n,m}\left(\mathbf{u}\right)\right)}\prod _{j=1}^{n}d{u}^j(i,j=1,\dots ,m)$$

(522)

is satisfied for all $n,m\in \mathbb{N}$, where $1<m<n$.

Principle 6, which is the principle of induced metric consistency in the product measure approach, is satisfied for the surface NIDS integrals in the product measure approach. In the next subsubsection, this fact will be proved explicitly.

2.10.5. Fulfillment of the Correspondence Principle in the Product Measure Approach

Let us give the equation of the surface NIDS integral over the surface that is described by the equation $F\left(\mathbf{x}\right)=0$ in the D-dimensional space $W=W_1\times \dots \times W_n$ with an arbitrary $n\in \mathbb{N}$.

Theorem 14. 

Let a surface in D-dimensional space $W=W_1\times \dots \times W_n$ with the dimension $D={\alpha }_1+\dots +{\alpha }_n\in (0,n]$ be given by the equation $F\left(\mathbf{x}\right)=0$, where $F_{x_n}^{\left(1\right)}\left(\mathbf{x}\right)\ne 0$, and let Ω be a bounded region on this surface.

Let us assume that, in the region Ω, there are no regions in which the functions $x^k=x^k\left(\mathbf{u}\right)$, $k=1,\dots ,n$ are constant.

Then, the surface NIDS integral of the function $f\left(\mathbf{x}\right)\in C^1(\Omega )$ over the region Ω on the surface in the D-dimensional space $W=W_1\times \dots \times W_n$ is defined by the equation

$$I_{\Omega }^{D}f\left(\mathbf{x}\right):={\int }_U{\left(f\left(\mathbf{x}\right)\right)}_{F\left(\mathbf{x}\right)=0}{\left(c_n(D,\mathbf{x})\right)}_{F\left(\mathbf{x}\right)=0}{\left({\displaystyle \frac{|grad(F\left(\mathbf{x}\right))|}{|F_{x_n}^{\left(1\right)}\left(\mathbf{x}\right)|}}\right)}_{F\left(\mathbf{x}\right)=0}\prod _{k=1}^{n-1}d{x}_k,$$

(523)

where $\mathbf{x}=(x_1,\dots ,x_n)\in W$, $(x_1,\dots ,x_{n-1})\in U$, and U is a projection of the surface region Ω on $W_1\times \dots \times W_{n-1}$.

Proof. 

The proof of Theorem 14 is similar to the proof of Corollary 2. □

Let us prove that Equation (523) for the surface NIDS integral satisfies the second correspondence principle for all D-dimensional spaces $W=W_1\times \dots \times W_n$ with all $n\in \mathbb{N}$ with $n⩾3$.

Theorem 15. 

Let $W=W_1\times \dots \times W_n$ be the NIDS with dimension $D={\alpha }_1+\dots +{\alpha }_n\in (0,n]$, where $n\in \mathbb{N}$ and $n⩾3$. Let the $m=n-1$-parameter surface ${\Omega }_S\subset W$ be the semi-sphere

$${\Omega }_S:=\{\mathbf{x}:{\mathbf{x}}^2-R^2=0,x_n⩾0\},$$

(524)

where $\mathbf{x}=(x_1,\dots ,x_n)\in W$, and let U be a projection of the surface region ${\Omega }_S$ on $W_1\times \dots \times W_{n-1}$ and $(x_1,\dots ,x_{n-1})\in U$.

Then, the surface NIDS integral in the form

$$S_D\left({\Omega }_S\right)={\int }_U{\left(c_n(D,\mathbf{x})\right)}_{F\left(\mathbf{x}\right)=0}{\left({\displaystyle \frac{|grad(F\left(\mathbf{x}\right))|}{|F_{x_n}^{\left(1\right)}\left(\mathbf{x}\right)|}}\right)}_{F\left(\mathbf{x}\right)=0}\prod _{k=1}^{n-1}d{x}_k,$$

(525)

where $F\left(\mathbf{x}\right)={\mathbf{x}}^2-R^2$, gives the NIDS area of the semi-sphere ${\Omega }_S$ in the well-known form

$$S_D\left({\Omega }_S\right)={\displaystyle \frac{{\pi }^{D/2}}{\Gamma (D/2)}}{R}^{D-1},$$

(526)

for all $R>0$, all $D\in (0,n]$, and all $n\in \mathbb{N}$, where $n⩾3$.

Proof. 

For the semi-sphere ${\Omega }_S$, the functions $x^k=x^k\left(\mathbf{u}\right)$, $k=1,\dots ,n$ are not constant. As a result, we can use Equation (525).

In the case $n⩾3$, the semi-sphere ${\Omega }_S$ is described by the equation $F\left(\mathbf{x}\right)=0$ and the inequality $x_n⩾0$, where

$$F\left(\mathbf{x}\right)=\sum _{k=1}^n{x}_k^2-R^2,$$

(527)

and

$$x_n=\sqrt{R^2-\sum _{k=1}^{n-1}{x}_k^2}.$$

(528)

Then,

$$grad\left(F\left(\mathbf{x}\right)\right)=2\sum _{k=1}^n{x}_k{\mathbf{e}}_k,$$

(529)

$$|grad\left(F\left(\mathbf{x}\right)\right)|=2\sqrt{\sum _{k=1}^n{x}_k^2},$$

(530)

$${|grad\left(F\left(\mathbf{x}\right)\right)|}_{F\left(\mathbf{x}\right)=0}=2{\left(\sqrt{\sum _{k=1}^n{x}_k^2}\right)}_{F\left(\mathbf{x}\right)=0}=2R,$$

(531)

and

$$F_{x_n}^{\left(1\right)}\left(\mathbf{x}\right)={\displaystyle \frac{\partial F\left(\mathbf{x}\right)}{\partial x_n}}=2x_n,$$

(532)

$$|F_{x_n}^{\left(1\right)}{\left(\mathbf{x}\right)|}_{F\left(\mathbf{x}\right)=0}=2{|x_n|}_{F\left(\mathbf{x}\right)=0}=2\sqrt{R^2-\sum _{k=1}^{n-1}{x}_k^2},$$

(533)

$$c_n(D,\mathbf{x})=\prod _{k=1}^n{c}_1({\alpha }_k,x_k)={\displaystyle \frac{{\pi }^{D/2}}{{\prod }_{k=1}^n\Gamma ({\alpha }_k/2)}}\prod _{k=1}^n{|x_k|}^{{\alpha }_k-1},$$

(534)

$$|c_n{(D,\mathbf{x})|}_{F\left(\mathbf{x}\right)=0}={\displaystyle \frac{{\pi }^{D/2}}{{\prod }_{k=1}^n\Gamma ({\alpha }_k/2)}}{\left(R^2-\sum _{k=1}^{n-1}{x}_k^2\right)}^{{\alpha }_n/2-1/2}\prod _{k=1}^{n-1}{|x_k|}^{{\alpha }_k-1}.$$

(535)

As a result, Equation (525) takes the form

$$S_D\left({\Omega }_S\right)={\displaystyle \frac{{\pi }^{D/2}}{{\prod }_{k=1}^n\Gamma ({\alpha }_k/2)}}R{\int }_U\prod _{k=1}^{n-1}{|x_k|}^{{\alpha }_k-1}{\left(R^2-\sum _{k=1}^{n-1}{x}_k^2\right)}^{{\alpha }_n/2-1}\prod _{k=1}^{n-1}d{x}_k,$$

(536)

where $n⩾3$, and we take into account Equation (533) that gives ${\alpha }_n/2-1/2$ instead of ${\alpha }_n/2-1$.

(A) Let us consider the case $n=3$.

For $n=3$, Equation (536) has the form

$$S_D\left({\Omega }_S\right)={\displaystyle \frac{{\pi }^{({\alpha }_1+{\alpha }_2+{\alpha }_3)/2}}{\Gamma ({\alpha }_1/2)\Gamma ({\alpha }_2/2)\Gamma ({\alpha }_3/2)}}R{\int }_U|x_1{|}^{{\alpha }_1-1}{|x_2|}^{{\alpha }_2-1}{\left(R^2-x_1^2-x_2^2\right)}^{{\alpha }_3/2-1}d{x}_{1}d{x}_2.$$

(537)

For the circle on the plane $X_1{X}_2$, using the polar coordinates $(t,\varphi )$, the equation

$$x_1=rcos\left(\varphi \right),x_2=rsin\left(\varphi \right),$$

(538)

and $J(r,\varphi )=r$, where $r\in [0,R]$ and $\varphi \in [0,2\pi )$, Equation (537) takes the form

$$S_D(\Omega )={\displaystyle \frac{{\pi }^{({\alpha }_1+{\alpha }_2+{\alpha }_3)/2}}{\Gamma ({\alpha }_1/2)\Gamma ({\alpha }_2/2)\Gamma ({\alpha }_3/2)}}{\int }_0^{2\pi }{|cos\left(\varphi \right)|}^{{\alpha }_1-1}{|sin\left(\varphi \right)|}^{{\alpha }_2-1}R{\int }_0^{R}dr{r}^{{\alpha }_1+{\alpha }_2-1}{\left(R^2-r^2\right)}^{{\alpha }_3/2-1}.$$

(539)

Then, we can use the equations

$${\int }_0^{2\pi }{|cos\left(\varphi \right)|}^{{\alpha }_1-1}{|sin\left(\varphi \right)|}^{{\alpha }_2-1}=4{\int }_0^{\pi /2}{|cos\left(\varphi \right)|}^{{\alpha }_1-1}{|sin\left(\varphi \right)|}^{{\alpha }_2-1}={\displaystyle \frac{2\Gamma ({\alpha }_1/2)\Gamma ({\alpha }_2/2)}{\Gamma ({\alpha }_1/2+{\alpha }_2/2)}},$$

(540)

and

$$R{\int }_0^R{r}^{{\alpha }_1+{\alpha }_2-2}{\left(R^2-r^2\right)}^{{\alpha }_3/2-1}rdr=$$

$${\displaystyle \frac{1}{2}}{R}^{{\alpha }_1+{\alpha }_2+{\alpha }_3-1}{\int }_0^1{t}^{{\alpha }_1/2+{\alpha }_2/2-1}{\left(1-t\right)}^{{\alpha }_3/2-1}dt=$$

$${\displaystyle \frac{1}{2}}{R}^{{\alpha }_1+{\alpha }_2+{\alpha }_3-1}{\displaystyle \frac{\Gamma ({\alpha }_1/2+{\alpha }_2/2)\Gamma ({\alpha }_3/2)}{\Gamma ({\alpha }_1/2+{\alpha }_2/2+{\alpha }_3/2)}},$$

(541)

where $t={(r/R)}^2$.

As a result, the substitution of Equations (540) and (541) into Equation (539) gives the NIDS area of the semi-sphere

$$S_D(\Omega )={\displaystyle \frac{{\pi }^{({\alpha }_1+{\alpha }_2+{\alpha }_3)/2}}{\Gamma ({\alpha }_1/2+{\alpha }_2/2+{\alpha }_3/2)}}{R}^{{\alpha }_1+{\alpha }_2+{\alpha }_3-1}={\displaystyle \frac{{\pi }^{D/2}}{\Gamma (D/2)}}{R}^{D-1}.$$

(542)

This result proves that the surface NIDS integral over a semi-sphere gives the well-known expression for the area of the semi-sphere in D-dimensional space $W=W_1\times W_2\times W_3$. One can see that Equation (537) satisfies Principle 4, which is the correspondence principle for volume, surface, and line NIDS integrals in the case $D\in (0,n]$ with $n=3$.

(B) Let us consider the case $n=4$. The Cartesian coordinate $(x_1,x_2,x_3)\in U$ can be transformed to the spherical coordinates $(r,\varphi ,\theta )$ by the equation

$$x_1=rsin\theta cos\varphi ,$$

(543)

$$x_2=rsin\theta sin\varphi ,$$

(544)

$$x_3=rcos\theta ,$$

(545)

where $r\in [0,R]$, $\varphi \in [0,2\pi )$, $\theta \in [0,\pi ]$. and $J(r,\varphi ,\theta )=r^{2}sin\theta$ is the Jacobian of the coordinate change. In these spherical coordinates, Equation (536) with $n=4$ takes the form

$$S_D\left({\Omega }_S\right)={\displaystyle \frac{{\pi }^{D/2}}{{\prod }_{k=1}^4\Gamma ({\alpha }_k/2)}}R{\int }_0^R{r}^{{\alpha }_1+{\alpha }_2+{\alpha }_3-1}{\left(R^2-r^2\right)}^{{\alpha }_4/2-1}dr·$$

$${\int }_0^{\pi }{|sin\theta |}^{{\alpha }_1+{\alpha }_2-1}{|cos\theta |}^{{\alpha }_3-1}d\theta ·{\int }_0^{2\pi }{|cos\varphi |}^{{\alpha }_1-1}{|sin\varphi |}^{{\alpha }_2-1}d\varphi .$$

(546)

Using the equation

$$R{\int }_0^R{r}^{{\alpha }_1+{\alpha }_2+{\alpha }_3-1}{\left(R^2-r^2\right)}^{{\alpha }_4/2-1}dr=$$

$$R{\int }_0^R{r}^{{\alpha }_1+{\alpha }_2+{\alpha }_3-2}{\left(R^2-r^2\right)}^{{\alpha }_4/2-1}rdr=$$

$${\displaystyle \frac{1}{2}}{R}^{{\alpha }_1+{\alpha }_2+{\alpha }_3+{\alpha }_4-1}{\int }_0^1{t}^{{\alpha }_1/2+{\alpha }_2/2+{\alpha }_3/2-1}{(1-t)}^{{\alpha }_4/2-1}dt=$$

$${\displaystyle \frac{1}{2}}{R}^{{\alpha }_1+{\alpha }_2+{\alpha }_3+{\alpha }_4-1}{\displaystyle \frac{\Gamma ({\alpha }_1/2+{\alpha }_2/2+{\alpha }_3/2)\Gamma ({\alpha }_4/2)}{\Gamma ({\alpha }_1/2+{\alpha }_2/2+{\alpha }_3/2+{\alpha }_4/2)}},$$

(547)

where $t={(r/R)}^2$, and

$${\int }_0^{\pi }{|sin\theta |}^{{\alpha }_1+{\alpha }_2-1}{|cos\theta |}^{{\alpha }_3-1}d\theta ={\displaystyle \frac{\Gamma ({\alpha }_1/2+{\alpha }_2/2)\Gamma ({\alpha }_3/2)}{\Gamma ({\alpha }_1/2+{\alpha }_2/2+{\alpha }_3/2)}},$$

(548)

$${\int }_0^{2\pi }{|cos\varphi |}^{{\alpha }_1-1}{|sin\varphi |}^{{\alpha }_2-1}d\varphi ={\displaystyle \frac{2\Gamma ({\alpha }_1/2)\Gamma ({\alpha }_2/2)}{\Gamma ({\alpha }_1/2+{\alpha }_2/2)}},$$

(549)

Equation (546) gives

$$S_D\left({\Omega }_S\right)={\displaystyle \frac{{\pi }^{{\alpha }_1/2+{\alpha }_2/2+{\alpha }_3/2+{\alpha }_4/2}}{\Gamma ({\alpha }_1/2+{\alpha }_2/2+{\alpha }_3/2+{\alpha }_4/2)}}{R}^{{\alpha }_1+{\alpha }_2+{\alpha }_3+{\alpha }_4-1}={\displaystyle \frac{{\pi }^{D/2}}{\Gamma (D/2)}}{R}^{D-1},$$

(550)

which describes the NIDS area of the D-dimensional semi-sphere with $D\in (0,n]$ with $n=4$.

(C) Let us consider the case $n⩾5$, which is similar to the proof for the case $n=4$. The theorem is proved for the case $n⩾5$, in general. Using the $(n-1)$-dimensional coordinate system, which is described by Equations (A12)–(A14), and the equations that are used in the proof of Theorem 6, Equation (536) of the NIDS area of the D-dimensional semi-sphere gives

$$S_D(\Omega )={\displaystyle \frac{{\pi }^{D/2}}{\Gamma (D/2)}}{R}^{D-1},$$

(551)

for all $D\in (0,n]$ and all $n\in \mathbb{N}$ with $n⩾5$. □

Theorem 15 proves that the surface NIDS integral over a semi-sphere gives the well-known expression for the area of the semi-sphere in the product measure approach. One can see that Equation (525) satisfies Principle 4, which is the correspondence principle for volume, surface, and line NIDS integrals, for all $D\in (0,n]$ with all $n\in \mathbb{N}$, $n⩾3$. For the case n = 2, we have the “one-dimensional surface” that is the boundary of the region in the form of a closed line. Therefore, one can state that the fulfillment of Principle 4 (the correspondence principle for volume, surface, and line NIDS integrals) is proved by Theorem 9 for the circle length in the D-dimensional space $W=W_1\times W_2$.

Remark 47. 

Theorem 15 is proved for the m-parametric semi-sphere with $m=n-1$ only. It is easy to prove that Theorem 15 can be generalized in the case of the arbitrary $m\in [2,n-1]$ with $n,m\in \mathbb{N}$.

To prove this generalization, we can consider the $(n-1)$-parametric semi-sphere

$${\Omega }_S^{(n,q)}:=\{\mathbf{x}:\sum _{k=1}^n{\left(x^k\right)}^1-R^2=0,x_n⩾0x^k=x_0^k=const\mathrm{for}k=n+1,\dots ,n+q\}$$

(552)

in the $(D+d)$-dimensional space $W=W_1\times \dots \times W_n\times W_{n+1}\times \dots \times W_{n+q}$. In this case, we have

$$c_1({\alpha }_k,x^k)=1\mathrm{for}k=n+1,\dots ,n+q,$$

(553)

$$c_{n+q}(D+d,\mathbf{x})=c_n(D+d,\mathbf{x}),$$

(554)

where $D={\alpha }_1+\dots +{\alpha }_n$, and $d={\alpha }_{n+1}+\dots +{\alpha }_{n+q}$.

Then, using the property of the surface NIDS integrals that is described in Remark 42 in the form

$$I_{{\Omega }_S^{(n,q)}}^{D+d}f\left(\mathbf{x}\right)=I_{{\Omega }_S}^{D}f\left(\mathbf{x}\right)$$

(555)

and Theorem 15, Equation (525) gives

$$S_D\left({\Omega }_S^{(n,q)}\right)={\displaystyle \frac{{\pi }^{D/2}}{\Gamma (D/2)}}{R}^{D-1},$$

(556)

for all $(n-1)$-parametric semi-spheres in the $(D+d)$-dimensional space $W=W_1\times \dots \times W_n\times W_{n+1}\times \dots \times W_{n+q}$.

Changing the notation ($n+q\to n$, $D+d\to D$), we find that the statement of Theorem 15 is also true in the D-dimensional space $W=W_1\times \dots \times W_n$ for the m-parametric semi-sphere with the arbitrary values $m\in [2,n-1]$ with $n,m\in \mathbb{N}$.

As a result, in the product measure approach, the correspondence principle is satisfied for the surface NIDS integration for the arbitrary $n,m\in \mathbb{N}$, such that $m\in [2,n-1]$. Note that the cases $m=1$ and $m=n$ relate to the line and volume NIDS integrals.

3. NIDS Differential Calculus

Let us first formulate the principles that will be used to define and specify the explicit form of derivatives and differential operators in spaces with non-integer dimensions.

3.1. Principles of NIDS Differential Calculus

To formulate the NIDS calculus so that it will be self-consistent, the NIDS differential operators must be consistent with the NIDS integral operators. For this, the definition and exact form of the NIDS derivative must be such that the fundamental theorems of the NIDS calculus can be satisfied. The NIDS generalization of the standard vector differential operators (gradient, divergence, and curl) must be such that there exist NIDS analogs of the standard gradient theorem, the divergence theorem (the Gauss–Ostrogradsky theorem), and the Stokes theorem.

Principle 7 

(Principle of Self-Consistency of NIDS Calculus). For the self-consistent formulation of the NIDS calculus, the definitions and explicit form of NIDS differential operators must be consistent with the definitions and explicit form of NIDS integral operators.

First, the definition and exact form of the NIDS derivative must be such that the first and second fundamental theorems of the NIDS calculus exist and hold.

Second, the definition and exact form of the NIDS generalizations of the vector differential operators (gradient, divergence, and curl) must be such that the NIDS generalizations of the gradient theorem, the Gauss–Ostrogradsky theorem (divergence theorem), and the Stokes theorem are satisfied for the NIDS.

In addition to the principle of self-consistency of NIDS calculus (Principle 7), the correspondence principles must be satisfied. According to the first principle, the NIDS derivatives (and other NIDS differential operators) for integer dimensions must give standard exact expressions of these operators for integer-dimensional space. According to the second correspondence principle, exact expressions of second-order NIDS differential operators of NIDS vector calculus must be consistent with the first-order NIDS differential operators. Note that Stillinger [14] and Palmer and Stavrinou [15] proposed the scalar Laplace operators, which do not satisfy this principle [17,18,19].

Principle 8 

(Correspondence Principles for NIDS Differential Calculus). For the correct formulation of the NIDS calculus, the correspondence principles must be satisfied.

According to the first principle, the NIDS derivatives (and other NIDS differential operators) in D-dimensional space for integer dimensions $D=n$ must give standard exact expressions of these operators that are used in integer-dimensional spaces. Symbolically, this can be written as the left-sided limit

$$\underset{\alpha \to 1-}{lim}{D}_x^{\alpha }f\left(x\right)={\displaystyle \frac{df\left(x\right)}{dx}},$$

(557)

where the symbol $D_x^{\alpha }$ denotes the NIDS derivative with respect to x in the NIDS with dimension $D=\alpha \in (0,1]$.

According to the second correspondence principle, exact expressions of second-order NIDS differential operators in the NIDS vector calculus (the NIDS scalar Laplacian and the NIDS vector Laplacian) must be consistent with the first-order NIDS vector differential operators.

3.2. NIDS Derivatives of a Single Variable

Let us give the definition of the NIDS derivative of the first order for $n=1$. Then, using characteristic properties and the principle of the self-consistency of NIDS calculus (Principle 7), we prove a theorem that gives the explicit form of the NIDS derivative proposed in the definition. We will prove that a first-order linear operator satisfying Principle 7, which requires the consistency between differential and integral operators, specifies an explicit form of the NIDS derivative.

Let us define the NIDS continuously differentiable functions for the NIDS derivatives of a single variable.

Definition 20 

(NIDS Continuously Differentiable Functions). Let $W=W_1$ be a D-dimensional space with $D=\alpha \in (0,1]$, and let Ω be a region of W.

The function $f\left(x\right)$ with $x\in \Omega$ will be called the NIDS continuously differentiable function, if

$${\displaystyle \frac{1}{c_1(\alpha ,x)}}{\displaystyle \frac{df\left(x\right)}{dx}}\in C(\Omega ),$$

(558)

for all $x\in \Omega$. The set of such functions will be denoted as $C_{\left(\alpha \right)}^1(\Omega )$.

The function $f\left(x\right)$ with $x\in \Omega$ will be called the $C_{\left(\alpha \right)}^m$-function, if all m-order NIDS derivatives

$$({\displaystyle \frac{1}{c_1(\alpha ,x)}}{\displaystyle \frac{df\left(x\right)}{dx}}{)}^{m}f\left(x\right)=\underset{m-times}{\underbrace{{\displaystyle \frac{1}{c_1(\alpha ,x)}}{\displaystyle \frac{d}{dx}}\dots {\displaystyle \frac{1}{c_1(\alpha ,x)}}{\displaystyle \frac{d}{dx}}}}f\left(x\right)$$

(559)

are continuous for all $x\in \Omega$. In this case, we will write $f\left(x\right)\in C_{\left(\alpha \right)}^m(\Omega )$.

Let us define the NIDS derivatives of a single variable.

Definition 21. 

Let $W=W_1$ be a D-dimensional space with $D=\alpha \in (0,1]$, and let Ω be a region of W.

If $f\left(x\right)\in C_{\left(\alpha \right)}^1(\Omega )$, the NIDS derivative of first order is defined by the equation

$$D_x^{\alpha }f\left(x\right):={\displaystyle \frac{1}{c_1(\alpha ,x)}}{\displaystyle \frac{df\left(x\right)}{dx}}=c^{-1}\left(\alpha \right){|x|}^{1-\alpha }{\displaystyle \frac{df\left(x\right)}{dx}},$$

(560)

where

$$c_1(\alpha ,x):=c\left(\alpha \right){|x|}^{\alpha -1},$$

(561)

$$c\left(\alpha \right):={\displaystyle \frac{{\pi }^{\alpha /2}}{\Gamma (\alpha /2)}},$$

(562)

with $\alpha \in (0,1]$.

Let us prove that the classical chain rule is violated for the first-order NIDS derivative (560).

Theorem 16 

(Chain Rule for NIDS Derivative). Let $W=W_1$ be a D-dimensional space with $D=\alpha \in (0,1]$, and let Ω be a region of W.

Let $g\left(x\right)\in C_{\left(\alpha \right)}^1(\Omega )$ and $f\left(x\right)\in C^1\left({\Omega }_g\right)$, where ${\Omega }_g:=\{g:g=g\left(x\right),x\in \Omega \}$.

Then, the chain rule for the first-order NIDS derivative (560) has the form

$$D_x^{\alpha }f\left(g\left(x\right)\right)={\left({\displaystyle \frac{df\left(g\right)}{dg}}\right)}_{g=g\left(x\right)}{D}_x^{\alpha }g\left(x\right),$$

(563)

for all $x\in \Omega$.

Proof. 

Using Equation (560) of Definition 21 and the standard chain rule for the first-order derivative, we get

$$D_x^{\alpha }f\left(g\left(x\right)\right)={\displaystyle \frac{1}{c_1(\alpha ,x)}}{\displaystyle \frac{df\left(g\right(x\left)\right)}{dx}}={\displaystyle \frac{1}{c_1(\alpha ,x)}}{\left({\displaystyle \frac{df\left(g\right)}{dg}}\right)}_{g=g\left(x\right)}{\displaystyle \frac{dg\left(x\right)}{dx}}=$$

$${\left({\displaystyle \frac{df\left(g\right)}{dg}}\right)}_{g=g\left(x\right)}\left({\displaystyle \frac{1}{c_1(\alpha ,x)}}{\displaystyle \frac{dg\left(x\right)}{dx}}\right)={\left({\displaystyle \frac{df\left(g\right)}{dg}}\right)}_{g=g\left(x\right)}{D}_x^{\alpha }g\left(x\right),$$

(564)

where Equation (560) was used. □

Let us prove that the operator (560) is linear and satisfies the classical Leibniz rule (the product rule) for NIDS continuously differentiable functions.

Theorem 17 

(Leibniz (Product) Rule for NIDS Derivative). Let $W=W_1$ be a D-dimensional space with $D=\alpha \in (0,1]$, and let Ω be a region of W.

If $f\left(x\right),g\left(x\right)\in C_{\left(\alpha \right)}^1(\Omega )$, then the first-order NIDS derivative (560) satisfies the Leibniz (product) rule

$$D_x^{\alpha }\left(f\left(x\right)g\left(x\right)\right)=\left(D_x^{\alpha }f\left(x\right)\right)g\left(x\right)+f\left(x\right)\left(D_x^{\alpha }g\left(x\right)\right),$$

(565)

for all $x\in \Omega$.

Proof. 

Using Equation (560) of Definition 21 and the standard product rule for the first-order derivative, we get

$$D_x^{\alpha }\left(f\left(x\right)g\left(x\right)\right)={\displaystyle \frac{1}{c_1(\alpha ,x)}}{\displaystyle \frac{d}{dx}}\left(f\left(x\right)g\left(x\right)\right)=$$

$${\displaystyle \frac{1}{c_1(\alpha ,x)}}\left({\displaystyle \frac{df\left(x\right)}{dx}}g\left(x\right)+f\left(x\right){\displaystyle \frac{dg\left(x\right)}{dx}}\right)=$$

$$\left({\displaystyle \frac{1}{c_1(\alpha ,x)}}{\displaystyle \frac{df\left(x\right)}{dx}}\right)g\left(x\right)+f\left(x\right)\left({\displaystyle \frac{1}{c_1(\alpha ,x)}}{\displaystyle \frac{dg\left(x\right)}{dx}}\right)=$$

$$\left(D_x^{\alpha }f\left(x\right)\right)g\left(x\right)+f\left(x\right)\left(D_x^{\alpha }g\left(x\right)\right),$$

(566)

where Equation (560) was used. □

Theorem 18 

(Linearity Property of NIDS Derivative). Let $W=W_1$ be a D-dimensional space with $D=\alpha \in (0,1]$, and let Ω be a region of W.

If $f\left(x\right),g\left(x\right)\in C_{\left(\alpha \right)}^1(\Omega )$ and $a,b\in \mathbb{R}$, then the first-order NIDS derivative (560) satisfies the equation

$$D_x^{\alpha }\left(af\left(x\right)+bg\left(x\right)\right)=a\left(D_x^{\alpha }f\left(x\right)\right)+b\left(D_x^{\alpha }g\left(x\right)\right),$$

(567)

for all $x\in \Omega$.

Proof. 

Using Equation (560) of Definition 21 and the linearity property of the first-order derivatives, we get

$$D_x^{\alpha }\left(af\left(x\right)+bg\left(x\right)\right)={\displaystyle \frac{1}{c_1(\alpha ,x)}}{\displaystyle \frac{d}{dx}}\left(af\left(x\right)+bg\left(x\right)\right)=$$

$${\displaystyle \frac{1}{c_1(\alpha ,x)}}\left(a{\displaystyle \frac{df\left(x\right)}{dx}}+b{\displaystyle \frac{dg\left(x\right)}{dx}}\right)=$$

$$a\left({\displaystyle \frac{1}{c_1(\alpha ,x)}}{\displaystyle \frac{df\left(x\right)}{dx}}\right)+b\left({\displaystyle \frac{1}{c_1(\alpha ,x)}}{\displaystyle \frac{dg\left(x\right)}{dx}}\right)=$$

$$a\left(D_x^{\alpha }f\left(x\right)\right)+b\left(D_x^{\alpha }g\left(x\right)\right),$$

(568)

where Equation (560) was used. □

Let us consider what restrictions on the explicit form of the differential operator $O_x^{\alpha }$ on $\mathbb{R}$ are imposed by the linearity conditions and the Leibniz rule in the standard calculus.

Theorem 19. 

Let $O_x^{\alpha }$ be an operator that satisfies the linearity property and the Leibniz rule

$$O_x^{\alpha }(af\left(x\right)+bg\left(x\right))=a{O}_x^{\alpha }f\left(x\right)+b{O}_x^{\alpha }g\left(x\right),$$

(569)

$$O_x^{\alpha }\left(f\left(x\right)g\left(x\right)\right)=g\left(x\right)O_x^{\alpha }f\left(x\right)+f\left(x\right)O_x^{\alpha }g\left(x\right),$$

(570)

for all $f\left(x\right),g\left(x\right)\in C^1\left(\mathbb{R}\right)$ and all $a,b\in \mathbb{R}$.

Then, the operator $O_x^{\alpha }$ can be represented as

$$O_x^{\alpha }f\left(x\right)=a(\alpha ,x){\displaystyle \frac{df\left(x\right)}{dx}},$$

(571)

where $a(\alpha ,x)$ is a function of the variable x and the parameter α.

Proof. 

Theorem 19 is proved in papers [149,150] and states that operators satisfying the linearity property (569) and the product rule (570) are differential operators of integer (first) order in the form (571). □

Let us consider what restrictions on the explicit form of the differential operator (571) are imposed by the condition that this operator changes the degree of the power function from $\beta$ to $\beta -\alpha$.

Lemma 3. 

Let $O_x^{\alpha }$ be a linear operator that satisfies the Leibniz rule, and let us assume that the property

$$O_x^{\alpha }{x}^{\beta }=b(\alpha ,\beta )x^{\beta -\alpha }$$

(572)

holds for all $\beta \in \mathbb{R}$ and all $x>0$, where the function $b(\alpha ,\beta )$ does not depend on x.

Then, the operator $O_x^{\alpha }$ can be represented in the form

$$O_x^{\alpha }f\left(x\right)=a\left(\alpha \right)x^{1-\alpha }{\displaystyle \frac{df\left(x\right)}{dx}}$$

(573)

for $x>0$, where $a\left(\alpha \right)$ is the function of the parameter $\alpha >0$.

Proof. 

Using Theorem 19, the operator $O_x^{\alpha }$ can be represented as

$$O_x^{\alpha }f\left(x\right)=a(\alpha ,x){\displaystyle \frac{df\left(x\right)}{dx}}.$$

(574)

Substituting Equation (574) into property (572) gives

$$a(\alpha ,x)\beta x^{\beta -1}=b(\alpha ,\beta )x^{\beta -\alpha }.$$

(575)

Equation (575) gives the expression of the function $a(\alpha ,x)$ in the form

$$a(\alpha ,x)={\displaystyle \frac{b(\alpha ,\beta )}{\beta }}{x}^{1-\alpha },$$

(576)

where $x>0$. Since the function $a(\alpha ,x)$ does not depend on the parameter $\beta$, one can state that ${\beta }^{-1}b(\alpha ,\beta )$ does not depend on the parameter $\beta$. Therefore, we can write

$$a(\alpha ,x)=a\left(\alpha \right)x^{1-\alpha },$$

(577)

where $x>0$, and $a\left(\alpha \right)$ is a function of the parameter alpha.

Substitution of Equation (577) into Equation (574) gives that the operator $O_x^{\alpha }$ has the form (573), where $x>0$. □

Note that the correspondence principle for NIDS differential calculus (Principle 8), in which operator (571) must be the standard derivative of integer order $\alpha =1$, gives the conditions $a(\alpha =1)=1$; therefore, $b(\alpha =1,\beta )=\beta$.

Remark 48. 

Note that property (572) is not necessary to derive explicit form of the NIDS derivative, since this form follows from the principle of the self-consistency of NIDS calculus (Principle 7).

For the self-consistent formulation of the NIDS calculus, the definitions and explicit form of NIDS differential operators must be consistent with the definitions and explicit form of NIDS integral operators. This consistency means the requirement that the NIDS generalizations of the standard fundamental theorems must be satisfied. The NIDS fundamental theorems lead to a fixed form of the function $a(\alpha ,x)$ by the condition

$$a(\alpha ,x)c_1(\alpha ,x)=1.$$

(578)

As a result, in the NIDS calculus, it is necessary to use the multiplier

$$a\left(\alpha \right)={\displaystyle \frac{\Gamma (\alpha /2)}{{\pi }^{\alpha /2}}},a(\alpha ,x)={\displaystyle \frac{\Gamma (\alpha /2)}{{\pi }^{\alpha /2}}}{|x|}^{1-\alpha }$$

(579)

with parameter $\alpha \in (0,1]$ that describes the dimension of the NIDS. This statement will be proved in the next theorem (Theorem 20) in this section.

The fundamental theorems of the NIDS calculus are proved in the following subsection.

Let us prove that the requirement of fulfilling the first fundamental theorem specifies the explicit form of the NIDS derivative based on the explicit form of the NIDS integral.

Theorem 20. 

Let $O_x^{\alpha }$ be a linear operator that satisfies the Leibniz rule, and let the equation

$$O_x^{\alpha }{I}_{[0,x]}^{\alpha }f\left(x\right)=f\left(x\right)$$

(580)

hold for all $x>0$ and all $f\left(x\right)\in C^1[0,\infty )$, where $I_{[0,x]}^{\alpha }$ is the NIDS integral that is defined by

$$I_{[0,x]}^{\alpha }f\left(x\right):={\int }_0^{x}f\left(\xi \right)d{\mu }_1(\alpha ,x)={\int }_0^{x}f\left(\xi \right)c_1(\alpha ,\xi )d\xi ,$$

(581)

where

$$c_1(\alpha ,x):={\displaystyle \frac{{\pi }^{\alpha /2}}{\Gamma (\alpha /2)}}{|x|}^{\alpha -1},$$

(582)

for all $\alpha \in (0,1]$.

Then, the operator $O_x^{\alpha }$ can be represented in the form

$$O_x^{\alpha }f\left(x\right)=c_1^{-1}(\alpha ,x){\displaystyle \frac{df\left(x\right)}{dx}}={\displaystyle \frac{\Gamma (\alpha /2)}{{\pi }^{\alpha /2}}}{|x|}^{1-\alpha }{\displaystyle \frac{df\left(x\right)}{dx}},$$

(583)

for all $x>0$ and all $f\left(x\right)\in C^1[0,\infty )$.

Proof. 

Using Theorem 19 and the theorem condition that $O_x^{\alpha }$ is the linear operator that satisfies the Leibniz rule, the operator $O_x^{\alpha }$ can be represented as

$$O_x^{\alpha }f\left(x\right)=a(\alpha ,x){\displaystyle \frac{df\left(x\right)}{dx}},$$

(584)

where $a(\alpha ,x)$ is a function of the variable x and the parameter $\alpha \in (0,1]$.

Substitution of Equations (581) and (584) into Equation (580) gives

$$a(\alpha ,x){\displaystyle \frac{d}{dx}}{\int }_0^{x}f\left(\xi \right)c_1(\alpha ,\xi )d\xi =f\left(x\right).$$

(585)

Using the fundamental theorems of the standard calculus, Equation (585) gives

$$a(\alpha ,x)f\left(x\right)c_1(\alpha ,x)=f\left(x\right).$$

(586)

Since Equation (580) should be satisfied for all $x>0$ and all $f\left(x\right)\in C^1[0,\infty )$, Equation (586) should also be satisfied for all $x>0$ and all $f\left(x\right)\in C^1[0,\infty )$. Therefore, Equation (586) gives the condition

$$a(\alpha ,x)c_1(\alpha ,x)=1,$$

(587)

which must be satisfied for all $x>0$. Using Equation (582) and $c_1(\alpha ,x)=c_1(\alpha ,|x|)$, we get Equation (583). □

Theorem 20 states that property (580), which is the first fundamental theorem of the NIDS calculus, gives the exact form of the NIDS derivative

$$D_x^{\alpha }f\left(x\right)={\pi }^{-\alpha /2}\Gamma (\alpha /2)f^{\left(1\right)}\left(x\right),$$

(588)

where $\alpha \in (0,1]$.

As a result, we prove that the principle of the self-consistency of NIDS calculus (Principle 7), in the form of the requirement to fulfill the NIDS generalizations of standard fundamental theorems, specifies an explicit form of the NIDS derivatives that is defined in Definition 21 by Equation (560).

3.3. NIDS Derivative via Limit

The standard derivative is defined as the limit of the ratio of the increment of a function to the increment of its argument, as the increment of the argument tends to zero, if this limit exists:

$$f^{\left(1\right)}\left(x\right)=\underset{h\to 0}{lim}{\displaystyle \frac{f(x+h)-f\left(x\right)}{h}}.$$

(589)

The NIDS derivative can also be represented by the limit of the ratio of the some increment of a function to the increment parameter.

Theorem 21 

(NIDS Derivative via Limit). Let $f\left(x\right)$ be a continuously differentiable function on the real axis $\mathbb{R}$, i.e., $f\left(x\right)\in C_{\left(\alpha \right)}^1\left(\mathbb{R}\right)$.

Then, the NIDS derivative of single variable in the NIDS with dimension $\alpha \in (0,1]$ can be represented in the form

$$D_x^{\alpha }f\left(x\right)=\underset{h\to 0}{lim}{\displaystyle \frac{f(x+ha(\alpha ,x\left)\right)-f\left(x\right)}{h}},$$

(590)

if

$$a(\alpha ,x):=c_1^{-1}(\alpha ,x):={\pi }^{-\alpha /2}\Gamma (\alpha /2){|x|}^{1-\alpha },$$

(591)

where $\alpha \in (0,1]$.

Proof. 

Consider the case $x\ne 0$ and $\alpha \in (0,1)$. Let us define $\eta :=ha(\alpha ,x)$. Then, $h=a^{-1}(\alpha ,x)\eta$, if $a(\alpha ,x)\ne 0$, and

$$\underset{h\to 0}{lim}{\displaystyle \frac{f(x+ha(\alpha ,x\left)\right)}{h}}=\underset{\eta \to 0}{lim}{\displaystyle \frac{f(x+\eta )-f\left(x\right)}{a^{-1}(\alpha ,x)\eta }}=$$

$$\underset{\eta \to 0}{lim}{\displaystyle \frac{f(x+\eta )-f\left(x\right)}{a^{-1}(\alpha ,x)\eta }}=a(\alpha ,x)\underset{\eta \to 0}{lim}{\displaystyle \frac{f(x+\eta )-f\left(x\right)}{\eta }}=a(\alpha ,x)f^{\left(1\right)}\left(x\right).$$

(592)

Therefore, if the function $a(\alpha ,x)$ has the form (591), then we get

$$\underset{h\to 0}{lim}{\displaystyle \frac{f(x+h{c}_1^{-1}(\alpha ,x))-f\left(x\right)}{h}}=D_x^{\alpha }f\left(x\right).$$

(593)

Let us consider the case $x=0$ and $\alpha \in (0,1)$. For $x=0$, we get $a(\alpha ,0)=0$, and

$$\underset{h\to 0}{lim}{\displaystyle \frac{f(0+ha(\alpha ,0\left)\right)-f\left(0\right)}{h}},=\underset{h\to 0}{lim}{\displaystyle \frac{f\left(0\right)-f\left(0\right)}{h}}=0,$$

(594)

since $h\ne 0$ in the standard $(\epsilon ,\delta )$-definition of a limit. The NIDS derivative is defined as

$$D_x^{\alpha }f\left(x\right)={\pi }^{-\alpha /2}\Gamma (\alpha /2){|x|}^{1-\alpha }{f}^{\left(1\right)}\left(x\right),$$

(595)

where $\alpha \in (0,1)$. Then, $\left(D_x^{\alpha }f\right)\left(0\right)=0$, if $f\left(x\right)\in C_{\left(\alpha \right)}^1\left(\mathbb{R}\right)$. Using the well-known fact that the function of one variable is differentiable at a point $x=0$, if and only if it has finite derivative at $x=0$, we get that the function $f^{\left(1\right)}\left(x\right)$ has a finite limit at the point $x=0$. Therefore, we get that equality (590) holds for $x=0$ and $\alpha \in (0,1)$.

For the case $\alpha =1$, we have

$$D_x^{1}f\left(x\right)=f^{\left(1\right)}\left(x\right),$$

(596)

and

$$\underset{h\to 0}{lim}{\displaystyle \frac{f(x+ha(\alpha ,x\left)\right)-f\left(x\right)}{h}}=\underset{h\to 0}{lim}{\displaystyle \frac{f(x+h)-f\left(x\right)}{h}}=f^{\left(1\right)}\left(x\right).$$

(597)

Therefore, Equation (590) holds for the case $\alpha =1$.

As a result, Equation (590) holds for $\alpha \in (0,1]$. □

Let us note that properties of the operators of the form (590) are described in papers [151,152].

Remark 49. 

It should be noted that Equation (590) gives the representation of the NIDS derivative as a differential operator of the first order that cannot be interpreted as a fractional derivative and as a fractal derivative [151,152,153].

3.4. Fundamental Theorems of NIDS Calculus

The NIDS integral in the D-dimensional space with $D=\alpha \in (0,1]$ is

$$I_{[a,x]}^{\alpha }f\left(x\right):={\int }_a^{x}f\left(\xi \right)d{\mu }_1(\alpha ,\xi )={\int }_a^{x}f\left(\xi \right)c_1(\alpha ,\xi )d\xi ={\displaystyle \frac{{\pi }^{\alpha /2}}{\Gamma (\alpha /2)}}{\int }_a^{x}f\left(\xi \right){|\xi |}^{\alpha -1}d\xi .$$

(598)

The NIDS derivative in the D-dimensional space with $D=\alpha \in (0,1]$ is

$$D_x^{\alpha }f\left(x\right):={\displaystyle \frac{1}{c_1(\alpha ,x)}}{\displaystyle \frac{d}{dx}}f\left(x\right):={\displaystyle \frac{\Gamma (\alpha /2)}{{\pi }^{\alpha /2}}}{|x|}^{1-\alpha }{\displaystyle \frac{df\left(x\right)}{dx}}.$$

(599)

The form of the first-order NIDS derivative (599) is clearly dictated by the requirement of the consistency of integral and differential NIDS calculi. This consistency is expressed in the fulfillment of the fundamental theorems of NIDS calculus in the form of the equations

$$D_x^{\alpha }{I}_{[a,x]}^{\alpha }f\left(x\right)=f\left(x\right),$$

(600)

$$I_{[a,b]}^{\alpha }{D}_x^{\alpha }f\left(x\right)=f\left(b\right)-f\left(a\right),$$

(601)

which follow from the standard fundamental theorems of standard calculus.

Let us prove the fundamental theorems of the NIDS calculus.

Theorem 22 

(First Fundamental Theorem of NIDS Calculus). Let $W=W_1$ be the D-dimensional space with $D=\alpha \in (0,1]$ and let $\Omega =[a,b]$ be region of W.

If $c_1(\alpha ,x)f\left(x\right)\in C(\Omega )$, the NIDS derivative $D_x^{\alpha }$ and the NIDS integral $I_{[a,x]}^{\alpha }$ with $[a,x]\subset [a,b]$ satisfy the equation

$$D_x^{\alpha }{I}_{[a,x]}^{\alpha }f\left(x\right)=f\left(x\right),$$

(602)

for all $x\in (a,b)$.

Proof. 

The first fundamental theorem (FT) for operators (598) and (599) follows from the standard FT of standard calculus by the equalities

$$D_x^{\alpha }{I}_{[a,x]}^{\alpha }f\left(x\right)={\displaystyle \frac{1}{c_1(\alpha ,x)}}{\displaystyle \frac{d}{dx}}{\int }_a^{x}f\left(\xi \right)d{\mu }_1(\alpha ,\xi )=$$

$${\displaystyle \frac{1}{c_1(\alpha ,x)}}{\displaystyle \frac{d}{dx}}{\int }_a^{x}f\left(\xi \right)c_1(\alpha ,\xi )d\xi ={\displaystyle \frac{1}{c_1(\alpha ,x)}}f\left(x\right)c_1(\alpha ,x)=f\left(x\right),$$

(603)

for all $x\in (a,b)$. □

Theorem 23 

(Second Fundamental Theorem of NIDS Calculus). Let $W=W_1$ be a D-dimensional space with $D=\alpha \in (0,1]$, and let $\Omega =[a,b]$ be a region of W.

If $f\left(x\right)\in C_{\left(\alpha \right)}^1[a,b]$, the NIDS derivative $D_x^{\alpha }$ and the NIDS integral $I_{[a,x]}^{\alpha }$ satisfy the equation

$$I_{[a,x]}^{\alpha }{D}_x^{\alpha }f\left(x\right)=f\left(x\right)-f\left(a\right),$$

(604)

for all $x\in (a,b)$.

Proof. 

The second fundamental theorem (FT) for operators (598) and (599) also follows from the standard FT of standard calculus by the equalities

$$I_{[a,x]}^{\alpha }{D}^{\alpha }f\left(x\right)={\int }_a^x\left(D_s^{\alpha }f\left(s\right)\right)d{\mu }_1(\alpha ,s)=$$

$${\int }_a^x{\displaystyle \frac{1}{c_1(\alpha ,s)}}{f}^{\left(1\right)}\left(s\right)d{\mu }_1(\alpha ,s)={\int }_a^x{\displaystyle \frac{1}{c_1(\alpha ,s)}}{f}^{\left(1\right)}\left(s\right)c_1(\alpha ,s)ds=$$

$${\int }_a^x{f}^{\left(1\right)}\left(s\right)ds=f\left(x\right)-f\left(a\right),$$

(605)

for all $x\in (a,b)$. □

The multi-variable generalizations of these fundamental theorems are represented by the NIDS generalizations of the standard gradient theorem, the Gauss–Ostrogradsky theorem, and the Stokes theorem [18,19,71,75,77,78].

3.5. NIDS Integration by Parts

Let us consider the NIDS integration by parts for the D-dimensional space with $D=\alpha \in (0,1]$ for the NIDS integration on the finite interval $[a,b]$ in the form

$$I_{[a,b]}^{\alpha }f\left(x\right):={\int }_a^{b}f\left(x\right)d{\mu }_1(\alpha ,x)={\int }_a^{b}f\left(x\right)c_1(\alpha ,x)dx,$$

(606)

where $c_1(\alpha ,x)f\left(x\right)\in C[a,b]$.

Theorem 24 

(First Theorem of NIDS Integration by Parts). Let $W=W_1$ be a D-dimensional space with $D=\alpha \in (0,1]$, and let Ω be a region of W.

If the functions $f\left(x\right)$ and $g\left(x\right)$ satisfy the conditions

$$f\left(x\right)\in C_{\left(\alpha \right)}^1(\Omega ),g\left(x\right)\in C_{\left(\alpha \right)}^1(\Omega ),$$

(607)

then the equation

$${\int }_a^{b}f\left(x\right)\left(D_x^{\alpha }g\left(x\right)\right)d{\mu }_1(\alpha ,x)=$$

$$f\left(b\right)g\left(b\right)-f\left(a\right)g\left(a\right)-{\int }_a^{b}g\left(x\right)\left(D_x^{\alpha }f\left(x\right)\right)d{\mu }_1(\alpha ,x)$$

(608)

holds for all $[a,b]\subset \Omega$, where the NIDS integral and the NIDS derivative are defined by Equations (606) and (560), respectively.

Proof. 

Using Equations (598) and (599) and the standard integration by parts, we get

$${\int }_a^{b}f\left(x\right)\left(D_x^{\alpha }g\right)\left(x\right)d{\mu }_1(\alpha ,x)={\int }_a^{b}f\left(x\right)\left(D_x^{\alpha }g\right)\left(x\right)c_1(\alpha ,x)dx=$$

$${\int }_a^{b}f\left(x\right){\displaystyle \frac{1}{c_1(\alpha ,x)}}{g}^{\left(1\right)}\left(x\right)c_1(\alpha ,x)dx={\int }_a^{b}f\left(x\right)g^{\left(1\right)}\left(x\right)dx=$$

$$f\left(b\right)g\left(b\right)-f\left(a\right)g\left(a\right)-{\int }_a^{b}g\left(x\right)f^{\left(1\right)}\left(x\right)dx=$$

$$f\left(b\right)g\left(b\right)-f\left(a\right)g\left(a\right)-{\int }_a^{b}g\left(x\right){\displaystyle \frac{1}{c_1(\alpha ,x)}}{f}^{\left(1\right)}\left(x\right)c_1(\alpha ,x)dx=$$

$$f\left(b\right)g\left(b\right)-f\left(a\right)g\left(a\right)-{\int }_a^{b}g\left(x\right)\left(D_x^{\alpha }f\right)\left(x\right)d{\mu }_1(\alpha ,x),$$

(609)

for all $[a,b]\subset \Omega$. □

Theorem 25 

(Second Theorem of NIDS Integration by Parts). Let $W=W_1$ be a D-dimensional space with $D=\alpha \in (0,1]$, and let Ω be a region of W.

If the functions $f\left(x\right)$ and $g\left(x\right)$ satisfy the conditions

$$\tilde{f}\left(x\right):=c_1(\alpha ,x)f\left(x\right)\in C^1(\Omega ),g\left(x\right)\in C^1(\Omega ),$$

(610)

then the equation

$${\int }_a^{b}f\left(x\right)g^{\left(1\right)}\left(x\right)d{\mu }_1(\alpha ,x)=c_1(\alpha ,b)f\left(b\right)g\left(b\right)-c_1(\alpha ,a)f\left(a\right)g\left(a\right)-$$

$${\int }_a^{b}g\left(x\right)D_x^{\alpha }\left(c_1(\alpha ,x)f\left(x\right)\right)d{\mu }_1(\alpha ,x)$$

(611)

holds for all $[a,b]\subset \Omega$, where the NIDS integral and the NIDS derivative are defined by Equations (606) and (560), respectively.

Proof. 

Using the standard derivative in the integrand and the standard integration by parts, we obtain

$${\int }_a^{b}f\left(x\right)g^{\left(1\right)}\left(x\right)d{\mu }_1(\alpha ,x)={\int }_a^b{c}_1(\alpha ,x)f\left(x\right)g^{\left(1\right)}\left(x\right)dx=$$

$$c_1(\alpha ,b)f\left(b\right)g\left(b\right)-c_1(\alpha ,a)f\left(a\right)g\left(a\right)-{\int }_a^{b}g\left(x\right){\displaystyle \frac{d}{dx}}\left(c_1(\alpha ,x)f\left(x\right)\right)dx=$$

$$c_1(\alpha ,b)f\left(b\right)g\left(b\right)-c_1(\alpha ,a)f\left(a\right)g\left(a\right)-{\int }_a^{b}g\left(x\right)c_1^{-1}(\alpha ,x){\displaystyle \frac{d}{dx}}\left(c_1(\alpha ,x)f\left(x\right)\right)c_1(\alpha ,x)dx=$$

$$c_1(\alpha ,b)f\left(b\right)g\left(b\right)-c_1(\alpha ,a)f\left(a\right)g\left(a\right)-{\int }_a^{b}g\left(x\right)D_x^{\alpha }\left(c_1(\alpha ,x)f\left(x\right)\right)d{\mu }_1(\alpha ,x),$$

(612)

for all $[a,b]\subset \Omega$. □

Note that for the case $\alpha =1$, Equations (608) and (611) give the standard equations of the integration by parts.

4. NIDS Vector Calculus

Generalizations of differential vector operators for NIDS were first proposed in [18,19], and then, they were actively used in various areas of physics to describe fractal media, fields, and particle distributions. In this section, we define vector NIDS differential and integral operators. The NIDS generalization of the standard Gauss–Ostrogradsky theorem, the Stoke theorem, and the gradient theorem for the NIDS are proposed. These NIDS theorems provide the consistency of the NIDS vector differential operators and the NIDS integral vector operators.

4.1. Principles of NIDS Vector Calculus

In order for the formulation of the NIDS vector calculus to be self-consistent, the NIDS differential operators of the first order (such as the NIDS gradient, curl, and divergence) must be consistent with the NIDS integral operators such as the line, surface, and volume NIDS integrals. Therefore the exact form of the NIDS gradient, curl, and divergence must be such that the NIDS generalizations of the standard gradient theorem, the divergence theorem (the Gauss–Ostrogradsky theorem), and the Stokes theorem are satisfied. Therefore, one can state that the form of differential NIDS vector operators is actually determined by the explicit form of the line, surface and volume NIDS integrals and by the requirement of the existence of the fundamental theorems of the NIDS vector calculus. In this way, explicit expressions defining differential operators will be obtained in this section of this paper.

Let us give an example of how the obvious requirements for the self-consistency of vector calculus are not actually obvious and, in fact, have not been implemented in other articles on this topic. For the NIDS differential NIDS vector calculus to be self-consistent, it is obvious that expressions of second-order NIDS differential operators of NIDS vector calculus must be consistent with the first-order NIDS differential operators. However, it should be noted that Stillinger [14] and Palmer-Stavrinou [15] proposed scalar Laplace operators for NIDS that do not satisfy this obvious requirement (for details see [17,18,19]).

Principle 9 

(Principle of Self-Consistency of NIDS Vector Calculus). For the self-consistency of NIDS vector calculus, the following principles of self-consistency of the integral and differential NIDS vector calculi and the consistency of the vector calculi must be satisfied.

1. 

Self-Consistency of Integral NIDS Vector Calculus: the line and surface NIDS integrals of vector functions must be consistent with the NIDS integrals of scalar functions.

2. 

Consistency between Differential and Integral NIDS Vector Calculi:

(a) For the self-consistent formulation of the NIDS vector calculus, the definitions and explicit form of NIDS differential operators must be consistent with the definitions and explicit form of NIDS integral operators. Therefore, the definition and exact form of the NIDS vector differential operators (the NIDS generalizations of the standard gradient, curl, and divergence) must be such that the NIDS generalizations of the gradient theorem, the Gauss–Ostrogradsky theorem (the divergence theorem), and the Stokes theorem (the curl theorem) are satisfied for the NIDS.

(b) The differential NIDS vector operators (the NIDS gradient, the NIDS divergence, and the NIDS curl operator) must be derived by using the NIDS derivative of the additive set function with respect to the D-dimensional volume $V_D(\Omega )$.

3. 

Self-Consistency of Differential NIDS Vector Calculus:

(a) The use of the same nabla operator (del operator) to define all differential NIDS vector operators of the first and second order (the NIDS gradient, the NIDS divergence, the NIDS curl operator, and the scalar and vector NIDS Laplacian operators) in the product measure approach is a necessary requirement for the self-consistency of differential NIDS vector calculus.

(b) The NIDS generalizations of the scalar and vector product rules (the Leibniz rules) must be satisfied, and they must relate first-order differential NIDS operators to each other.

(c) First, the exact expressions of second-order NIDS vector operators (the scalar NIDS Laplacian and the vector NIDS Laplacian) must be consistent with the first-order NIDS vector differential operators (NIDS gradient, NIDS divergence, and NIDS curl). Second, the NIDS generalizations of the identities of vector calculus must be satisfied.

This principle of the self-consistency of NIDS vector calculus must be satisfied for the metric and product measure approaches.

The following correspondence principle of the NIDS vector calculus must also be satisfied.

Principle 10 

(Correspondence Principle of NIDS Vector Calculus). For integer values of ${\alpha }_k=1$ for all $k=1,\dots ,n$, the NIDS gradient, curl, and divergence must give standard expressions of these operators in the integer-dimensional space.

For integer values of ${\alpha }_k=1$ for all $k=1,\dots ,n$, the line and surface NIDS integrals of vector functions must give standard expressions of line and surface integrals in the integer-dimensional space.

In fact, Principle 10 is fulfilled, due to the fact that the functions $c_1({\alpha }_k,x^k)=1$ for all $k=1,\dots ,n$, if ${\alpha }_k=1$ for all $k=1,\dots ,n$.

4.2. NIDS Differential Vector Operators in the Metric Approach

The NIDS vector calculus of the differential vector operators for NIDS, including the NIDS del operator, the NIDS gradient, the NIDS divergence, and the NIDS curl operator, was first proposed in the 2014/5 papers [18,19] and then used in various branches of physics to describe fractal media and fields.

Using the natural NIDS metric $g_{kl}\left(\mathbf{x}\right)=g_{kk}\left(\mathbf{x}\right){\delta }_{kl}$ and the orthogonal coordinates $\mathbf{x}=(x_1,\dots ,x_n)$, we can express the NIDS gradient, NIDS divergence, and NIDS curl operator in terms of the normalized bases ${\mathbf{e}}^k$ and ${\mathbf{e}}_k$ and the scale factors

$$h_k=\sqrt{g_{kk}}=c_1({\alpha }_k,x_k)(k=1,\dots ,n).$$

(613)

Therefore, in the metric approach, we can use the well-known equations

$${Grad}_{\left(\alpha \right)}U=\sum _{k=1}^n{\displaystyle \frac{1}{h_k}}{\displaystyle \frac{\partial U}{\partial x^k}}{\mathbf{e}}^k,$$

(614)

$${Div}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right):=\sum _{k=1}^n{\displaystyle \frac{1}{{\prod }_{j=1}^n{h}_j}}{\displaystyle \frac{\partial }{\partial x^k}}\left(A^k\left(\mathbf{x}\right)\prod _{j=1,j\ne k}^n{h}_j\right),$$

(615)

$${Curl}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right):=\sum _{i,j,k=1}^3{\mathbf{e}}_i{\epsilon }_{ijk}{\displaystyle \frac{h_i}{{\prod }_{s=1}^3{h}_s}}{\displaystyle \frac{\partial }{\partial x^j}}\left(A^k\left(\mathbf{x}\right)h_k\right),(n=3),$$

(616)

where ${\epsilon }_{ijk}$ is the Levi–Civita symbol. If any two indices are equal, the symbol is equal to zero. If all indices are unequal, then ${\epsilon }_{ijk}:={(-1)}^p{\epsilon }_{123}={(-1)}^p$, where p is the parity of the permutation, and

$${\epsilon }_{ijk}=\left\{\begin{array}{cc}+1\hfill & \mathit{if}\left(ijk\right)\mathrm{is}\mathrm{an}\mathrm{even}\mathrm{permutation}\mathrm{of}\left(123\right),\hfill \\ -1\hfill & \mathit{if}\left(ijk\right)\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{permutation}\mathrm{of}\left(123\right),\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(617)

Using $h_k=c_1({\alpha }_k,x_k)$, we have

$${\displaystyle \frac{\partial h_k}{\partial x^j}}=0\left(\mathit{if}k\ne j\right),$$

(618)

and Equations (614)–(616) can be simplified.

For the D-dimensional space $W:=W_1\times \dots \times W_n$, the metric approach was proposed in [18] to define the NIDS nabla operator (the NIDS del operator) in the form

$${\nabla }_{\left(\alpha \right)}=\sum _{k=1}^n{\mathbf{e}}_k{\displaystyle \frac{1}{c_1({\alpha }_k,x^k)}}{\displaystyle \frac{\partial }{\partial x_k}},$$

(619)

where

$$c_1({\alpha }_k,x_k)={\displaystyle \frac{{\pi }^{{\alpha }_k/2}}{\Gamma ({\alpha }_k/2)}}{|x_k|}^{{\alpha }_k-1},$$

(620)

$\left(\alpha \right)=({\alpha }_1,\dots ,{\alpha }_n)$ is the multi-index, and $c_1({\alpha }_k,x_k)$ is defined by (620) for the non-integer dimensionality along the $X_k$-axis. Note that, from a physical point of view, the functions $c_1({\alpha }_k,x_k)$ describe the change in the density of states [18,19,69,87] in D-dimensional spaces compared to the density of states in the spaces with integer dimensions.

The NIDS generalization of the gradient, divergence, curl operator, and the scalar and vector Laplace operators [154] was defined in [18,19]. Note that all these NIDS operators are expressed through function (620) and the NIDS del operator [18].

Let us define the NIDS continuously differentiable functions in the metric approach.

Definition 22 

(NIDS Continuously Differentiable Functions). Let $W=W_1\times \dots \times W_n$ be a D-dimensional space with $D={\alpha }_1+\dots +{\alpha }_n$, and let Ω be a region of W.

In the metric approach, the function $f\left(\mathbf{x}\right)$ with $\mathbf{x}\in \Omega$ will be called the NIDS continuously differentiable function, if

$$D_{x^k}^{{\alpha }_k}f\left(\mathbf{x}\right):={\displaystyle \frac{1}{c_1({\alpha }_k,x_k)}}{\displaystyle \frac{\partial f\left(\mathbf{x}\right)}{\partial x^k}}\in C(\Omega ),$$

(621)

for all $\mathbf{x}\in \Omega$ and all $k=1,\dots ,n$. The set of such functions will be denoted as $C_{\left(\alpha \right)}^1(\Omega )$, where $\left(\alpha \right)=({\alpha }_1,\dots ,{\alpha }_n)$.

The function $f\left(\mathbf{x}\right)$ with $\mathbf{x}\in \Omega$ will be called the $C_{\left(\alpha \right)}^m$-function, if all m-order partial NIDS derivatives

$$D_{\mathbf{x}}^{\left(\alpha \right)}f\left(\mathbf{x}\right):=D_{x^{k1}}^{{\alpha }_{k1}}\dots D_{x^{km}}^{{\alpha }_{km}}f\left(\mathbf{x}\right)\in C(\Omega )$$

(622)

are continuous for all $\mathbf{x}\in \Omega$. In this case, we will write $f\left(\mathbf{x}\right)\in C_{\left(\alpha \right)}^m(\Omega )$.

Let us give definitions of the NIDS differential vector operators in the D-dimensional space in the metric approach that are proposed in [18].

Definition 23 

(Vector Operators in Metric Approach). Let $W=W_1\times \dots \times W_n$ be a D-dimensional space with $D={\alpha }_1+\dots +{\alpha }_n$, and let Ω be a region of W. Then, in the metric approach, the first order NIDS differential operators of the NIDS vector calculus are defined in the following forms.

The NIDS gradient of scalar function $U=U\left(\mathbf{x}\right)\in C_{\left(\alpha \right)}^1(\Omega )$ is the vector field

$${Grad}_{\left(\alpha \right)}U\left(\mathbf{x}\right)=\sum _{k=1}^n{\displaystyle \frac{1}{c_1({\alpha }_k,x_k)}}{\displaystyle \frac{\partial U\left(\mathbf{x}\right)}{\partial x_k}}{\mathbf{e}}_k,$$

(623)

where $\left(\alpha \right)=({\alpha }_1,\dots ,{\alpha }_n)$.

The NIDS divergence of the vector field $\mathbf{A}\left(\mathbf{x}\right)={\sum }_{k=1}^n{A}^k\left(\mathbf{x}\right){\mathbf{e}}_k$ with all $A^k\left(\mathbf{x}\right)\in C_{\left(\alpha \right)}^1(\Omega )$ is the scalar field

$${Div}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right)=\sum _{k=1}^n{\displaystyle \frac{1}{c_1({\alpha }_k,x_k)}}{\displaystyle \frac{\partial A_k\left(\mathbf{x}\right)}{\partial x_k}},$$

(624)

where $\left(\alpha \right)=({\alpha }_1,\dots ,{\alpha }_n)$.

The NIDS curl operator for vector field $\mathbf{A}\left(\mathbf{x}\right)={\sum }_{k=1}^3{A}^k\left(\mathbf{x}\right){\mathbf{e}}_k$ with all $A^k\left(\mathbf{x}\right)\in C_{\left(\alpha \right)}^1(\Omega )$ in D-dimensional space $W=W_1\times W_3\times W_3$ is the vector field

$${Curl}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right)=\sum _{i,j,k=1}^3{\mathbf{e}}_i{\epsilon }_{ijk}{\displaystyle \frac{1}{c_1({\alpha }_j,x_j)}}{\displaystyle \frac{\partial A_k\left(\mathbf{x}\right)}{\partial x_j}},$$

(625)

where $\left(\alpha \right)=({\alpha }_1,{\alpha }_2,{\alpha }_3)$, ${\epsilon }_{ijk}$ is the Levi–Civita symbol, and we take into account that $\partial c_1({\alpha }_k,x_k)/\partial x_l=0$ for $k\ne l$.

Using (623) and (624), we can define the second-order NIDS differential operators such as the scalar NIDS Laplacian

$${}^S\Delta _{\left(\alpha \right)}U\left(\mathbf{x}\right):={Div}_{\left(\alpha \right)}{Grad}_{\left(\alpha \right)}U\left(\mathbf{x}\right)=$$

$$\sum _{k=1}^n{\displaystyle \frac{1}{c_1({\alpha }_k,x_k)}}{\displaystyle \frac{\partial }{\partial x_k}}\left({\displaystyle \frac{1}{c_1({\alpha }_k,x_k)}}{\displaystyle \frac{\partial U\left(\mathbf{x}\right)}{\partial x_k}}\right)=$$

$$\sum _{k=1}^3{\displaystyle \frac{1}{c_1^2({\alpha }_k,x_k)}}\left({\displaystyle \frac{{\partial }^{2}U\left(\mathbf{x}\right)}{\partial x_k^2}}-{\displaystyle \frac{{\alpha }_k-1}{x_k}}{\displaystyle \frac{\partial U\left(\mathbf{x}\right)}{\partial x_k}}\right)=$$

$$\sum _{k=1}^3{\displaystyle \frac{1}{c_1^2({\alpha }_k,x_k)}}\left({\displaystyle \frac{{\partial }^{2}U\left(\mathbf{x}\right)}{\partial x_k^2}}+{\displaystyle \frac{1-{\alpha }_k}{x_k}}{\displaystyle \frac{\partial U\left(\mathbf{x}\right)}{\partial x_k}}\right),$$

(626)

where we use (620) and $U\left(\mathbf{x}\right)\in C_{\left(\alpha \right)}^2(\Omega )$.

Using (623)–(625), we can define the vector NIDS Laplacian

$${}^V\Delta _{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right):={Grad}_{\left(\alpha \right)}{Div}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right)-{Curl}_{\left(\alpha \right)}{Curl}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right)$$

(627)

for $\Omega \subset W=W_1\times W_2\times W_3$ and $\mathbf{A}\left(\mathbf{x}\right)\in C_{\left(\alpha \right)}^2(\Omega )$. For $\Omega \subset W=W_1\times \dots \times W_n$ and $\mathbf{A}\left(\mathbf{x}\right)\in C_{\left(\alpha \right)}^2(\Omega )$, the vector NIDS Laplacian is the vector field

$${}^V\Delta _{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right):=\sum _{k=1}^n{\mathbf{e}}_k{}^S\Delta _{\left(\alpha \right)}Ak\left(\mathbf{x}\right).$$

(628)

For $n=3$, using the NIDS del operator (619) and the NIDS generalization of the Lagrange formula

$$\left[{\nabla }_{\left(\alpha \right)}\times \left[{\nabla }_{\left(\alpha \right)}\times \mathbf{A}\right]\right]={\nabla }_{\left(\alpha \right)}\left({\nabla }_{\left(\alpha \right)},\mathbf{A}\right)-\left({\nabla }_{\left(\alpha \right)},{\nabla }_{\left(\alpha \right)}\right)\mathbf{A},$$

(629)

we get Equation (627).

Remark 50. 

Note that definitions of the NIDS differential operators of the NIDS vector calculus for spherical and cylindrical coordinates are proposed in [19].

In the metric approach, the following NIDS analogs of the standard vector calculus identities

$${Curl}_{\left(\alpha \right)}{Grad}_{\left(\alpha \right)}U\left(\mathbf{x}\right)=0,$$

(630)

$${Div}_{\left(\alpha \right)}{Curl}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right)=0$$

(631)

hold for all $U\left(\mathbf{x}\right)\in C_{\left(\alpha \right)}^2(\Omega )$ and all $\mathbf{A}\left(\mathbf{x}\right)\in C_{\left(\alpha \right)}^2(\Omega )$. The proof of identities (630) and (631) follows immediately from the anti-symmetry property of the Christoffel symbol. Detailed proofs of these identities are given in Section 4.5.3 for the product measure and metric approaches.

Remark 51. 

Note that operators (623)–(627) are differential operators of the first and second order that are defined for differentiable functions. The NIDS gradient (623), the NIDS divergence (624), and the NIDS curl operator (625) are differential operators of the first order. The scalar NIDS Laplacian (626) and the vector NIDS Laplacian (627) are the second-order differential operators.

It should be emphasized that the NIDS operators (623)–(627) do not coincide with the Laplace operators proposed by Stillinger [14] and by Palmer and Stavrinou [15]. It should also be emphasized that the NIDS vector calculus was not proposed in the work [14,15]. The NIDS vector calculus was first proposed in [18,19] and was subsequently applied to describe fractal physical processes and media.

The first advantage of the suggested approach and the proposed NIDS Laplace operators (626) and (627) are defined via the NIDS gradient, NIDS divergence, and the NIDS curl operator. This fact allows us to state the consistency between the first-order and second-order NIDS differential operators.

The second advantage is based on the consistency between the NIDS differential vector operators and the NIDS integral operators. This consistency is realized by the NIDS generalizations of the Gauss–Ostrogradsky theorem, the Stoke theorem, and the gradient theorem for the NIDS vector operators. These theorems are considered and proved in the following subsection for the metric approach.

Remark 52. 

In the case of parallelism of all lines to the coordinate axes and all surfaces to coordinate planes, the definitions of the NIDS gradient, the NIDS curl, and the NIDS divergence coincide in the product measure and metric approaches.

It should be emphasized that the product measure approach, unlike the metric approach, allows one to obtain standard expressions for the circle length and area of semi-sphere in the D-dimensional space for all values of $D\in (0,n]$ and all $n\in \mathbb{N}$.

4.3. Line and Surface NIDS Integral of Vector Fields

In the principle of self-consistency of NIDS vector calculus (Principle 9), we proposed that the line and surface NIDS integrals of vector fields must be consistent with the NIDS integrals of scalar fields. Using this requirement, we can define the line and surface NIDS integrals of vector fields.

4.3.1. Line NIDS Integral of Vector Fields

The line NIDS integration of a vector function can be defined via the line NIDS integration of a scalar function by using Definition 12. Let us give a definition of the line NIDS integration of a vector function in the product measure and metric approaches.

Definition 24 

(Line NIDS Integral of Vector Field). Let the function $\mathbf{F}\left(\mathbf{x}\right)$ be defined and integrable along the curve L in the region U of the D-dimensional space $W=W_1\times \dots \times W_n$ with $D={\alpha }_1+\dots +{\alpha }_n$. Let $L\subset U$ be a piecewise smooth curve that is described by the equations $\mathbf{x}=\mathbf{x}\left(t\right)$ with $t\in [a,b]$, such that $|{\mathbf{x}}_t^{\left(1\right)}\left(t\right)|\ne 0$ for all $t\in [a,b]$.

(1) In the product measure approach, the line NIDS integral of vector field $\mathbf{F}=\mathbf{F}\left(\mathbf{x}\right)$ is defined as the line NIDS integral of the scalar field

$$F_{\tau }\left(\mathbf{x}\right)=\left(\mathbf{F},\mathbf{t}\right)=\sum _{k=1}^n{F}_k\left(\mathbf{x}\right)t_k\left(\mathbf{x}\right),$$

(632)

which is the tangential component of the vector field $\mathbf{F}\left(\mathbf{x}\right)$, by the equations

$$\left({\mathbf{I}}_L^D,\mathbf{F}\right):=I_L^D{F}_{\tau }\left(\mathbf{x}\right):={\int }_a^b{F}_{\tau }\left(\mathbf{x}\left(t\right)\right)\sqrt{\sum _{k=1}^n{\left({\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt,$$

(633)

where

$${\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)):=c_1({\alpha }_k,x^k\left(t\right))\prod _{j=1,j\ne k}^n{\widehat{c}}_1({\alpha }_j,x^j\left(t\right)),$$

(634)

with

$${\widehat{c}}_1({\alpha }_j,x^j\left(t\right)):=\left\{\begin{array}{cc}1\hfill & {\displaystyle \mathit{if}d{x}^j\left(t\right)/dt=0,}\hfill \\ c_1({\alpha }_j,x^j\left(t\right))\hfill & {\displaystyle \mathit{if}d{x}^j\left(t\right)/dt\ne 0.}\hfill \end{array}\right.$$

(635)

The vector $\mathbf{t}\left(t\right)$ is the unit tangent vector

$$\mathbf{t}\left(t\right)={\displaystyle \frac{\mathbf{T}\left(\mathbf{x}\right(t\left)\right)}{|\mathbf{T}(\mathbf{x}\left(t\right))|}}={\displaystyle \frac{{\mathbf{x}}_t^{\left(1\right)}\left(t\right)}{|{\mathbf{x}}_t^{\left(1\right)}\left(t\right)|}},|{\mathbf{x}}_t^{\left(1\right)}\left(t\right)|=\sqrt{\sum _{k=1}^n{\left({\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2},$$

(636)

and $|\mathbf{T}\left(x\left(t\right)\right)|=|{\mathbf{x}}_t^{\left(1\right)}\left(t\right)|\ne 0$ for all $t\in [a,b]$, where $\mathbf{T}\left(\mathbf{x}\left(t\right)\right)={\mathbf{x}}_t^{\left(1\right)}\left(t\right)$ is the unit tangent vector of the line $\mathbf{x}\left(t\right)$.

(2) In the metric approach, the line NIDS integral of the vector field is defined by Equation (633), in which we must use the function $c_1({\alpha }_k,x^k\left(t\right))$ instead of ${\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right))$ for all $k=1,\dots ,n$ and all $t\in [a,b]$. The line NIDS integral of vector field $\mathbf{F}\left(\mathbf{x}\right)$ is defined as

$$\left({\mathbf{I}}_L^{D,\ast },\mathbf{F}\right):=I_L^{D.\ast }{F}_{\tau }\left(\mathbf{x}\right):={\int }_a^b{F}_{\tau }\left(\mathbf{x}\left(t\right)\right)\sqrt{\sum _{k=1}^n{\left(c_k({\alpha }_k,x^k\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt,$$

(637)

for all $D={\alpha }_1+\dots +{\alpha }_n$.

Let us give an example of the line NIDS integral of a vector field in the case $n=2$ for the product measure and the metric approaches.

Example 8. 

Let the function $\mathbf{F}(x,y)$ be defined and integrable along the curve L in the region U of the D-dimensional space $W=W_1\times W_2$ with $D={\alpha }_1+{\alpha }_2$. Let $L\subset U$ be a piecewise smooth curve that is described by the equations $x=x\left(t\right)$ and $y=y\left(t\right)$ with $t\in [a,b]$, such that $|{\mathbf{x}}_t^{\left(1\right)}\left(t\right)|\ne 0$ for all $t\in [a,b]$.

In the product measure approach, the line NIDS integral of a vector field is described as

$$\left({\mathbf{I}}_L^D,\mathbf{F}\right):={\int }_a^b{F}_{\tau }(x\left(t\right),y\left(t\right))·$$

$$\sqrt{c_1^2({\alpha }_1,x\left(t\right)){\widehat{c}}_1^2({\alpha }_2,y\left(t\right)){\left({\displaystyle \frac{dx\left(t\right)}{dt}}\right)}^2+{\widehat{c}}_1^2({\alpha }_1,x\left(t\right))c_1^2({\alpha }_2,y\left(t\right)){\left({\displaystyle \frac{dy\left(t\right)}{dt}}\right)}^2}dt$$

(638)

and in the metric approach as

$$\left({\mathbf{I}}_L^{D,\ast },\mathbf{F}\right):={\int }_a^b{F}_{\tau }(x\left(t\right),y\left(t\right))\sqrt{c_1^2({\alpha }_1,x\left(t\right)){\left({\displaystyle \frac{dx\left(t\right)}{dt}}\right)}^2+c_1^2({\alpha }_2,y\left(t\right)){\left({\displaystyle \frac{dy\left(t\right)}{dt}}\right)}^2}dt,$$

(639)

where

$${\widehat{c}}_1({\alpha }_1,x\left(t\right)):=\left\{\begin{array}{cc}1\hfill & {\displaystyle \mathit{if}dx\left(t\right)/dt=0,}\hfill \\ c_1({\alpha }_1,x\left(t\right))\hfill & {\displaystyle \mathit{if}dx\left(t\right)/dt\ne 0,}\hfill \end{array}\right.$$

(640)

$${\widehat{c}}_1({\alpha }_2,y\left(t\right)):=\left\{\begin{array}{cc}1\hfill & {\displaystyle \mathit{if}dy\left(t\right)/dt=0,}\hfill \\ c_1({\alpha }_2,y\left(t\right))\hfill & {\displaystyle \mathit{if}dy\left(t\right)/dt\ne 0,}\hfill \end{array}\right.$$

(641)

$$c_1({\alpha }_1,x\left(t\right))={\displaystyle \frac{{\pi }^{{\alpha }_1}}{\Gamma \left({\alpha }_1\right)}}{|x\left(t\right)|}^{{\alpha }_2-1},c_1({\alpha }_2,y\left(t\right))={\displaystyle \frac{{\pi }^{{\alpha }_2}}{\Gamma \left({\alpha }_2\right)}}{|y\left(t\right)|}^{{\alpha }_2-1}.$$

(642)

The tangential component $F_{\tau }(x\left(t\right),y\left(t\right))$ of the vector field $\mathbf{F}\left(\mathbf{x}\right)$ is described by the standard equation, that is, the same equation as in the calculus for the integer dimension.

Let us prove theorems about other representations of the line NIDS integrals for the vector field, whose equations are equivalent to Equation (633).

Theorem 26 

(First Equivalent Representation of Line NIDS integral). Let the function $\mathbf{F}\left(\mathbf{x}\right)$ be defined and integrable along the curve L in the region U of the D-dimensional space $W=W_1\times \dots \times W_n$ with $D={\alpha }_1+\dots +alph{a}_n$. Let $L\subset U$ be a piecewise smooth curve that is described by the equations $\mathbf{x}=\mathbf{x}\left(t\right)$ with $t\in [a,b]$, such that $|{\mathbf{x}}_t^{\left(1\right)}\left(t\right)|\ne 0$ for all $t\in [a,b]$.

In the product measure approach, the line NIDS integral of vector field $\mathbf{F}=\mathbf{F}\left(\mathbf{x}\right)$, which is defined by Equation (633), can be represented as

$$\left({\mathbf{I}}_L^D,\mathbf{F}\right):=I_L^D{F}_{\tau }\left(\mathbf{x}\right):={\int }_a^b{\widehat{c}}_n(D,\mathbf{x}\left(t\right))F_{\tau }\left(\mathbf{x}\left(t\right)\right)\sqrt{\sum _{k=1}^n{\left({\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt,$$

(643)

where

$${\widehat{c}}_n(D,\mathbf{x}\left(t\right)):=\prod _{j=1}^n{\widehat{c}}_1({\alpha }_j,x^j\left(t\right)),$$

(644)

and ${\widehat{c}}_1({\alpha }_j,x^j\left(t\right))$ is defined by Equation (635).

Proof. 

Using Equation (635), we get the equality

$${\widehat{c}}_1({\alpha }_k,x^k\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}=c_1({\alpha }_k,x^k\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}.$$

(645)

Then, from Equations (634) and (645), we get

$${\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}={\widehat{c}}_n(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}.$$

(646)

As a result,

$$I_L^D{F}_{\tau }\left(\mathbf{x}\right)={\int }_a^b{F}_{\tau }\left(\mathbf{x}\left(t\right)\right)\sqrt{\sum _{k=1}^n{\left({\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt=$$

$${\int }_a^b{F}_{\tau }\left(\mathbf{x}\left(t\right)\right)\sqrt{\sum _{k=1}^n{\left({\widehat{c}}_n(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt=$$

$${\int }_a^b{F}_{\tau }\left(\mathbf{x}\left(t\right)\right){\widehat{c}}_n(D,\mathbf{x}\left(t\right))\sqrt{\sum _{k=1}^n{\left({\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt$$

(647)

for all $D\in (0,n]$. □

Let us prove the theorem about another representation of the line NIDS integrals for the vector field, where these integrals can be described as

$$\left({\mathbf{I}}_L^D,\mathbf{F}\right)={\int }_L{\widehat{c}}_n(D,\mathbf{x})\left(\mathbf{F}\left(\mathbf{x}\right),d\mathbf{x}\right):={\int }_a^b{\widehat{c}}_n(D,\mathbf{x}\left(t\right))\left(\mathbf{F}\left(\mathbf{x}\left(t\right)\right),{\displaystyle \frac{d\mathbf{x}\left(t\right)}{dt}}\right)dt,$$

(648)

where $(\mathbf{A},\mathbf{B})$ is the scalar product of the vector fields $\mathbf{A}\left(\mathbf{x}\right)$ and $\mathbf{B}\left(\mathbf{x}\right)$.

Theorem 27 

(Second Equivalent Representation of Line NIDS integral). Let $\mathbf{F}\left(\mathbf{x}\right)$ be a vector field in the region U of the D-dimensional space $W=W_1\times \dots \times W_n$ with $D={\alpha }_1+\dots +{\alpha }_n$, and let $L\subset U$ be a piecewise smooth curve that is described by the equations $\mathbf{x}=\mathbf{x}\left(t\right)$ with $t\in [a,b]$, such that $|{\mathbf{x}}_t^{\left(1\right)}\left(t\right)|\ne 0$ for all $t\in [a,b]$.

(1) In the product measure approach, the line NIDS integral of the vector field $\mathbf{F}\left(\mathbf{x}\right)$ along the line L can be represented as

$$\left({\mathbf{I}}_L^D,\mathbf{F}\right)={\int }_a^b\left(\sum _{k=1}^n{F}_k\left(\mathbf{x}\left(t\right)\right){\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)dt,$$

(649)

or

$$\left({\mathbf{I}}_L^D,\mathbf{F}\right)={\int }_a^b{\widehat{c}}_n(D,\mathbf{x}\left(t\right))\left(\mathbf{F}\left(\mathbf{x}\left(t\right)\right),{\displaystyle \frac{d\mathbf{x}\left(t\right)}{dt}}\right)dt,$$

(650)

where

$$\left(\mathbf{F}\left(\mathbf{x}\left(t\right)\right),{\displaystyle \frac{d\mathbf{x}\left(t\right)}{dt}}\right):=\sum _{k=1}^n{F}_k\left(\mathbf{x}\left(t\right)\right){\displaystyle \frac{d{x}^k\left(t\right)}{dt}},$$

(651)

and ${\widehat{c}}_n(D,\mathbf{x})$ and ${\widehat{c}}_{n,k}(D,\mathbf{x})$ are defined by Equations (634) and (644).

(2) In the metric approach, the line NIDS integral of the vector field is defined by Equation (649), in which we must use the function $c_1({\alpha }_k,x^k\left(t\right))$ instead of ${\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right))$ for all $k=1,\dots ,n$ and all $t\in [a,b]$. The line NIDS integral of vector field $\mathbf{F}\left(\mathbf{x}\right)$ is defined as

$$\left({\mathbf{I}}_L^{D,\ast },\mathbf{F}\right)={\int }_a^b\left(\sum _{k=1}^n{F}_k\left(\mathbf{x}\left(t\right)\right)c_1({\alpha }_k,x^k\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)dt,$$

(652)

for all $D={\alpha }_1+\dots +{\alpha }_n$.

Proof. 

Using unit tangent vector (636), we get

$${\int }_L{\widehat{c}}_n(D,\mathbf{x})\left(\mathbf{F}\left(\mathbf{x}\right),d\mathbf{x}\right):=$$

$${\int }_a^b{\widehat{c}}_n(D,\mathbf{x}\left(t\right))\left(\mathbf{F}\left(\mathbf{x}\left(t\right)\right),{\displaystyle \frac{d\mathbf{x}\left(t\right)}{dt}}\right)dt=$$

$${\int }_a^b{\widehat{c}}_n(D,\mathbf{x}\left(t\right))\left(\mathbf{F}\left(\mathbf{x}\left(t\right)\right),{\displaystyle \frac{{\mathbf{x}}_t^{\left(1\right)}\left(t\right)}{|{\mathbf{x}}_t^{\left(1\right)}\left(t\right)|}}\right)|{\mathbf{x}}_t^{\left(1\right)}\left(t\right)|dt=$$

$${\int }_a^b{\widehat{c}}_n(D,\mathbf{x}\left(t\right))\left(\mathbf{F}\left(\mathbf{x}\right(t\left)\right),\mathbf{t}\left(t\right)\right)\sqrt{\sum _{k=1}^n{\left({\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt=$$

(653)

$${\int }_a^b{\widehat{c}}_n(D,\mathbf{x}\left(t\right))F_{\tau }\left(\mathbf{x}\left(t\right)\right)\sqrt{\sum _{k=1}^n{\left({\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt,$$

(654)

where $\mathbf{x}\left(a\right)$ and $\mathbf{x}\left(b\right)$ are the endpoints of the line L. This proves Equation (650).

Equation (649) follows from Equation (650) by the transformations similar to transformations in the proof of Theorem 26. □

Remark 53. 

Let us consider the special case, when the piecewise smooth line L in D-dimensional space consists only of segments parallel to the coordinate axes $X_k$. Then,

$${\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right))=c_1({\alpha }_k,x^k\left(t\right)),$$

(655)

for all $k=1,\dots ,n$ by definition of ${\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right))$, since $d{x}^j\left(t\right)/dt=0$ for all $j\ne k$, and $t\in [a_k,b_k]\subset [a,b]$ for the case when the segments $\mathbf{x}\left(t\right)$ with $t\in [a_k,b_k]$ are parallel to the coordinate axes $X_k$.

In this case, the line NIDS integral of the vector field is written as

$$\left({\mathbf{I}}_L^D,\mathbf{F}\right)={\int }_a^b\left(\sum _{k=1}^n{F}_k\left(\mathbf{x}\left(t\right)\right){\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)dt=$$

$${\int }_a^b\left(\sum _{k=1}^n{F}_k\left(\mathbf{x}\left(t\right)\right)c_1({\alpha }_k,x_k\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)dt.$$

(656)

Let us assume that the piecewise smooth line L in D-dimensional space consists only of segments $L_k$ that are parallel to the coordinate axes $X_k$, and

$$L=\bigcup _{k=1}^n{L}_k,$$

(657)

where

$$L_k:=\{\mathbf{x}:\mathbf{x}=\mathbf{x}\left(t\right)t\in [a_k,b_k]\}.$$

(658)

If the segment $L_k$ is parallel to the coordinate axes $X_k$, then $d{x}^k\left(t\right)/dt\ne 0$ for $t\in [a_k,b_k]\subset [a,b]$, and $d{x}^j\left(t\right)/dt=0$ for all $j\ne k$ and $t\in [a_k,b_k]\subset [a,b]$. Then, Equation (656) gives

$$\left({\mathbf{I}}_L^D,\mathbf{F}\right)=\sum _{k=1}^n{\int }_{L_k}\left(F_k\left(\mathbf{x}\right)d{\mu }_1({\alpha }_k,x_k)\right).$$

(659)

In the metric approach, the line NIDS integral of the vector field is written by Equation (656),

$$\left({\mathbf{I}}_L^{D,\ast },\mathbf{F}\right)={\int }_a^b\left(\sum _{k=1}^n{F}_k\left(\mathbf{x}\left(t\right)\right)c_1({\alpha }_k,x_k\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)dt$$

(660)

in the general case, even when the line segments are not parallel to the coordinate axes.

Remark 54. 

Note that, in the metric approach, we can use the equation with ${\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right))$ of the product measure approach. In this case, the equations of the line NIDS integral must use ${\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right))=c_1({\alpha }_k,x^k\left(t\right))$ for all cases, even when the line segments are not parallel to the coordinate axes.

As a result, notations and equations with ${\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right))$, which is used in the product measure approach, can be considered as universal notations that can be used in both approaches, implying by this the following designations.

• For the product measure approach,

  $${\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)):=c_1({\alpha }_k,x^k\left(t\right))\prod _{j=1,j\ne k}^n{\widehat{c}}_1({\alpha }_j,x^j\left(t\right)).$$

  (661)
• For the metric approach,

  $${\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)):=c_1({\alpha }_k,x^k\left(t\right)).$$

  (662)

This allows us to prove some statements for both approaches simultaneously, using only the notations of the product measure approach.

Using Equation (649), one can give a new representation of the line NIDS integration in the product measure approach. To do this, we propose a new tangent vector along a line that will be called the NIDS tangent vector along the curved line in D-dimensional space.

Definition 25. 

Let $W=W_1\times \dots \times W_n$ be the D-dimensional space with $D={\alpha }_1+\dots +{\alpha }_n$, and let $L\subset U$ be a piecewise smooth curve in W that is described by the equations $\mathbf{x}=\mathbf{x}\left(t\right)$ with $t\in [a,b]$, such that $|{\mathbf{x}}_t^{\left(1\right)}\left(t\right)|\ne 0$ for all $t\in [a,b]$.

Then, the NIDS tangent vector along a curved line L is defined as the vector $\mathbf{t}\left(t\right)$

$$\widehat{\mathbf{T}}\left(t\right):={\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)){\mathbf{x}}_t^{\left(1\right)}\left(t\right)={\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}},$$

(663)

the unit NIDS tangent vector is

$$\widehat{\mathbf{t}}\left(t\right):={\displaystyle \frac{\widehat{\mathbf{T}}\left(t\right)}{|\widehat{\mathbf{T}}\left(t\right)|}},|\widehat{\mathbf{T}}\left(t\right)|=\sqrt{\sum _{k=1}^n{\left({\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2},$$

(664)

and $|\widehat{\mathbf{T}}\left(t\right)|\ne 0$ for all $t\in [a,b]$, where ${\widehat{c}}_{n,k}(D,\mathbf{x})$ is defined by the equation and (634).

The NIDS tangent vectors allow us to propose a new representation of the line NIDS integral of the vector field in the product measure approach.

Theorem 28 

(Third Equivalent Representation of Line NIDS integral). Let $\mathbf{F}\left(\mathbf{x}\right)$ be a vector field in the region U of the D-dimensional space $W=W_1\times \dots \times W_n$ with $D={\alpha }_1+\dots +{\alpha }_n$, and let $L\subset U$ be a piecewise smooth curve that is described by the equations $\mathbf{x}=\mathbf{x}\left(t\right)$ with $t\in [a,b]$, such that $|{\mathbf{x}}_t^{\left(1\right)}\left(t\right)|\ne 0$ for all $t\in [a,b]$.

Then, in the product measure approach, the line NIDS integral of the vector field $\mathbf{F}\left(\mathbf{x}\right)$ along the line L can be represented as

$$\left({\mathbf{I}}_L^D,\mathbf{F}\right)=I_L^D\left(\mathbf{F},\widehat{\mathbf{T}}\right)={\int }_a^b{\widehat{F}}_{\tau }\left(\mathbf{x}\left(t\right)\right)\sqrt{\sum _{k=1}^n{\left({\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt,$$

(665)

where ${\widehat{F}}_{\tau }\left(\mathbf{x}\left(t\right)\right)$ is projection of the vector $\mathbf{F}$ onto the NIDS tangent vector

$${\widehat{F}}_{\tau }\left(\mathbf{x}\left(t\right)\right):=\left(\mathbf{F}\left(\mathbf{x}\left(t\right)\right),\widehat{\mathbf{t}}\left(t\right)\right)=\left(\mathbf{F}\left(\mathbf{x}\left(t\right)\right),{\displaystyle \frac{\widehat{\mathbf{T}}\left(t\right)}{|\widehat{\mathbf{T}}\left(t\right)|}}\right),$$

(666)

and the vectors $\widehat{\mathbf{T}}\left(t\right)$ and $\widehat{\mathbf{t}}\left(t\right)$ are defined by Equations (663) and (664).

Proof. 

Using unit tangent vector (663) and unit tangent vector (664), Equation (649) gives

$$\left({\mathbf{I}}_L^D,\mathbf{F}\right):={\int }_a^b\left(\mathbf{F}\left(\mathbf{x}\left(t\right)\right),{\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d\mathbf{x}\left(t\right)}{dt}}\right)dt=$$

$${\int }_a^b\left(\mathbf{F}\left(\mathbf{x}\left(t\right)\right),\widehat{\mathbf{T}}\left(t\right))dt={\int }_a^b(\mathbf{F}\left(\mathbf{x}\left(t\right)\right),{\displaystyle \frac{\widehat{\mathbf{T}}\left(t\right)}{|\widehat{\mathbf{T}}\left(t\right)|}}\right)|\widehat{\mathbf{T}}\left(t\right)|dt=$$

$${\int }_a^b\left(\mathbf{F}\left(\mathbf{x}\left(t\right)\right),\widehat{\mathbf{t}}\left(t\right)\right)\sqrt{\sum _{k=1}^n{\left({\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt=$$

$${\int }_a^b{\widehat{F}}_{\tau }\left(\mathbf{x}\left(t\right)\right)\sqrt{\sum _{k=1}^n{\left({\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt,$$

(667)

where $\mathbf{x}\left(a\right)$ and $\mathbf{x}\left(b\right)$ are the endpoints of the line L. □

Remark 55. 

In the product measure approach, for piecewise smooth lined L in D-dimensional space, we have the equality

$${\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}={\widehat{c}}_n(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}},$$

(668)

for all $k=1,\dots ,n$ and all $t\in [a,b]$. This fact leads to the equations

$$\widehat{\mathbf{T}}\left(t\right)={\widehat{c}}_n(D,\mathbf{x}\left(t\right))\mathbf{T}\left(t\right),$$

(669)

$$|\widehat{\mathbf{T}}\left(t\right)|={\widehat{c}}_n(D,\mathbf{x}\left(t\right))|\mathbf{T}\left(t\right)|.$$

(670)

Therefore, we get the equalities

$$\widehat{\mathbf{t}}\left(t\right)=\mathbf{t}\left(t\right),$$

(671)

$${\widehat{F}}_{\tau }\left(\mathbf{x}\left(t\right)\right)=F_{\tau }\left(\mathbf{x}\left(t\right)\right),$$

(672)

for $t\in [a,b]$, in the product measure approach.

Let us give examples of the line NIDS integrals of the vector fields.

Example 9. 

If the line L of the D-dimensional space $W=W_1\times W_2$ with $D={\alpha }_1+{\alpha }_2$ and $n=2$ does not have finite segments parallel to the coordinate axis, then

$${\widehat{c}}_{2,k}(D,\mathbf{x}\left(t\right))=c_1({\alpha }_1,x)c_1({\alpha }_2,y),$$

(673)

for $k=1$ and $k=2$, and the line NIDS integral is

$$\left({\mathbf{I}}_L^D,\mathbf{F}\right):={\int }_L{c}_2(D,\mathbf{x})\left(\sum _{k=1}^2{F}_k\left(\mathbf{x}\right)d{x}^k\right),$$

(674)

where all ${\alpha }_k\in (0,1]$.

Example 10. 

If the line L of the D-dimensional space $W=W_1\times W_2\times W_3$ with $D={\alpha }_1+{\alpha }_2+{\alpha }_3$ and $n=3$ does not have finite segments parallel to the coordinate axis and parallel to the coordinate planes, then

$${\widehat{c}}_{3,k}(D,\mathbf{x}\left(t\right))=c_1({\alpha }_1,x)c_1({\alpha }_2,y)c_1({\alpha }_3,z)=C_3(D,\mathbf{x}),$$

(675)

for $k=1$, $k=2$, and $k=3$. In this case, the line NIDS integral is

$$\left({\mathbf{I}}_L^D,\mathbf{F}\right):={\int }_L{c}_3(D,\mathbf{x})\left(\sum _{k=1}^3{F}_k\left(\mathbf{x}\right)d{x}^k\right),$$

(676)

where all ${\alpha }_k\in (0,1]$.

Example 11. 

If the line L of the D-dimensional space $W=W_1\times \dots \times W_n$ with $D={\alpha }_1+\dots +{\alpha }_n$ consists only of finite segments parallel to the coordinate axes, then

$${\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right))=c_1({\alpha }_k,x^k),$$

(677)

for all $k=1,\dots ,n$, and the line NIDS integral is

$$\left({\mathbf{I}}_L^D,\mathbf{F}\right):={\int }_L\left(\sum _{k=1}^n{F}_k\left(\mathbf{x}\right)c_1({\alpha }_k,x^k)d{x}^k\right)={\int }_L\left(\sum _{k=1}^n{F}_k\left(\mathbf{x}\right)d{\mu }_1({\alpha }_k,x^k)\right),$$

(678)

where all ${\alpha }_k\in (0,1]$.

In physics, circulation is the line integral of a vector field around a closed curve. Therefore, we can define the NIDS circulation ${\mathcal{E}}_L^D\left(\mathbf{F}\right)$ as the line NIDS integral of a vector field $\mathbf{F}$ around a closed curve L in the D-dimensional space.

4.3.2. Surface NIDS Integral of Vector Fields

The surface NIDS integration of vector function can be defined via the surface NIDS integration of scalar function (see Definition 16). If $\Omega$ is a surface region in the D-dimensional space $W=W_1\times W_2\times W_3$, such that

$$\Omega =\{\mathbf{x}:x^k=x^k(u,v),k=1,2,3(u,v)\in U\},$$

(679)

then the surface NIDS integral of the scalar function $f\left(\mathbf{x}\right(u,v\left)\right)\in C\left(U\right)$ over the surface region $\Omega$ is given by the equation

$$I_{\Omega }^{D}f\left(\mathbf{x}\right):={\int }_{U}f\left(\mathbf{x}(u,v)\right)\sqrt{{\widehat{G}}^{3,2}(u,v)}dudv,$$

(680)

where the NIDS integral is taken over the region U in the $(u,v)$-plane, and

$${\widehat{G}}^{3,2}(u,v):=det\left({\widehat{G}}_{ij}^{3,2}(u,v)\right)={\widehat{G}}_{11}^{3,2}(u,v){\widehat{G}}_{22}^{3,2}(u,v)-{\left({\widehat{G}}_{12}^{3,2}(u,v)\right)}^2$$

(681)

is a determinant of the metric

$${\widehat{G}}_{ij}^{3,2}(u,v):=\sum _{k3=1}^3{\widehat{c}}_{3,(k1,k2})(D,\mathbf{x}(u,v)){\displaystyle \frac{\partial x^{k3}(u,v)}{\partial u^i}}{\displaystyle \frac{\partial x^{k3}(u,v)}{\partial u^j}}(u^1=u,u^2=v),$$

(682)

where $k3\ne k1,k2$ and $k1\ne k2$, which is induced on the surface $\mathbf{x}=\mathbf{x}(u,v)$ by the D-dimensional space W.

In the classical (n-dimensional) vector calculus, the surface integral of the vector field $\mathbf{F}\left(\mathbf{x}\right)$ on a surface $\Omega$, which is described by the equation $\mathbf{x}=\mathbf{x}(u,v)$ with $(u,v)\in U$, is defined as the surface NIDS integral of the scalar function

$$F_n\left(\mathbf{x}(u,v)\right):=\left(\mathbf{F}\left(\mathbf{x}\right(u,v\left)\right),\mathbf{n}\left(\mathbf{x}\right(u,v\left)\right)\right),$$

(683)

which is the normal component of the vector field $\mathbf{F}$, where the surface normal $\mathbf{N}\left(\mathbf{x}\right(u,v\left)\right)$ is described by the vector

$$\mathbf{N}\left(\mathbf{x}(u,v)\right):=\left[{\displaystyle \frac{\partial \mathbf{x}}{\partial u}}\times {\displaystyle \frac{\partial \mathbf{x}}{\partial v}}\right]=\left({\displaystyle \frac{\partial (y,z)}{\partial (u,v)}};{\displaystyle \frac{\partial (z,x)}{\partial (u,v)}};{\displaystyle \frac{\partial (x,y)}{\partial (u,v)}}\right),$$

(684)

and the unit surface normal $\mathbf{n}\left(\mathbf{x}\right(u,v\left)\right)$ is defined as

$$\mathbf{n}\left(\mathbf{x}(u,v)\right):={\displaystyle \frac{\mathbf{N}\left(\mathbf{x}\right(u,v\left)\right)}{|\mathbf{N}(\mathbf{x}(u,v))|}},$$

(685)

where

$$|\mathbf{N}\left(\mathbf{x}(u,v)\right)|:=|\left[{\displaystyle \frac{\partial \mathbf{x}}{\partial u}}\times {\displaystyle \frac{\partial \mathbf{x}}{\partial v}}\right]|=\sqrt{{\left({\displaystyle \frac{\partial (y,z)}{\partial (u,v)}}\right)}^2+{\left({\displaystyle \frac{\partial (z,x)}{\partial (u,v)}}\right)}^2+{\left({\displaystyle \frac{\partial (x,y)}{\partial (u,v)}}\right)}^2}.$$

(686)

Let us define the vector that will be called the NIDS surface normal.

Definition 26. 

Let Ω be a two-parameter surface region in the D-dimensional space $W=W_1\times W_2\times W_3$, such that

$$\Omega =\{\mathbf{x}:x^k=x^k(u,v)\in C^1\left(U\right),k=1,2,3(u,v)\in U\}.$$

(687)

Then, the following vector will be called the NIDS surface normal

$$\widehat{\mathbf{N}}\left(\mathbf{x}(u,v)\right):=\left({\widehat{c}}_{3,yz}(D,\mathbf{x}){\displaystyle \frac{\partial (y,z)}{\partial (u,v)}};{\widehat{c}}_{3,zx}(D,\mathbf{x}){\displaystyle \frac{\partial (z,x)}{\partial (u,v)}};{\widehat{c}}_{3,xy}(D,\mathbf{x}){\displaystyle \frac{\partial (x,y)}{\partial (u,v)}}\right),$$

(688)

and the unit NIDS surface normal is defined as

$$\widehat{\mathbf{n}}\left(\mathbf{x}(u,v)\right):={\displaystyle \frac{\widehat{\mathbf{N}}\left(\mathbf{x}(u,v)\right)}{|\widehat{\mathbf{N}}\left(\mathbf{x}(u,v)\right)|}},$$

(689)

where

$$|\widehat{\mathbf{N}}\left(\mathbf{x}(u,v)\right)|:=$$

$$\sqrt{{\left({\widehat{c}}_{3,yz}(D,\mathbf{x}){\displaystyle \frac{\partial (y,z)}{\partial (u,v)}}\right)}^2+{\left({\widehat{c}}_{3,zx}(D,\mathbf{x}){\displaystyle \frac{\partial (z,x)}{\partial (u,v)}}\right)}^2+{\left({\widehat{c}}_{3,xy}(D,\mathbf{x}){\displaystyle \frac{\partial (x,y)}{\partial (u,v)}}\right)}^2},$$

(690)

and

$${\widehat{c}}_{3,yz}(D,\mathbf{x}):={\widehat{c}}_1({\alpha }_1,x)c_1({\alpha }_2,y)c_1({\alpha }_3,z),$$

(691)

$${\widehat{c}}_{3,zx}(D,\mathbf{x}):=c_1({\alpha }_1,x){\widehat{c}}_1({\alpha }_2,y)c_1({\alpha }_3,z),$$

(692)

$${\widehat{c}}_{3,xy}(D,\mathbf{x}):=c_1({\alpha }_1,x)c_1({\alpha }_2,y){\widehat{c}}_1({\alpha }_3,z),$$

(693)

with all ${\alpha }_k\in (0,1]$.

The surface NIDS integral of vector field is defined as the surface NIDS integral of the scalar field in the following form.

Definition 27 

(Surface NIDS Integral of Vector Function). Let Ω be a two-parameter surface region in the D-dimensional space $W=W_1\times W_2\times W_3$, such that

$$\Omega =\{\mathbf{x}:x^k=x^k(u,v)\in C^1\left(U\right),k=1,2,3(u,v)\in U\}.$$

(694)

Then, in the product measure approach, the surface NIDS integral of the vector function $\mathbf{F}\left(\mathbf{x}\right(u,v\left)\right)\in C\left(U\right)$ over the surface region Ω is given by the equation

$$\left({\mathbf{I}}_{\Omega }^D,\mathbf{F}\right):={\int }_U\widehat{F_n}\left(\mathbf{x}(u,v)\right)\sqrt{{\widehat{G}}^{3,2}(u,v)}dudv,$$

(695)

where the NIDS integral is taken over the region U in the $(u,v)$-plane, the scalar function $\widehat{F_n}(u,v))$ is the NIDS normal component of the vector field $\mathbf{F}\left(\mathbf{x}\right)$ such that

$$\widehat{F_n}\left(\mathbf{x}(u,v)\right):=\left(\mathbf{F}\left(\mathbf{x}(u,v)\right),\widehat{\mathbf{n}}\left(\mathbf{x}(u,v)\right)\right),$$

(696)

and the vector $\widehat{\mathbf{n}}\left(\mathbf{x}(u,v)\right)$ is the unit surface normal (689). The function

$${\widehat{G}}^{3,2}(u,v):={|\widehat{\mathbf{N}}\left(\mathbf{x}(u,v)\right)|}^2$$

(697)

can be represented as the determinant

$${\widehat{G}}^{3,2}(u,v):=det\left({\widehat{G}}_{ij}^{3,2}(u,v)\right)={\widehat{G}}_{11}^{3,2}(u,v){\widehat{G}}_{22}^{3,2}(u,v)-{\left({\widehat{G}}_{12}^{3,2}(u,v)\right)}^2$$

(698)

of the metric

$${\widehat{G}}_{ij}^{3,2}(u,v):=\sum _{k3=1}^3{\widehat{c}}_{3,(k1,k2})(D,\mathbf{x}(u,v)){\displaystyle \frac{\partial x^{k3}(u,v)}{\partial u^i}}{\displaystyle \frac{\partial x^{k3}(u,v)}{\partial u^j}}(u^1=u,u^2=v)$$

(699)

that is induced on the surface $\mathbf{x}=\mathbf{x}(u,v)$, where

$${\widehat{c}}_{3,(k1,k2)}(D,\mathbf{x}(u,v)):=c_1({\alpha }_{k1},x^{k1}(u,v))c_1({\alpha }_{k2},x^{k2}(u,v)){\widehat{c}}_1({\alpha }_{k3},x^{k3}(u,v),$$

(700)

with $k3\ne k1,k2$ and $k1\ne k2$, (see also Equations (691)–(693)), characterizing the measures induced by the D-dimensional space W.

In the metric approach, the surface NIDS integral of the vector function $\mathbf{F}\left(\mathbf{x}\right(u,v\left)\right)\in C\left(U\right)$ over the surface region Ω is described by same equations, in which we must use all

$${\widehat{c}}_1({\alpha }_{k3},x^{k3}(u,v)=1,$$

(701)

$${\widehat{c}}_{3,(k1,k2)}(D,\mathbf{x}(u,v)):=c_1({\alpha }_{k1},x^{k1}(u,v))c_1({\alpha }_{k2},x^{k2}(u,v)).$$

(702)

Let us write the equation of the surface NIDS integral in explicit form with $x^1=x$, $x^2=y$, and $x^3=z$.

Example 12. 

Let Ω be a two-parameter surface region in the D-dimensional space $W=W_1\times W_2\times W_3$ in the form (694).

In the product measure approach, the surface NIDS integral of the vector function $\mathbf{F}\left(\mathbf{x}\right(u,v\left)\right)\in C\left(U\right)$ over the surface region Ω is given as

$$\left({\mathbf{I}}_{\Omega }^D,\mathbf{F}\right):={\int }_U\widehat{F_n}\left(\mathbf{x}(u,v)\right)\sqrt{{\widehat{G}}_{11}^{3,2}(u,v){\widehat{G}}_{22}^{3,2}(u,v)-{\left({\widehat{G}}_{12}^{3,2}(u,v)\right)}^2}dudv,$$

(703)

where

$${\widehat{G}}_{11}^{3,2}(u,v):=$$

$$c_1({\alpha }_1,x(u,v))c_1({\alpha }_2,y(u,v)){\widehat{c}}_1({\alpha }_3,z(u,v)){\displaystyle \frac{\partial z(u,v)}{\partial u}}{\displaystyle \frac{\partial z(u,v)}{\partial u}}+$$

$$c_1({\alpha }_1,x(u,v))c_1({\alpha }_3,z(u,v)){\widehat{c}}_1({\alpha }_2,y(u,v)){\displaystyle \frac{\partial y(u,v)}{\partial u}}{\displaystyle \frac{\partial y(u,v)}{\partial u}}+$$

$$c_1({\alpha }_2,y(u,v))c_1({\alpha }_3,z(u,v)){\widehat{c}}_1({\alpha }_1,x(u,v)){\displaystyle \frac{\partial x(u,v)}{\partial u}}{\displaystyle \frac{\partial x(u,v)}{\partial u}},$$

(704)

$${\widehat{G}}_{22}^{3,2}(u,v)):=$$

$$c_1({\alpha }_1,x(u,v))c_1({\alpha }_2,y(u,v)){\widehat{c}}_1({\alpha }_3,z(u,v)){\displaystyle \frac{\partial z(u,v)}{\partial v}}{\displaystyle \frac{\partial z(u,v)}{\partial v}}+$$

$$c_1({\alpha }_1,x(u,v))c_1({\alpha }_3,z(u,v)){\widehat{c}}_1({\alpha }_2,y(u,v)){\displaystyle \frac{\partial y(u,v)}{\partial v}}{\displaystyle \frac{\partial y(u,v)}{\partial v}}+$$

$$c_1({\alpha }_2,y(u,v))c_1({\alpha }_3,z(u,v)){\widehat{c}}_1({\alpha }_1,x(u,v)){\displaystyle \frac{\partial x(u,v)}{\partial v}}{\displaystyle \frac{\partial x(u,v)}{\partial v}},$$

(705)

$${\widehat{G}}_{12}^{3,2}(u,v):=$$

$$c_1({\alpha }_1,x(u,v))c_1({\alpha }_2,y(u,v)){\widehat{c}}_1({\alpha }_3,z(u,v)){\displaystyle \frac{\partial z(u,v)}{\partial u}}{\displaystyle \frac{\partial z(u,v)}{\partial v}}+$$

$$c_1({\alpha }_1,x(u,v))c_1({\alpha }_3,z(u,v)){\widehat{c}}_1({\alpha }_2,y(u,v)){\displaystyle \frac{\partial y(u,v)}{\partial u}}{\displaystyle \frac{\partial y(u,v)}{\partial v}}+$$

$$c_1({\alpha }_2,y(u,v))c_1({\alpha }_3,z(u,v)){\widehat{c}}_1({\alpha }_1,x(u,v)){\displaystyle \frac{\partial x(u,v)}{\partial u}}{\displaystyle \frac{\partial x(u,v)}{\partial v}},$$

(706)

and $\widehat{F_n}(u,v))$ is defined by Equations (689) and (696).

In the metric approach, the surface NIDS integral of the vector function $\mathbf{F}\left(\mathbf{x}\right(u,v\left)\right)\in C\left(U\right)$ over the surface region Ω is described by same equations, in which we must use

$${\widehat{c}}_1({\alpha }_1,x(u,v))=1,$$

(707)

$${\widehat{c}}_1({\alpha }_2,y(u,v))=1,$$

(708)

$${\widehat{c}}_1({\alpha }_3,z(u,v))=1,$$

(709)

and

$$\widehat{F_n}\left(\mathbf{x}(u,v)\right)=F_n\left(\mathbf{x}(u,v)\right),$$

(710)

where $(u,v)\in U$, and $x(u,v),y(u,v),z(u,v)\in C^1\left(U\right)$.

Let us give some representations of the surface NIDS integral of the vector field.

Theorem 29 

(First Representation of Surface NIDS Integral of Vector Field). Let $\Omega =\{\mathbf{x}:x^k=x^k(u,v),k=1,2,3(u,v)\in U\}$ be a surface region in the D-dimensional space $W=W_1\times W_2\times W_3$, such that it does not have regions in which the functions $x^k=x^k\left(\mathbf{u}\right)$, $k=1,\dots ,n$ are constant.

Then, the surface NIDS integral of the vector function $\mathbf{F}\left(\mathbf{x}\right(u,v\left)\right)\in C\left(U\right)$ over the surface region Ω is given by the equations

$$\left({\mathbf{I}}_{\Omega }^D,\mathbf{F}\right):={\int }_U{F}_n\left(\mathbf{x}(u,v)\right)c_3(D,\mathbf{x}(u,v))\sqrt{G(u,v)}dudv,$$

(711)

where the NIDS integral is taken over the region U in the $(u,v)$-plane, the scalar function $F_n\left(\mathbf{x}(u,v)\right)$ is the normal component of the vector field $\mathbf{F}\left(\mathbf{x}\right)$, such that

$$F_n\left(\mathbf{x}(u,v)\right):=\left(\mathbf{F}\left(\mathbf{x}\right(u,v\left)\right),\mathbf{n}\left(\mathbf{x}\right(u,v\left)\right)\right),$$

(712)

and the vector $\mathbf{n}\left(\mathbf{x}\right(u,v\left)\right)$ is the unit surface normal (689). The function

$$G(u,v):=det\left(G_{ij}(u,v)\right)=G_{11}(u,v)G_{22}(u,v)-{\left(G_{12}(u,v)\right)}^2$$

(713)

is a determinant of the metric

$$G_{ij}(u,v):=\sum _{k=1}^3{\displaystyle \frac{\partial x^k(u,v)}{\partial u^i}}{\displaystyle \frac{\partial x^k(u,v)}{\partial u^j}}(u^1=u,u^2=v)$$

(714)

that is induced on the surface $\mathbf{x}=\mathbf{x}(u,v)$ by the D-dimensional space W.

Proof. 

If there are no regions on the surface $\Omega$ in which the functions $x^k=x^k\left(\mathbf{u}\right)$, $k=1,\dots ,n$, are constant, then

$${\widehat{c}}_1({\alpha }_k,x^k(u,v))=c_1({\alpha }_k,x^k(u,v)),$$

(715)

for all $k=1,2,3$ and all $(u,v)\in U$. Therefore,

$${\widehat{c}}_{3,(k1,k2)}(D,\mathbf{x}(u,v)):=c_3(D,\mathbf{x}(u,v)),$$

(716)

for all $k1,k2=1,2,3$ and all $(u,v)\in U$.

As a result, in this case, we get the induced metric as

$${\widehat{G}}_{ij}^{3,2}(u,v):=c_3^2(D,\mathbf{x}(u,v))G_{ij}(u,v)$$

(717)

and the NIDS surface normal as

$$\widehat{\mathbf{N}}\left(\mathbf{x}(u,v)\right)=c_3(D,\mathbf{x}(u,v))\mathbf{N}\left(\mathbf{x}(u,v)\right),$$

(718)

and

$$\widehat{\mathbf{n}}\left(\mathbf{x}(u,v)\right)=\mathbf{n}\left(\mathbf{x}(u,v)\right),$$

(719)

for all $(u,v)\in U$. □

Let us consider a surface $\Omega$, that is described by the equation $\mathbf{x}=\mathbf{x}(u,v)$. The surface NIDS integral of a vector field $\mathbf{F}\left(\mathbf{x}\right)$ on the surface $\Omega$, which is also called the NIDS flux of the vector field passing through the surface $\Omega$, is defined as the integration of the normal component $F_n\left(\mathbf{x}\right):=\left(\mathbf{F},\mathbf{n}\right)$ of the vector field $\mathbf{F}\left(\mathbf{x}\right)$ over the surface, and

$$\left({\mathbf{I}}_{\Omega }^D,\mathbf{F}\right):={\int }_{\Omega }\left(\mathbf{F},d{\mathbf{S}}_D\right)={\int }_{\Omega }\left(\mathbf{F},\mathbf{n}\right)d{S}_D={\int }_{\Omega }{F}_n\left(\mathbf{x}\right)d{S}_D,$$

(720)

where $d{\mathbf{S}}_D$ is the vector area of the elementary (infinitesimal) surface $d{S}_D$, directed as the surface normal,

$$\left(\mathbf{F},d{\mathbf{S}}_D\right)=F_{x}d{S}_{D,23}+F_{y}d{S}_{D,31}+F_{z}d{S}_{D,12}=$$

$$F_x\left(\mathbf{x}\right){\widehat{c}}_{3,(2,3)}(D,\mathbf{x})dydz+F_y\left(\mathbf{x}\right){\widehat{c}}_{3,(1,3)}(D,\mathbf{x})dzdx+F_z\left(\mathbf{x}\right){\widehat{c}}_{3,(1,2)}(D,\mathbf{x})dxdy,$$

(721)

where we use the notation

$$d{S}_{D,ij}={\widehat{c}}_{3,(i,j)}(D,\mathbf{x})d{x}^{i}d{x}^j={\widehat{c}}_1({\alpha }_k,x^k){\widehat{c}}_1({\alpha }_i,x^i){\widehat{c}}_1({\alpha }_j,x^j)d{x}^{i}d{x}^j,$$

(722)

where $k\ne i,j$ and $j\ne i$.

Let us prove a theorem that defines another representation of the surface NIDS integral.

Theorem 30 

(Second Representation of the Surface NIDS Integral of the Vector Field). Let Ω be a surface region in the D-dimensional space $W=W_1\times W_2\times W_3$, such that

$$\Omega =\{\mathbf{x}:x^k=x^k(u,v),k=1,2,3(u,v)\in U\}.$$

(723)

Then, in the product measure approach, the surface NIDS integral of the vector function $\mathbf{F}\left(\mathbf{x}\right(u,v\left)\right)\in C\left(U\right)$ over the surface region Ω is given by the equation

$$\left({\mathbf{I}}_{\Omega }^D,\mathbf{F}\right):={\int }_U\left(\mathbf{F},d{\mathbf{S}}_D\right)=$$

$${\int }_{\Omega }\left(F_x\left(\mathbf{x}\right){\widehat{c}}_{3,yz}(D,\mathbf{x})dydz+F_y\left(\mathbf{x}\right){\widehat{c}}_{3,zx}(D,\mathbf{x})dzdx+F_z\left(\mathbf{x}\right){\widehat{c}}_{3,xy}(D,\mathbf{x})dxdy\right),$$

(724)

where

$${\widehat{c}}_{3,x_i{x}_j}(D,\mathbf{x})={\widehat{c}}_{3,(i,j)}(D,\mathbf{x}):=$$

$$c_1({\alpha }_i,x^i(u,v))c_1({\alpha }_j,x^j(u,v)){\widehat{c}}_1({\alpha }_k,x^k(u,v))(k\ne i,j),$$

(725)

with coordinates $x^1=x$, $x^2=y$, and $x^3=z$.

Proof. 

Let $\mathbf{x}=\mathbf{x}(u,v)$ be an orientation-preserving parametrization of the surface $\Omega$ with $(u,v)\in U$. Let us use the coordinate change

$$dxdy={\displaystyle \frac{\partial (x,y)}{\partial (u,v)}}dudv,dydz={\displaystyle \frac{\partial (y,z)}{\partial (u,v)}}dudv,dzdx={\displaystyle \frac{\partial (z,x)}{\partial (u,v)}}dudv,$$

(726)

where $\partial (x,y)/\partial (u,v)$, $\partial (y,z)/\partial (u,v)$, and $\partial (z,x)/\partial (u,v)$ are the determinants of the Jacobian matrix of the transition functions.

Then, the surface NIDS integral of vector function $\mathbf{F}$ on the surface $\Omega$ is given by

$$\left({\mathbf{I}}_{\Omega }^D,\mathbf{F}\right):={\int }_U\left(\mathbf{F},d{\mathbf{S}}_D\right)=$$

$${\int }_{\Omega }\left(F_x\left(\mathbf{x}\right){\widehat{c}}_{3,(2,3)}(D,\mathbf{x})dydz+F_y\left(\mathbf{x}\right){\widehat{c}}_{3,(1,3)}(D,\mathbf{x})dzdx+F_z\left(\mathbf{x}\right){\widehat{c}}_{3,(1,2)}(D,\mathbf{x})dxdy\right)=$$

$${\int }_U(F_x\left(\mathbf{x}(u,v)\right){\widehat{c}}_{3,yz}(D,\mathbf{x}){\displaystyle \frac{\partial (y,z)}{\partial (u,v)}}+$$

$$F_y\left(\mathbf{x}(u,v)\right){\widehat{c}}_{3,zx}(D,\mathbf{x}){\displaystyle \frac{\partial (z,x)}{\partial (u,v)}}+F_z\left(\mathbf{x}(u,v)\right){\widehat{c}}_{3,xy}(D,\mathbf{x}){\displaystyle \frac{\partial (x,y)}{\partial (u,v)}})dudv=$$

$${\int }_U\left(\mathbf{F}\left(\mathbf{x}(u,v)\right),\widehat{N}\left(\mathbf{x}(u,v)\right)\right)dudv,$$

(727)

where

$$\widehat{N}\left(\mathbf{x}(u,v)\right)=\left({\displaystyle \frac{{\widehat{c}}_{3,yz}(D,\mathbf{x})\partial (y,z)}{\partial (u,v)}};{\widehat{c}}_{3,zx}(D,\mathbf{x}){\displaystyle \frac{\partial (z,x)}{\partial (u,v)}};{\widehat{c}}_{3,xy}(D,\mathbf{x}){\displaystyle \frac{\partial (x,y)}{\partial (u,v)}}\right)$$

(728)

is the surface element normal to the surface $\Omega$.

Using

$$|\widehat{\mathbf{N}}\left(\mathbf{x}(u,v)\right)|=\sqrt{{\left({\displaystyle \frac{\partial (y,z)}{\partial (u,v)}}\right)}^2+{\left({\displaystyle \frac{\partial (z,x)}{\partial (u,v)}}\right)}^2+{\left({\displaystyle \frac{\partial (x,y)}{\partial (u,v)}}\right)}^2},$$

(729)

the surface NIDS integral of vector function $\mathbf{F}$ on the surface $\Omega$ is given by

$$\left({\mathbf{I}}_{\Omega }^D,\mathbf{F}\right):={\int }_U\left(\mathbf{F},d{\mathbf{S}}_D\right)={\int }_U\left(\mathbf{F}\left(\mathbf{x}(u,v)\right),\widehat{\mathbf{N}}\left(\mathbf{x}(u,v)\right)\right)dudv=$$

$${\int }_U\left(\mathbf{F}\left(\mathbf{x}(u,v)\right),\widehat{\mathbf{n}}\left(\mathbf{x}(u,v)\right)\right)|\widehat{\mathbf{N}}\left(\mathbf{x}(u,v)\right)|dudv=$$

$${\int }_U{\widehat{c}}_3(D,\mathbf{x}(u,v))F_n\left(\mathbf{x}(u,v)\right)\sqrt{\widehat{G^{3,2}}(u,v)}dudv,$$

(730)

with coordinates $x^1=x$, $x^2=y$, $x^3=z$. □

Theorem 31 

(Third Representation of Surface NIDS Integral of Vector Field). Let $\Omega =\{\mathbf{x}:x^k=x^k(u,v),k=1,2,3(u,v)\in U\}$ be a surface region in the D-dimensional space $W=W_1\times W_2\times W_3$, such that it does not have regions in which the functions $x^k=x^k\left(\mathbf{u}\right)$, $k=1,\dots ,n$ are constant.

Then, the surface NIDS integral of the vector function $\mathbf{F}\left(\mathbf{x}\right(u,v\left)\right)\in C\left(U\right)$ over the surface region Ω is given by the equations

$$\left({\mathbf{I}}_{\Omega }^D,\mathbf{F}\right):={\int }_U\left(\mathbf{F},d{\mathbf{S}}_D\right)=$$

$${\int }_{\Omega }{\widehat{c}}_3(D,\mathbf{x})\left(F_x\left(\mathbf{x}\right)dydz+F_y\left(\mathbf{x}\right)dzdx+F_z\left(\mathbf{x}\right)dxdy\right)$$

(731)

in the product measure approach.

Proof. 

The surface NIDS integral of vector function $\mathbf{F}$ on the surface $\Omega$ is given as

$$\left({\mathbf{I}}_{\Omega }^D,\mathbf{F}\right):={\int }_U\left(\mathbf{F},d{\mathbf{S}}_D\right)=$$

$${\int }_{\Omega }\left(F_x\left(\mathbf{x}\right){\widehat{c}}_{3,(2,3)}(D,\mathbf{x})dydz+F_y\left(\mathbf{x}\right){\widehat{c}}_{3,(1,3)}(D,\mathbf{x})dzdx+F_z\left(\mathbf{x}\right){\widehat{c}}_{3,(1,2)}(D,\mathbf{x})dxdy\right)=$$

$${\int }_{\Omega }{\widehat{c}}_3(D,\mathbf{x})\left(F_x\left(\mathbf{x}\right)dydz+F_y\left(\mathbf{x}\right)dzdx+F_z\left(\mathbf{x}\right)dxdy\right)=$$

$${\int }_U{c}_3(D,\mathbf{x}(u,v))\left(F_x\left(\mathbf{x}(u,v)\right){\displaystyle \frac{\partial (y,z)}{\partial (u,v)}}+F_y\left(\mathbf{x}(u,v)\right){\displaystyle \frac{\partial (z,x)}{\partial (u,v)}}+F_z\left(\mathbf{x}(u,v)\right){\displaystyle \frac{\partial (x,y)}{\partial (u,v)}}\right)dudv=$$

$${\int }_U{c}_3(D,\mathbf{x}(u,v))\left(\mathbf{F}\left(\mathbf{x}(u,v)\right),\left[{\displaystyle \frac{\partial \mathbf{x}}{\partial u}}\times {\displaystyle \frac{\partial \mathbf{x}}{\partial v}}\right]\right)dudv,$$

(732)

where

$$\left[{\displaystyle \frac{\partial \mathbf{x}}{\partial u}}\times {\displaystyle \frac{\partial \mathbf{x}}{\partial v}}\right]=\left({\displaystyle \frac{\partial (y,z)}{\partial (u,v)}},{\displaystyle \frac{\partial (z,x)}{\partial (u,v)}},{\displaystyle \frac{\partial (x,y)}{\partial (u,v)}}\right)$$

(733)

is the surface element normal to the surface $\Omega$.

Using

$$\left|\left[{\displaystyle \frac{\partial \mathbf{x}}{\partial u}}\times {\displaystyle \frac{\partial \mathbf{x}}{\partial v}}\right]\right|=\sqrt{{\left({\displaystyle \frac{\partial (y,z)}{\partial (u,v)}}\right)}^2+{\left({\displaystyle \frac{\partial (z,x)}{\partial (u,v)}}\right)}^2+{\left({\displaystyle \frac{\partial (x,y)}{\partial (u,v)}}\right)}^2},$$

(734)

the surface integral of vector function $\mathbf{F}$ on the surface $\Omega$ is given by

$$\left({\mathbf{I}}_{\Omega }^D,\mathbf{F}\right):={\int }_U\left(\mathbf{F},d{\mathbf{S}}_D\right)={\int }_U{c}_3(D,\mathbf{x}(u,v))\left(\mathbf{F}\left(\mathbf{x}(u,v)\right),\left[{\displaystyle \frac{\partial \mathbf{x}}{\partial u}}\times {\displaystyle \frac{\partial \mathbf{x}}{\partial v}}\right]\right)dudv=$$

$${\int }_U{c}_3(D,\mathbf{x}(u,v))\left(\mathbf{F}\left(\mathbf{x}(u,v)\right),{\displaystyle \frac{{\displaystyle \left[\frac{\partial \mathbf{x}}{\partial u}\times \frac{\partial \mathbf{x}}{\partial v}\right]}}{{\displaystyle \left|\left[\frac{\partial \mathbf{x}}{\partial u}\times \frac{\partial \mathbf{x}}{\partial v}\right]\right|}}}\right)\left|\left[{\displaystyle \frac{\partial \mathbf{x}}{\partial u}}\times {\displaystyle \frac{\partial \mathbf{x}}{\partial v}}\right]\right|dudv=$$

$${\int }_U{c}_3(D,\mathbf{x}(u,v))\left(\mathbf{F}\left(\mathbf{x}\right(u,v\left)\right),\mathbf{n}\left(\mathbf{x}\right(u,v\left)\right)\right)\left|\left[{\displaystyle \frac{\partial \mathbf{x}}{\partial u}}\times {\displaystyle \frac{\partial \mathbf{x}}{\partial v}}\right]\right|dudv=$$

$${\int }_U{c}_3(D,\mathbf{x}(u,v))F_n\left(\mathbf{x}(u,v)\right)\sqrt{G(u,v)}dudv,$$

(735)

where

$$G(u,v):=det\left(G_{ij}(u,v)\right)=G_{11}(u,v)G_{22}(u,v)-{\left(G_{12}(u,v)\right)}^2,$$

(736)

$$G_{ij}(u,v):=\sum _{k=1}^3{\displaystyle \frac{\partial x^k(u,v)}{\partial u^i}}{\displaystyle \frac{\partial x^k(u,v)}{\partial u^j}},$$

(737)

with $u^1=u$ and $u^2=v$. □

As an example, let us consider the surface NIDS integral of the vector field for the boundary $\partial {\Omega }_P$ of the parallelepiped region ${\Omega }_P$ by using the product measure approach.

Example 13. 

Let ${\Omega }_P$ be a parallelepiped region with edges parallel to the coordinate axes in the D-dimensional space $W=W_1\times W_2\times W_3$ with $D={\alpha }_1+{\alpha }_2+{\alpha }_3$. Let the piecewise smooth boundary $\partial {\Omega }_P$ of the region ${\Omega }_P$ be described as (478).

Then, the surface NIDS integral of a continuously differentiable vector field $\mathbf{F}\left(\mathbf{x}\right)$ over the piecewise smooth boundary (478) is

$${\int }_{\partial {\Omega }_P}(\mathbf{F}\left(\mathbf{x}\right),d{\mathbf{S}}_d)={\int }_{\partial {\Omega }_{23}}{F}_1\left(\mathbf{x}\right)d{S}_{d,23}+$$

$${\int }_{\partial {\Omega }_{13}}{F}_2\left(\mathbf{x}\right)d{S}_{d,13}+{\int }_{\partial {\Omega }_{12}}{F}_3\left(\mathbf{x}\right)d{S}_{d,12},$$

(738)

where

$$d{S}_{d,ij}=d{\mu }_1({\alpha }_i,x_i)d{\mu }_1({\alpha }_j,x_j)=c_2(d_{ij},\mathbf{x})d{S}_{ij}(i\ne j)$$

(739)

with

$$c_2(d_{ij},\mathbf{x})=c_1({\alpha }_i,x_i)c_1({\alpha }_j,x_j),(i\ne j).$$

(740)

For the NIDS, the dimensions $d_{ij}$ are

$$d_{ij}:={\alpha }_i+{\alpha }_j(i\ne j),D=\sum _{k=1}^3{\alpha }_k,$$

(741)

where all ${\alpha }_k\in (0,1]$.

4.4. Fundamental Theorems of NIDS Vector Calculus

The NIDS generalization of the standard Gauss–Ostrogradsky theorem, the Stoke theorem, and the gradient theorem must be satisfied to provide a consistency of the volume, surface and, line NIDS integrals and the NIDS vector differential operators. This requirement allows us to formulate a self-consistent NIDS vector calculus.

4.4.1. NIDS Gradient Theorem and NIDS Gradient Operators

In the integer-dimensional space, the gradient theorem can be interpreted as the second fundamental theorem of calculus for line integrals. This theorem ensures consistency between integral and differential vector operators, namely the line integral of vector fields and the gradient (del operator). This statement can be generalized to obtain the correct definition of the gradient in non-integer-dimensional space. As a result, we propose the following self-consistency principle of the NIDS vector calculus.

Principle 11 

(Principle of the Self-Consistency of NIDS Vector Calculus). The gradient (del operator) in NIDS must be defined, such that the NIDS generalization of the gradient theorem is satisfied in the NIDS.

Note that the NIDS vector calculus of the vector operators for NIDS, including the NODS del operator, the NIDS gradient, the NIDS divergence, and the NIDS curl operators, has been proposed in [18,19] and then actively used in various areas of physics to describe fractal media, particle distributions, and fields.

Let us define the NIDS continuously differentiable functions in the product measure approach. The NIDS continuously differentiable functions in the metric approach are proposed in definition 22.

Definition 28 

(NIDS Continuously Differentiable Functions). Let $W=W_1\times \dots \times W_n$ be a D-dimensional space with $D={\alpha }_1+\dots +{\alpha }_n$, and let Ω be a region of W.

In the product measure approach, the function $f\left(\mathbf{x}\right)$ with $\mathbf{x}\in \Omega$ will be called the NIDS continuously differentiable function if

$${\widehat{D}}_{x^k}^{{\alpha }_k}f\left(\mathbf{x}\right):={\displaystyle \frac{1}{c_1({\alpha }_k,x^k)}}{\displaystyle \frac{\partial \left({\widehat{c}}_1({\alpha }_k,x^k)f\left(\mathbf{x}\right)\right)}{\partial x^k}}\in C(\Omega ),$$

(742)

for all $\mathbf{x}\in \Omega$ and all $k=1,\dots ,n$. The set of such functions will be denoted as ${\widehat{C}}_{\left(\alpha \right)}^1(\Omega )$, where $\left(\alpha \right)=({\alpha }_1,\dots ,{\alpha }_n)$.

The function $f\left(\mathbf{x}\right)$ with $\mathbf{x}\in \Omega$ will be called the ${\widehat{C}}_{\left(\alpha \right)}^m$-function, if all m-order partial NIDS derivatives are continuous

$${\widehat{D}}_{x^{k1}}^{{\alpha }_{k1}}\dots {\widehat{D}}_{x^{km}}^{{\alpha }_{km}}f\left(\mathbf{x}\right)\in C(\Omega ),$$

(743)

for all $\mathbf{x}\in \Omega$. In this case, we will write $f\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^m(\Omega )$.

Let us give the formulation of the NIDS gradient theorem in the product measure approach and metric approach.

Theorem 32 

(Gradient Theorem for NIDS). Let Ω the region of the D-dimensional space be $W=W_1\times \dots \times W_n$, and let $\mathbf{x}=\mathbf{x}\left(t\right)$ with $t\in [a,b]$ be a piecewise continuously differentiable curve L in Ω, which starts at the point $\mathbf{x}\left(a\right)$ and ends at the point $\mathbf{x}\left(b\right)$.

(1) The Product Measure Approach: the line NIDS integral of the vector field

$$F_k\left(\mathbf{x}\right)={\displaystyle \frac{1}{{\widehat{c}}_{n,k}(D,\mathbf{x})}}{\displaystyle \frac{\partial \widehat{U}\left(\mathbf{x}\right)}{\partial x^k}}$$

(744)

with ${\widehat{c}}_n^{-1}(D,\mathbf{x})\widehat{U}\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^1(\Omega )$ along the piecewise continuously differentiable curve $\mathbf{x}=\mathbf{x}\left(t\right)$ for $a⩽t⩽b$ in the region Ω of the NIDS satisfies the equation

$$\left({\mathbf{I}}_L^D,\mathbf{F}\right):={\int }_L\left(\sum _{k=1}^n{F}_k\left(\mathbf{x}\right){\widehat{c}}_{n,k}(D,\mathbf{x})d{x}^k\right)=\widehat{U}\left(\mathbf{x}\left(b\right)\right)-\widehat{U}\left(\mathbf{x}\left(a\right)\right).$$

(745)

(2) The Metric Approach: the line NIDS integral of the vector field

$$F_k\left(\mathbf{x}\right)={\displaystyle \frac{1}{c_1({\alpha }_k,x^k)}}{\displaystyle \frac{\partial U\left(\mathbf{x}\right)}{\partial x^k}}$$

(746)

with $U\left(\mathbf{x}\right)\in C_{\left(\alpha \right)}^1(\Omega )$ along the piecewise continuously differentiable curve $\mathbf{x}=\mathbf{x}\left(t\right)$ for $a⩽t⩽b$ in the region Ω of the NIDS satisfies the equation

$$\left({\mathbf{I}}_L^{D,\ast },\mathbf{F}\right):={\int }_L\left(\sum _{k=1}^n{F}_k\left(\mathbf{x}\right)c_1({\alpha }_k,x^k)d{x}^k\right)=U\left(\mathbf{x}\left(b\right)\right)-U\left(\mathbf{x}\left(a\right)\right).$$

(747)

Proof. 

(1) In the product measure approach, the proof of the NIDS gradient theorem is based on the multivariable chain rule for the classical derivative with respect to $t\in [a,b]$ of the function composition

$${\displaystyle \frac{d\widehat{U}\left(\mathbf{x}\left(t\right)\right)}{dt}}={\left({\displaystyle \frac{\partial \widehat{U}\left(\mathbf{x}\right)}{\partial x^k}}\right)}_{\mathbf{x}=\mathbf{x}\left(t\right)}{\displaystyle \frac{d{x}^k\left(t\right)}{dt}}.$$

(748)

If the vector field $\mathbf{F}\left(\mathbf{x}\right)$ is defined as

$$F_k\left(\mathbf{x}\right)={\displaystyle \frac{1}{{\widehat{c}}_{n,k}(D,\mathbf{x})}}{\displaystyle \frac{\partial \widehat{U}\left(\mathbf{x}\right)}{\partial x^k}},$$

(749)

then the line NIDS integral (633) of this field (749) along the piecewise continuously differentiable curve $\mathbf{x}=\mathbf{x}\left(t\right)$ for $a⩽t⩽b$ gives

$$\left({\mathbf{I}}_L^D,\mathbf{F}\right):={\int }_L\left(\sum _{k=1}^n{F}_k\left(\mathbf{x}\right){\widehat{c}}_{n,k}(D,\mathbf{x})d{x}^k\right)=$$

$${\int }_a^b\left(\sum _{k=1}^n{F}_k\left(\mathbf{x}\left(t\right)\right){\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)dt=$$

$${\int }_a^b\left(\sum _{k=1}^n{\displaystyle \frac{1}{{\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right))}}{\left({\displaystyle \frac{\partial \widehat{U}\left(\mathbf{x}\right)}{\partial x^k}}\right)}_{\mathbf{x}=\mathbf{x}\left(t\right)}{\widehat{c}}_{n,k}(D,\mathbf{x}\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)dt=$$

$${\int }_a^b\left(\sum _{k=1}^n{\left({\displaystyle \frac{\partial \widehat{U}\left(\mathbf{x}\right)}{\partial x^k}}\right)}_{\mathbf{x}=\mathbf{x}\left(t\right)}{\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)dt=$$

$${\int }_a^b{\displaystyle \frac{d\widehat{U}\left(\mathbf{x}\left(t\right)\right)}{dt}}dt={\int }_{\widehat{U}\left(\mathbf{x}\left(a\right)\right)}^{\widehat{U}\left(\mathbf{x}\left(b\right)\right)}d\widehat{U}\left(\mathbf{x}\right)=\widehat{U}\left(\mathbf{x}\left(b\right)\right)-\widehat{U}\left(\mathbf{x}\left(a\right)\right).$$

(750)

This proves Equation (745).

(2) In the metric approach, the gradient theorem is proved similarly.

The proof of the NIDS gradient theorem is based on the multivariable chain rule for the classical derivative of the function composition

$${\displaystyle \frac{dU\left(\mathbf{x}\right(t\left)\right)}{dt}}={\left({\displaystyle \frac{\partial U\left(\mathbf{x}\right)}{\partial x^k}}\right)}_{\mathbf{x}=\mathbf{x}\left(t\right)}{\displaystyle \frac{d{x}^k\left(t\right)}{dt}}.$$

(751)

If the vector field $\mathbf{F}\left(\mathbf{x}\right)$ is defined as

$$F_k\left(\mathbf{x}\right)={\displaystyle \frac{1}{c_1({\alpha }_k,x_k)}}{\displaystyle \frac{\partial U\left(\mathbf{x}\right)}{\partial x^k}},$$

(752)

then the line NIDS integral (637) of this field gives

$$\left({\mathbf{I}}_L^{D,\ast },\mathbf{F}\right):={\int }_L\left(\sum _{k=1}^n{F}_k\left(\mathbf{x}\right)d{\mu }_1({\alpha }_k,x^k)\right)=$$

$${\int }_L\left(\sum _{k=1}^n{F}_k\left(\mathbf{x}\right))c_1({\alpha }_k,x^k)d{x}^k\right)=$$

$${\int }_a^b\left(\sum _{k=1}^n{F}_k\left(\mathbf{x}\left(t\right)\right)c_1({\alpha }_k,x^k){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)dt=$$

$${\int }_a^b\left(\sum _{k=1}^n{\displaystyle \frac{1}{c_1({\alpha }_k,x_k\left(t\right))}}{\displaystyle \frac{\partial U\left(\mathbf{x}\right(t\left)\right)}{\partial x^k\left(t\right)}}{c}_1({\alpha }_k,x^k\left(t\right)){\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)dt=$$

$${\int }_a^b\left(\sum _{k=1}^n{\left({\displaystyle \frac{\partial U\left(\mathbf{x}\right)}{\partial x^k}}\right)}_{\mathbf{x}=\mathbf{x}\left(t\right)}{\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)dt=$$

$${\int }_a^b{\displaystyle \frac{dU\left(\mathbf{x}\right(t\left)\right)}{dt}}dt={\int }_{U\left(x\right(a\left)\right)}^{U\left(x\right(b\left)\right)}dU\left(\mathbf{x}\right)=U\left(\mathbf{x}\left(b\right)\right)-U\left(\mathbf{x}\left(a\right)\right).$$

(753)

This proves the theorem. □

Remark 56. 

One can see that to prove Equation (747) in the metric approach, we can repeat the proof for the product measure approach with the functions

$${\widehat{c}}_1({\alpha }_j,x^j)=1$$

(754)

for all $j=1,\dots ,n$ in Equations (748)–(750), and $\widehat{U}\left(\mathbf{x}\right)=U\left(\mathbf{x}\right)$. This allows us to obtain Equations (751)–(753) in the proof of the metric approach. Here, we use (754) to mean

$${\widehat{c}}_{n,k}(D,\mathbf{x})=c_k({\alpha }_k,x^k)$$

(755)

for all $k=1,\dots ,n$. This proves Equation (747). Therefore, the proof of theorems in the metric approach can be derived from the proof of the theorems in the product measure approach.

Remark 57. 

According to the principle of the self-consistency of NIDS vector calculus (Principle 9), the differential and integral NIDS vector calculi must be consistent. Then, the form of differential operators ${\widehat{Grad}}_{\left(\alpha \right)}$ and ${Grad}_{\left(\alpha \right)}$ must be determined by the fact that the gradient theorem must be satisfied for the product measure approach and the metric approach, respectively. It can be said that the explicit expression of the NIDS gradient is derived in the proof of Theorem 32.

The NIDS gradient of the scalar field must be defined such that some NIDS generalization of the gradient theorems is satisfied and can be proved for the line NIDS integrals that satisfy the correspondence principles of the line NIDS integrals. Theorem 32 allows us to define the NIDS gradient. Therefore, we can give the following corollary and definition.

Corollary 3. 

In order for the gradient theorem for NIDS (Theorem 32) to be satisfied, the NIDS grad operators ${\widehat{Grad}}_{\left(\alpha \right)}$ and ${Grad}_{\left(\alpha \right)}$ should be defined by the following definition.

Definition 29. 

Let Ω be a region of the D-dimensional space $W=W_1\times \dots \times W_n$, where $dim\left(W_k\right)={\alpha }_k\in (0,1]$ for all $k=1,\dots ,n$.

(1) The Product Measure Approach: the NIDS gradient of the scalar function $U=U\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^1(\Omega )$ is defined by the equation

$${\widehat{Grad}}_{\left(\alpha \right)}U\left(\mathbf{x}\right):=\sum _{k=1}^n{\mathbf{e}}^k\widehat{D_{x^k}^{{\alpha }_k}}U\left(\mathbf{x}\right)=$$

$$\sum _{k=1}^n{\displaystyle \frac{1}{{\widehat{c}}_{n,k}(D,\mathbf{x})}}{\displaystyle \frac{\partial \widehat{U}\left(\mathbf{x}\right)}{\partial x^k}}{\mathbf{e}}^k=\sum _{k=1}^n{\mathbf{e}}^k{\displaystyle \frac{1}{c_1({\alpha }_k,x^k)}}{\displaystyle \frac{\partial \left({\widehat{c}}_1({\alpha }_k,x^k)U\left(\mathbf{x}\right)\right)}{\partial x^k}},$$

(756)

where

$$\widehat{U}\left(\mathbf{x}\right):={\widehat{c}}_n(D,\mathbf{x})U\left(\mathbf{x}\right),$$

(757)

$$\widehat{D_{x^k}^{{\alpha }_k}}U\left(\mathbf{x}\right):={\displaystyle \frac{1}{{\widehat{c}}_{n,k}(D,\mathbf{x})}}{\displaystyle \frac{\partial \widehat{U}\left(\mathbf{x}\right)}{\partial x^k}}={\displaystyle \frac{1}{c_1({\alpha }_k,x^k)}}{\displaystyle \frac{\partial \left({\widehat{c}}_1({\alpha }_k,x^k)U\left(\mathbf{x}\right)\right)}{\partial x^k}},$$

(758)

$${\widehat{c}}_{n,k}(D,\mathbf{x})=c_1({\alpha }_k,x^k)\prod _{j=1,j\ne k}^n{\widehat{c}}_1({\alpha }_j,x^k),$$

(759)

and $\left(\alpha \right)$ denotes the set $({\alpha }_1,\dots ,{\alpha }_n)$.

(2) The Metric Approach: the NIDS gradient of the scalar function $U=U\left(\mathbf{x}\right)\in C_{\left(\alpha \right)}^1(\Omega )$ is defined by the equation

$${Grad}_{\left(\alpha \right)}U\left(\mathbf{x}\right):=\sum _{k=1}^n{\displaystyle \frac{1}{c_1({\alpha }_k,x^k)}}{\displaystyle \frac{\partial U\left(\mathbf{x}\right)}{\partial x^k}}{\mathbf{e}}^k:=\sum _{k=1}^n{\mathbf{e}}^k{D}_{x^k}^{{\alpha }_k}U\left(\mathbf{x}\right),$$

(760)

where $\left(\alpha \right)$ denotes the set $({\alpha }_1,\dots ,{\alpha }_n)$.

Using the definition of the line NIDS integrals ${\mathbf{I}}_L^D$ and ${\mathbf{I}}_L^{D,\ast }$ and the NIDS gradients (756) and (760), the gradient theorem for NIDS (Theorem 32) can be formulated by the following corollary.

Corollary 4. 

The gradient theorem for NIDS (Theorem 32) can be represented by the following Equations.

(1) The Product Measure Approach: Equation (745) of the NIDS gradient theorem can be written as

$$\left({\mathbf{I}}_L^D,{\widehat{Grad}}_{\left(\alpha \right)}U\right)=\widehat{U}\left(\mathbf{x}\left(b\right)\right)-\widehat{U}\left(\mathbf{x}\left(a\right)\right)$$

(761)

for the piecewise continuously differentiable curve L described by the equation $\mathbf{x}=\mathbf{x}\left(t\right)$ for $a⩽t⩽b$ in the D-dimensional space.

(2) The Metric Approach: Equation (747) of the NIDS Gradient Theorem can be written as

$$\left({\mathbf{I}}_L^{D,\ast },{Grad}_{\left(\alpha \right)}U\right)=U\left(\mathbf{x}\left(b\right)\right)-U\left(\mathbf{x}\left(a\right)\right)$$

(762)

for the piecewise continuously differentiable curve L described by the equation $\mathbf{x}=\mathbf{x}\left(t\right)$ for $a⩽t⩽b$ in the D-dimensional space.

Remark 58. 

Let us consider the arguments justifying the importance of the function

$$\widehat{U}\left(\mathbf{x}\left(t\right)\right)={\widehat{c}}_n(D,\mathbf{x}\left(t\right))U\left(\mathbf{x}\left(t\right)\right),$$

(763)

where

$${\widehat{c}}_n(D,\mathbf{x}\left(t\right)):=\prod _{j=1}^n{\widehat{c}}_1({\alpha }_j,x^j\left(t\right)),$$

(764)

$${\widehat{c}}_1({\alpha }_j,x^j\left(t\right)):=\left\{\begin{array}{cc}1\hfill & {\displaystyle \mathit{if}d{x}^j\left(t\right)/dt=0,}\hfill \\ c_1({\alpha }_j,x^j\left(t\right))\hfill & {\displaystyle \mathit{if}d{x}^j\left(t\right)/dt\ne 0.}\hfill \end{array}\right.$$

(765)

Function (763) is actually used in the following equations.

(1) First, the function $\widehat{U}\left(\mathbf{x}\right)$ is explicitly used in the definition of the line integral of a scalar function. Let us consider the D-dimensional space $W=W_1\times \dots \times W_n$ with $D\in (0,n]$, $n\in \mathbb{N}$, and a continuously differentiable curved line L described by equations $\mathbf{x}=\mathbf{x}\left(t\right)$ with $t\in [a,b]$, where $\mathbf{x}\left(t\right)=(x_1\left(t\right),\dots ,x_n\left(t\right))$ are points of the NIDS for all $t\in [a,b]$. In definition 12, the line NIDS integral of the scalar field $U\left(\mathbf{x}\right(t)$ along a curve L is defined as

$$I_L^{D}U\left(\mathbf{x}\right)={\int }_a^b\widehat{U}\left(\mathbf{x}\left(t\right)\right)\sqrt{\sum _{k=1}^n{\left({\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt=$$

(766)

$${\int }_a^{b}U\left(\mathbf{x}\left(t\right)\right){\widehat{c}}_n(D,\mathbf{x}\left(t\right))\sqrt{\sum _{k=1}^n{\left({\displaystyle \frac{d{x}^k\left(t\right)}{dt}}\right)}^2}dt.$$

(767)

(2) Second, the use of the function (763) allows us to correctly connect the concept of differential and derivative in the product measure approach.

In the integer-dimensional space, the differential of the function $U\left(\mathbf{x}\right)$ is related to the partial derivatives

$$dU\left(\mathbf{x}\right)=\sum _{i=1}^n{\displaystyle \frac{\partial U\left(\mathbf{x}\right)}{\partial x^i}}d{x}^i.$$

(768)

In the NIDS, this statement is not true in the product measure approach. In the D-dimensional space $W=W_1\times \dots \times W_n$ with $D={\alpha }_1+\dots +{\alpha }_n$, the differential of the function $U\left(\mathbf{x}\right)$ has the form

$$dU\left(\mathbf{x}\right)=\sum _{i=1}^n{\displaystyle \frac{\partial U\left(\mathbf{x}\right)}{\partial x^i}}d{x}^i=$$

$$\sum _{i=1}^n{\displaystyle \frac{1}{c_1({\alpha }_i,x^i)}}{\displaystyle \frac{\partial U\left(\mathbf{x}\right)}{\partial x^i}}{c}_1({\alpha }_i,x^i)d{x}^i=\sum _{i=1}^n{D}_{x^i}^{{\alpha }_i}U\left(\mathbf{x}\right)c_1({\alpha }_i,x^i)d{x}^i,$$

(769)

where $D_{x^i}^{{\alpha }_i}$ is used instead of the NIDS derivative ${\widehat{D}}_{x^i}^{{\alpha }_i}$, which should be used in the product measure approach.

Therefore, in the product measure approach, we must use the function $\widehat{U}\left(\mathbf{x}\right)$ instead of $U\left(\mathbf{x}\right)$. Then, the differential of the function (763) is

$$d\widehat{U}\left(\mathbf{x}\right):=\sum _{i=1}^n{\displaystyle \frac{\partial \widehat{U}\left(\mathbf{x}\right)}{\partial x^i}}d{x}^i=$$

$$\sum _{i=1}^n{\displaystyle \frac{\partial }{\partial x^i}}\left(U\left(\mathbf{x}\right)\prod _{j=1}^n{\widehat{c}}_1({\alpha }_j,x^j)\right)d{x}^i=$$

$$\sum _{i=1}^n\left(\prod _{p=1,p\ne i}^n{\widehat{c}}_1({\alpha }_p,x^p)\right){\displaystyle \frac{\partial \left({\widehat{c}}_1({\alpha }_i,x^i)U\left(\mathbf{x}\right)\right)}{\partial x^i}}d{x}^i=$$

$$\sum _{i=1}^n\left(\prod _{p=1,p\ne i}^n{\widehat{c}}_1({\alpha }_p,x^p)\right){\displaystyle \frac{1}{c_1({\alpha }_i,x^i)}}{\displaystyle \frac{\partial \left({\widehat{c}}_1({\alpha }_i,x^i)U\left(\mathbf{x}\right)\right)}{\partial x^i}}{c}_1({\alpha }_i,x^i)d{x}^i=$$

$$\sum _{i=1}^n\widehat{D_{x^i}^{{\alpha }_i}}U\left(\mathbf{x}\right){\widehat{c}}_{n,i}(D,\mathbf{x})d{x}^i,$$

(770)

where

$${\widehat{c}}_{n,i}(D,\mathbf{x})=c_1({\alpha }_i,x^i)\prod _{p=1,p\ne i}^n{\widehat{c}}_1({\alpha }_p,x^p).$$

(771)

As a result, we get the differential

$$d\widehat{U}\left(\mathbf{x}\right)=\sum _{i=1}^n\widehat{D_{x^i}^{{\alpha }_i}}U\left(\mathbf{x}\right){\widehat{c}}_{n,i}(D,\mathbf{x})d{x}^i$$

(772)

that is connected with the NIDS derivative $\widehat{D_{x^i}^{{\alpha }_i}}$.

(3) Third, in the following section on differential forms in NIDS (Section 4.8), the importance of this type of functions for a unified description of integrands will be demonstrated. Note that not only are the functions

$$\widehat{U}\left(\mathbf{x}\right)=\left(\prod _{p=1}^n{\widehat{c}}_1({\alpha }_p,x^p)\right)U\left(\mathbf{x}\right)$$

(773)

important, but the functions

$$\widehat{F_k}\left(\mathbf{x}\right)=\left(\prod _{p=1,p\ne i}^n{\widehat{c}}_1({\alpha }_p,x^p)\right)F_k\left(\mathbf{x}\right),$$

(774)

$$\widehat{F_{ij}}\left(\mathbf{x}\right)=\left(\prod _{p=1,p\ne i,j}^n{\widehat{c}}_1({\alpha }_p,x^p)\right)F_{ij}\left(\mathbf{x}\right),$$

(775)

and so on are also important. The differential forms give a unified approach to define line, surface, and volume NIDS integrations in the D-dimensional spaces with $D\in (0,n]$, $n\in \mathbb{N}$.

4.4.2. NIDS Green Theorem

In this section, we prove the NIDS generalization of the Green theorem, which connects a line NIDS integral taken over the boundary of a certain region with a surface NIDS integral over that region itself.

The NIDS generalization of the Green theorem will be proved for the simple region G of the D-dimensional space $W=W_1\times W_2$ with $D={\alpha }_1+{\alpha }_2\in (0,2]$. Let us give the definition of the simple region.

Definition 30. 

Let $W=W_1\times W_2$ be a D-dimensional space with $D={\alpha }_1+{\alpha }_2\in (0,2]$.

The region $G\subset W$ is called simple, if the following conditions are satisfied.

(1) The region G is bounded from below and above by the piecewise smooth curves

$$G=\{(x,y):a⩽x⩽b,y_1\left(x\right)⩽y⩽y_2\left(x\right)\}$$

(776)

with the boundary $L=\partial G$, such that $L=L_1\bigcup L_2$,

$$L_1:=\{(x,y):a⩽x⩽b,y=y_1\left(x\right)\},$$

(777)

$$L_2:=\{(x,y):a⩽x⩽b,y=y_2\left(x\right)\},$$

(778)

where $y_1\left(a\right)=y_2\left(a\right)$, and $y_1\left(b\right)=y_2\left(b\right)$.

(2) The region G is bounded from the right and left by the piecewise smooth curves

$$G=\{(x,y):c⩽y⩽d,x_1\left(y\right)⩽x⩽x_2\left(y\right)\},$$

(779)

and the boundary $L=\partial G$ can be represented as $L=L_3\bigcup L_4$ with

$$L_3:=\{(x,y):c⩽y⩽d,x=x_1\left(y\right)\},$$

(780)

$$L_4:=\{(x,y):c⩽y⩽d,x=x_2\left(y\right)\},$$

(781)

where $x_1\left(c\right)=x_2\left(c\right)$ and $x_1\left(d\right)=x_2\left(d\right)$.

Under the assumptions, which are given in Definition 30, the following NIDS generalization of the Green theorem holds.

Theorem 33 

(Green Theorem for NIDS). Let $W=W_1\times W_2$ be a D-dimensional space with $D={\alpha }_1+{\alpha }_2\in (0,2]$, and let G be a simple closed region.

Then, the following equations are satisfied.

(1) In the product measure approach, we have the equation

$${\int }_{\partial G}\left(F_x(x,y){\widehat{c}}_{2,x}(D,\mathbf{x})dx+F_y(x,y){\widehat{c}}_{2,y}(D,\mathbf{x})dy\right)=$$

$${\int }_G\left({\displaystyle \frac{1}{c_1({\alpha }_1,x)}}{\displaystyle \frac{\partial \left({\widehat{c}}_1({\alpha }_1,x)F_y\right)}{\partial x}}-{\displaystyle \frac{1}{c_1({\alpha }_2,y)}}{\displaystyle \frac{\partial \left({\widehat{c}}_1({\alpha }_2,y)F_x\right)}{\partial y}}\right)c_1({\alpha }_1,x)c_1({\alpha }_2,y)dxdy,$$

(782)

if $F_x(x,y),F_y(x,y)\in {\widehat{C}}_{\left(\alpha \right)}^1\left(G\right)$, where

$${\widehat{c}}_{2,x}(D,\mathbf{x})=c_1({\alpha }_1,x){\widehat{c}}_1({\alpha }_2,y),$$

(783)

$${\widehat{c}}_{2,y}(D,\mathbf{x})={\widehat{c}}_1({\alpha }_1,x)c_1({\alpha }_2,y),$$

(784)

and ${\alpha }_1,{\alpha }_2\in (0,1]$.

(2) In the metric approach, we have the equation

$${\int }_{\partial G}\left(F_x(x,y)c_1({\alpha }_1,x)dx+F_y(x,y)c_1({\alpha }_2,y)dy\right)=$$

$${\int }_G\left({\displaystyle \frac{1}{c_1({\alpha }_1,x)}}{\displaystyle \frac{\partial F_y}{\partial x}}-{\displaystyle \frac{1}{c_1({\alpha }_2,y)}}{\displaystyle \frac{\partial F_x}{\partial y}}\right)c_1({\alpha }_1,x)c_1({\alpha }_2,y)dxdy,$$

(785)

if $F_x(x,y),F_y(x,y)\in C_{\left(\alpha \right)}^1\left(G\right)$.

Proof. 

Let us consider the boundary $L=\partial G$ of the domain G to be positively oriented, i.e., we will consider the direction is such that the domain G itself remains on the left all the time.

(1) In the product measure approach, let us define the function

$$\widehat{P}(x,y):=c_1({\alpha }_1,x){\widehat{c}}_1({\alpha }_2,y)F_x(x,y),$$

(786)

$$\widehat{Q}(x,y):={\widehat{c}}_1({\alpha }_1,x)c_1({\alpha }_2,y)F_y(x,y).$$

(787)

Note that, if ${\widehat{c}}_1({\alpha }_1,x)=1$, and ${\widehat{c}}_1({\alpha }_2,y)=1$, then Equation (782) gives Equation (785). Therefore, using the notations (786) and (787), it is possible to prove the Equations (782) and (785) together.

Let the function $\widehat{P}(x,y)$ be defined and continuous together with its partial derivative $\partial \widehat{P}(x,y)/\partial y$ in the entire domain G, including its boundary $\partial G$. Using the function $\widehat{P}(x,y)$, the simple region (776) and its boundary (777) and (778), we can perform the standard transformations of the classical Green theorem proof:

$${\int }_{\partial G}{F}_x(x,y){\widehat{c}}_{2,x}(D,\mathbf{x})dx={\int }_L\widehat{P}(x,y)dx=$$

$${\int }_{L_1}\widehat{P}(x,y_1\left(x\right))dx+{\int }_{L_2}\widehat{P}(x,y_2\left(x\right))dx=$$

$${\int }_a^b\widehat{P}(x,y_1\left(x\right))dx-{\int }_a^b\widehat{P}(x,y_2\left(x\right))dx={\int }_a^b\left(\widehat{P}(x,y_1\left(x\right))-\widehat{P}(x,y_2\left(x\right))\right)dx=$$

$$-{\int }_a^{b}dx{\int }_{y_1\left(x\right)}^{y_2\left(x\right)}{\displaystyle \frac{\partial \widehat{P}(x,y)}{\partial y}}dy=-{\int }_G{\displaystyle \frac{\partial \widehat{P}(x,y)}{\partial y}}dxdy.$$

(788)

Using Equation (786), we get

$${\int }_G{\displaystyle \frac{\partial \widehat{P}(x,y)}{\partial y}}dxdy={\int }_G{\displaystyle \frac{1}{c_2(D,\mathbf{x})}}{\displaystyle \frac{\partial \widehat{P}(x,y)}{\partial y}}{c}_2(D,\mathbf{x})dxdy=$$

$${\int }_G{\displaystyle \frac{1}{c_1({\alpha }_1,x)c_1({\alpha }_2,y)}}{\displaystyle \frac{\partial \left(c_1({\alpha }_1,x){\widehat{c}}_1({\alpha }_2,y)F_x(x,y)\right)}{\partial y}}{c}_2(D,\mathbf{x})dxdy=$$

(789)

$${\int }_G{\displaystyle \frac{1}{c_1({\alpha }_2,y)}}{\displaystyle \frac{\partial \left({\widehat{c}}_1({\alpha }_2,y)F_x(x,y)\right)}{\partial y}}{c}_2(D,\mathbf{x})dxdy,$$

(790)

where we use

$$c_2(D,\mathbf{x}):=c_1({\alpha }_1,x)c_1({\alpha }_2,y).$$

(791)

Let the function $\widehat{Q}(x,y)$ be defined and continuous together with its partial derivative $\partial \widehat{Q}(x,y)/\partial x$ in the entire domain G, including its boundary $L=\partial G$. Using the function $\widehat{Q}(x,y)$, the simple region (779), and its boundary (780) and (781), we can perform the standard transformations of the classical Green theorem proof:

$${\int }_{\partial G}{F}_y(x,y){\widehat{c}}_{2,y}(D,\mathbf{x})dy={\int }_L\widehat{Q}(x,y)dy=$$

$${\int }_{L_4}\widehat{Q}(x_2\left(y\right),y)dy+{\int }_{L_3}\widehat{Q}(x_1,y))dy=$$

$${\int }_c^d\widehat{Q}(x_2\left(y\right),y)dy-{\int }_c^d\widehat{Q}(x_1\left(y\right),y)dy={\int }_c^d\left(\widehat{Q}(x_2\left(y\right),y)-\widehat{Q}(x_1\left(y\right),y)\right)dy=$$

$$-{\int }_c^{d}dy{\int }_{x_1\left(y\right)}^{x_2\left(y\right))}{\displaystyle \frac{\partial \widehat{Q}(x,y)}{\partial x}}dx={\int }_G{\displaystyle \frac{\partial \widehat{Q}(x,y)}{\partial x}}dxdy.$$

(792)

Using Equation (787), we get

$${\int }_G{\displaystyle \frac{\partial \widehat{Q}(x,y)}{\partial x}}dxdy={\int }_G{\displaystyle \frac{1}{c_2(D,\mathbf{x})}}{\displaystyle \frac{\partial \widehat{Q}(x,y)}{\partial x}}{c}_2(D,\mathbf{x})dxdy=$$

$${\int }_G{\displaystyle \frac{1}{c_1({\alpha }_1,x)c_1({\alpha }_2,y)}}{\displaystyle \frac{\partial \left({\widehat{c}}_1({\alpha }_1,x)c_1({\alpha }_2,y)F_y(x,y)\right)}{\partial x}}{c}_2(D,\mathbf{x})dxdy=$$

(793)

$${\int }_G{\displaystyle \frac{1}{c_1({\alpha }_1,x)}}{\displaystyle \frac{\partial \left({\widehat{c}}_1({\alpha }_1,x)F_y(x,y)\right)}{\partial x}}{c}_2(D,\mathbf{x})dxdy.$$

(794)

As a result, Equations (788) and (792) give Equation (782). Equation (785) is a special case of Equation (782), when ${\widehat{c}}_1({\alpha }_1,x)={\widehat{c}}_1({\alpha }_2,y)=1$.

(2) In the metric approach, the Green theorem is proved similarly. In fact, it is sufficient to use

$${\widehat{c}}_1({\alpha }_j,x^j)=1$$

(795)

for all $j=1,2$ and equations in the proof. In this case (824), we get

$${\widehat{c}}_{2,x}(D,\mathbf{x})=c_1({\alpha }_1,x),$$

(796)

$${\widehat{c}}_{2,y}(D,\mathbf{x})=c_1({\alpha }_2,y),$$

(797)

where $k=1,2$ and $x_1=x$, $x_2=y$. □

Remark 59. 

Let us note the property of functions ${\widehat{c}}_1({\alpha }_1,x)$ and ${\widehat{c}}_1({\alpha }_2,y)$ on the boundary $L=\partial G$.

The function ${\widehat{c}}_1({\alpha }_2,y)$ on the line $L_1$ can be briefly written as

$${\widehat{c}}_1({\alpha }_2,y_1\left(x\right))=\left\{\begin{array}{cc}1\hfill & \mathit{if}{y}_1\left(x\right)=const,\hfill \\ c_1({\alpha }_2,y_1\left(x\right))\hfill & \mathit{if}{y}_1\left(x\right)\ne const.\hfill \end{array}\right.$$

(798)

This means that if there exists a finite interval $T_j\subset [a,b]$, such that $y_1\left(x\right)=const$ for all $x\in T_j$, then ${\widehat{c}}_1({\alpha }_2,y_1\left(x\right))=1$ for $x\in T_j$. If there are no finite intervals $T_j\subset [a,b]$, such that $y_1\left(x\right)=const$ for $x\in T_j$, then ${\widehat{c}}_1({\alpha }_2,y_1\left(x\right))=c_1({\alpha }_2,y_1\left(x\right))$ for all $x\in [a,b]$. The constancy of the function $y=y_1\left(x\right)$ means that the finite segment of the line $L_1$ is parallel to the coordinate Y-axis.

The property of the function ${\widehat{c}}_1({\alpha }_2,y_2\left(x\right))$ on the line $L_2$ is described similarly. The properties of the functions ${\widehat{c}}_1({\alpha }_1,x_1\left(y\right))$ and ${\widehat{c}}_1({\alpha }_1,x_2\left(y\right))$ on the line $L_2$ are also described similarly.

Remark 60. 

The Green theorem for NIDS (Theorem 33) can be represented in the following forms.

(1) In the product measure approach, we have

$$\left({\mathbf{I}}_{\partial G}^D,\mathbf{F}\right)=I_G^D\left({\widehat{D}}_x^{{\alpha }_1}{F}_y\left(\mathbf{x}\right)-{\widehat{D}}_y^{{\alpha }_2}{F}_x\left(\mathbf{x}\right)\right),$$

(799)

where $I_G^D$ is the volume NIDS integral in the D-dimensional space with $D={\alpha }_1+{\alpha }_2\in (0,2]$, and

$${\widehat{D}}_{x^k}^{{\alpha }_k}f\left(\mathbf{x}\right):={\displaystyle \frac{1}{c_1({\alpha }_k,x^k)}}{\displaystyle \frac{\partial \left({\widehat{c}}_1({\alpha }_k,x^k)f\left(\mathbf{x}\right)\right)}{\partial x^k}}$$

(800)

with $k=1,2$, and $x_1=x$, $x_2=y$.

(2) In the metric approach, we have

$$\left({\mathbf{I}}_{\partial G}^{D,\ast },\mathbf{F}\right)=I_G^D\left(D_x^{{\alpha }_1}{F}_y\left(\mathbf{x}\right)-D_y^{{\alpha }_2}{F}_x\left(\mathbf{x}\right)\right),$$

(801)

where $I_G^D$ is the volume NIDS integral in the D-dimensional space with $D={\alpha }_1+{\alpha }_2\in (0,2]$, and

$$D_{x^k}^{{\alpha }_k}f\left(\mathbf{x}\right):={\displaystyle \frac{1}{c_1({\alpha }_k,x^k)}}{\displaystyle \frac{\partial f\mathbf{x})}{\partial x^k}}$$

(802)

with $k=1,2$, and $x_1=x$, $x_2=y$.

4.4.3. NIDS Stokes Theorem and NIDS Curl Operators

Let $\Omega$ be a surface without singular points in the NIDS such that

$$\Omega :=\{\mathbf{x}=(x,y,z):x^k=x^k(u,v),k=1,2,3,(u,v)\in Ux(u,v)\in C^2\left(U\right)\},$$

(803)

where $x^k(u,v)$ are twice continuously differentiable functions in the flat bounded domain U, for which Green’s formula is valid. Let $L_0$ be a positively oriented contour bounding the domain U that is described by the equations $u=u\left(t\right)$, $v=v\left(t\right)$, and $t\in [a,b]$. Let $\mathbf{n}=\mathbf{n}(u,v)$ be the normal to the surface $\Omega$, defined by the equation

$$\mathbf{n}(u,v):={\displaystyle \frac{\mathbf{N}(u,v)}{|\mathbf{N}(u,v)|}},$$

(804)

where

$$\mathbf{N}(u,v):=[{\mathbf{x}}_u^{\left(1\right)}\times {\mathbf{x}}_v^{\left(1\right)}]={\displaystyle \frac{\partial (y,z)}{\partial (u,v)}}\mathbf{i}+{\displaystyle \frac{\partial (z,x)}{\partial (u,v)}}\mathbf{j}+{\displaystyle \frac{\partial (x,y)}{\partial (u,v)}}\mathbf{k},$$

(805)

and $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are unit coordinate vectors, and $\partial (y,z)/\partial (u,v)$, $\partial (z,x)/\partial (u,v)$, and $\partial (x,y)/\partial (u,v)$ are the Jacobians of the coordinate transformations.

Under these assumptions, the following NIDS generalization of the Stokes theorem holds.

Theorem 34 

(Stokes Theorem for NIDS). Let $W=W_1\times W_2\times W_3$ be a D-dimensional space with $D={\alpha }_1+{\alpha }_2+{\alpha }_3\in (0,3]$. Let L be a contour bounding region (803) on the surface, which is described by the equations $x^k=x^k(u\left(t\right),v\left(t\right))$ with $t\in [a,b]$ ($k=1,2,3$) in the region G of the space W, and $\Omega \subset G$.

(1) The Product Measure Approach: for the vector field $\mathbf{F}=\mathbf{F}\left(\mathbf{x}\right)$, which satisfies the conditions

$$F_k\left(\mathbf{x}\right),{\displaystyle \frac{\partial \left({\widehat{c}}_{3,j}(D,\mathbf{x})F_k\left(\mathbf{x}\right)\right)}{\partial x^j}}\in C(\Omega )(j\ne k)$$

(806)

for all $k,l=1,2,3$, the following equation is satisfied in the coordinate form

$${\int }_L\left({\widehat{c}}_{3,x}(D,\mathbf{x})F_x\left(\mathbf{x}\right)dx+{\widehat{c}}_{3,y}(D,\mathbf{x})F_y\left(\mathbf{x}\right)dy+{\widehat{c}}_{3,z}(D,\mathbf{x})F_z\left(\mathbf{x}\right)dz\right)=$$

$${\int }_{\Omega }\left({\widehat{D}}_x^{{\alpha }_1}{F}_y\left(\mathbf{x}\right)-{\widehat{D}}_y^{{\alpha }_2}{F}_x\left(\mathbf{x}\right)\right){\widehat{c}}_{3,xy}(D,\mathbf{x})dxdy+$$

$${\int }_{\Omega }\left({\widehat{D}}_y^{{\alpha }_2}{F}_z\left(\mathbf{x}\right)-{\widehat{D}}_z^{{\alpha }_3}{F}_y\left(\mathbf{x}\right)\right){\widehat{c}}_{3,yz}(D,\mathbf{x})dydz+$$

$${\int }_{\Omega }\left({\widehat{D}}_z^{{\alpha }_3}{F}_x\left(\mathbf{x}\right)-{\widehat{D}}_x^{{\alpha }_1}{F}_z\left(\mathbf{x}\right)\right){\widehat{c}}_{3,xz}(D,\mathbf{x})dzdx,$$

(807)

where

$${\widehat{D}}_{x^k}^{{\alpha }_k}f\left(\mathbf{x}\right):={\displaystyle \frac{1}{c_1({\alpha }_k,x^k)}}{\displaystyle \frac{\partial \left({\widehat{c}}_1({\alpha }_k,x^k)f\left(\mathbf{x}\right)\right)}{\partial x^k}}$$

(808)

with $k=1,2,3$, and $x_1=x$, $x_2=y$, and $x_3=z$.

(2) The Metric Approach: For the vector field $\mathbf{F}=\mathbf{F}\left(\mathbf{x}\right)$, which satisfies the conditions

$$F_k\left(\mathbf{x}\right),{\displaystyle \frac{\partial F_k\left(\mathbf{x}\right)}{\partial x^j}}\in C(\Omega )(j\ne k)$$

(809)

for all $k,l=1,2,3$, the following equation is satisfied in the coordinate form

$${\int }_L\left(c_1({\alpha }_1,x)F_x\left(\mathbf{x}\right)dx+c_1({\alpha }_2,y)F_y\left(\mathbf{x}\right)dy+c_1({\alpha }_3,z)F_z\left(\mathbf{x}\right)dz\right)=$$

$${\int }_{\Omega }\left(D_x^{{\alpha }_1}{F}_y\left(\mathbf{x}\right)-D_y^{{\alpha }_2}{F}_x\left(\mathbf{x}\right)\right)c_2(d_{xy},\mathbf{x})dxdy+$$

$${\int }_{\Omega }\left(D_y^{{\alpha }_2}{F}_z\left(\mathbf{x}\right)-D_z^{{\alpha }_3}{F}_y\left(\mathbf{x}\right)\right)c_2(d_{yz},\mathbf{x})dydz+$$

$${\int }_{\Omega }\left(D_z^{{\alpha }_3}{F}_x\left(\mathbf{x}\right)-D_x^{{\alpha }_1}{F}_z\left(\mathbf{x}\right)\right)c_2(d_{xz},\mathbf{x})dzdx,$$

(810)

where

$$D_{x^k}^{{\alpha }_k}f\left(\mathbf{x}\right):={\displaystyle \frac{1}{c_1({\alpha }_k,x^k)}}{\displaystyle \frac{\partial f\left(\mathbf{x}\right)}{\partial x^k}}$$

(811)

with $k=1,2,3$, and $x_1=x$, $x_2=y$, and $x_3=z$.

Proof. 

(1) In the product measure approach, let us prove that

$${\int }_L{\widehat{c}}_{3,x}(D,\mathbf{x})F_x\left(\mathbf{x}\right)dx=$$

$${\int }_{\Omega }({\displaystyle \frac{1}{c_1({\alpha }_3,z)}}{\displaystyle \frac{\partial \left({\widehat{c}}_1({\alpha }_3,z)F_x\left(\mathbf{x}\right)\right)}{\partial z}}{\widehat{c}}_{3,xz}(D,\mathbf{x})dzdx-$$

$${\displaystyle \frac{1}{c_1({\alpha }_2,y)}}{\displaystyle \frac{\partial \left({\widehat{c}}_1({\alpha }_2,y)F_x\left(\mathbf{x}\right)\right)}{\partial y}}{\widehat{c}}_{3,xy}(D,\mathbf{x})dxdy).$$

(812)

Equations with $F_y\left(\mathbf{x}\right)$ and $F_z\left(\mathbf{x}\right)$ are proved similarly to the proof of Equation (812).

Let the function

$$\widehat{P}\left(\mathbf{x}\right)=\widehat{P}(x,y,z):={\widehat{c}}_{3,x}(D,\mathbf{x})F_x\left(\mathbf{x}\right)$$

(813)

be defined and continuous together with its partial derivative $\partial \widehat{P}(x,y,z)/\partial y$, $\partial \widehat{P}(x,y,z)/\partial z$ in the entire domain $\Omega$, including its boundary. Function (813) allows us perform the standard transformations (for example, see Section 52.4 in [143]) to prove the Stokes theorem in NIDS.

Using function (813), the line NIDS integral is

$${\int }_L{\widehat{c}}_{3,x}(D,\mathbf{x})F_x\left(\mathbf{x}\right)dx={\int }_a^b\left(\widehat{P}\left(\mathbf{x}(u\left(t\right),v\left(t\right))\right){\displaystyle \frac{dx\left(u\right(t),v(t\left)\right)}{dt}}\right)dt=$$

$${\int }_{L_0}\widehat{P}\left(\mathbf{x}(u,v)\right)\left({\displaystyle \frac{\partial x(u,v)}{\partial u}}du+{\displaystyle \frac{\partial x(u,v)}{\partial v}}dv\right)=$$

$${\int }_{L_0}\left(\widehat{P}\left(\mathbf{x}(u,v)\right){\displaystyle \frac{\partial x(u,v)}{\partial u}}du+\widehat{P}\left(\mathbf{x}(u,v)\right){\displaystyle \frac{\partial x(u,v)}{\partial v}}dv\right),$$

(814)

where $L_0=\partial U$ is a positively oriented contour that bounds the region U, in which the Green theorem holds, $L=\partial \Omega$, and we use

$${\displaystyle \frac{dx\left(u\right(t),v(t\left)\right)}{dt}}={\left({\displaystyle \frac{\partial x(u,v)}{\partial u}}\right)}_{\mathbf{u}=\mathbf{u}\left(t\right)}{\displaystyle \frac{du\left(t\right)}{dt}}+{\left({\displaystyle \frac{\partial x(u,v)}{\partial v}}\right)}_{\mathbf{u}=\mathbf{u}\left(t\right)}{\displaystyle \frac{dv\left(t\right)}{dt}}.$$

(815)

Using the classical Green theorem in the form

$${\int }_{L_0}\left(A(u,v)du+B(u,v)dv\right)={\int }_U\left({\displaystyle \frac{\partial B(u,v)}{\partial u}}-{\displaystyle \frac{\partial A(u,v)}{\partial v}}\right)dudv$$

(816)

for the functions

$$A(u,v)=\widehat{P}\left(\mathbf{x}(u,v)\right){\displaystyle \frac{\partial x(u,v)}{\partial u}},B(u,v)=\widehat{P}\left(\mathbf{x}(u,v)\right){\displaystyle \frac{\partial x(u,v)}{\partial v}},$$

(817)

we get Equation (814) in the form

$${\int }_{L_0}\left(\widehat{P}\left(\mathbf{x}(u,v)\right){\displaystyle \frac{\partial x(u,v)}{\partial u}}du+\widehat{P}\left(\mathbf{x}(u,v)\right){\displaystyle \frac{\partial x(u,v)}{\partial v}}dv\right)=$$

$${\int }_U\left({\displaystyle \frac{\partial }{\partial u}}\left(\widehat{P}\left(\mathbf{x}(u,v)\right){\displaystyle \frac{\partial x(u,v)}{\partial v}}\right)-{\displaystyle \frac{\partial }{\partial v}}\left(\widehat{P}\left(\mathbf{x}(u,v)\right){\displaystyle \frac{\partial x(u,v)}{\partial u}}\right)\right)dudv=$$

$${\int }_U(\left(\sum _{k=1}^3{\displaystyle \frac{\partial \widehat{P}\left(\mathbf{x}(u,v)\right)}{\partial x^k}}{\displaystyle \frac{\partial x^k(u,v)}{\partial u}}\right){\displaystyle \frac{\partial x(u,v)}{\partial v}}+{\displaystyle \frac{{\partial }^{2}x(u,v)}{\partial u\partial v}}-$$

$$\left(\sum _{k=1}^3{\displaystyle \frac{\partial \widehat{P}\left(\mathbf{x}(u,v)\right)}{\partial x^k}}{\displaystyle \frac{\partial x^k(u,v)}{\partial v}}\right){\displaystyle \frac{\partial x(u,v)}{\partial u}}-{\displaystyle \frac{{\partial }^{2}x(u,v)}{\partial v\partial u}})dudv=$$

$${\int }_U(\left(\sum _{k=1}^3{\displaystyle \frac{\partial \widehat{P}\left(\mathbf{x}(u,v)\right)}{\partial x^k}}{\displaystyle \frac{\partial x^k(u,v)}{\partial u}}\right){\displaystyle \frac{\partial x(u,v)}{\partial v}}-$$

$$\left(\sum _{k=1}^3{\displaystyle \frac{\partial \widehat{P}\left(\mathbf{x}(u,v)\right)}{\partial x^k}}{\displaystyle \frac{\partial x^k(u,v)}{\partial v}}\right){\displaystyle \frac{\partial x(u,v)}{\partial u}})dudv=$$

$${\int }_U({\displaystyle \frac{\partial \widehat{P}\left(\mathbf{x}(u,v)\right)}{\partial y}}\left({\displaystyle \frac{\partial y(u,v)}{\partial u}}{\displaystyle \frac{\partial x(u,v)}{\partial v}}-{\displaystyle \frac{\partial x(u,v)}{\partial u}}{\displaystyle \frac{\partial y(u,v)}{\partial v}}\right)+$$

$${\displaystyle \frac{\partial \widehat{P}\left(\mathbf{x}(u,v)\right)}{\partial z}}\left({\displaystyle \frac{\partial z(u,v))}{\partial u}}{\displaystyle \frac{\partial x(u,v)}{\partial v}}-{\displaystyle \frac{\partial x(u,v))}{\partial u}}{\displaystyle \frac{\partial z(u,v)}{\partial v}}\right))dudv=$$

$${\int }_U\left(-{\displaystyle \frac{\partial \widehat{P}\left(\mathbf{x}(u,v)\right)}{\partial y}}{\displaystyle \frac{\partial (x,y)}{\partial (u,v)}}+{\displaystyle \frac{\partial \widehat{P}\left(\mathbf{x}(u,v)\right)}{\partial z}}{\displaystyle \frac{\partial (z,x)}{\partial (u,v)}}\right)dudv=$$

$${\int }_{\Omega }\left(-{\displaystyle \frac{\partial \widehat{P}\left(\mathbf{x}\right)}{\partial y}}dxdy+{\displaystyle \frac{\partial \widehat{P}\left(\mathbf{x}\right)}{\partial z}}dzdx\right),$$

(818)

where $\partial (x,y)/\partial (u,v)$ and $\partial (z,x)/\partial (u,v)$ are the Jacobians of the coordinate transformations, and we used the identity

$${\displaystyle \frac{{\partial }^{2}x(u,v)}{\partial u\partial v}}-{\displaystyle \frac{{\partial }^{2}x(u,v)}{\partial v\partial u}}=0,$$

(819)

which is satisfied, since $x(u,v)\in C^2\left(U\right)$.

As a result, we prove that

$${\int }_L{\widehat{c}}_{3,x}(D,\mathbf{x})F_x\left(\mathbf{x}\right)dx={\int }_{\Omega }\left({\displaystyle \frac{\partial \left({\widehat{c}}_{3,x}(D,\mathbf{x})F_x\left(\mathbf{x}\right)\right)}{\partial z}}dzdx-{\displaystyle \frac{\partial \left({\widehat{c}}_{3,x}(D,\mathbf{x})F_x\left(\mathbf{x}\right)\right)}{\partial y}}dxdy\right).$$

(820)

The first integral on the right side of Equation (820) is

$${\int }_{\Omega }{\displaystyle \frac{\partial \left({\widehat{c}}_{3,x}(D,\mathbf{x})F_x\left(\mathbf{x}\right)\right)}{\partial z}}dzdx=$$

$${\int }_{\Omega }{\displaystyle \frac{1}{{\widehat{c}}_{3,xz}(D,\mathbf{x})}}{\displaystyle \frac{\partial \left({\widehat{c}}_{3,x}(D,\mathbf{x})F_x\left(\mathbf{x}\right)\right)}{\partial z}}{\widehat{c}}_{3,xz}(D,\mathbf{x})dzdx=$$

$${\int }_{\Omega }{\displaystyle \frac{1}{c_1({\alpha }_1,x){\widehat{c}}_1({\alpha }_2,y)c_1({\alpha }_3,z)}}{\displaystyle \frac{\partial \left(c_1({\alpha }_1,x){\widehat{c}}_1({\alpha }_2,y){\widehat{c}}_1({\alpha }_3,z)F_x\left(\mathbf{x}\right)\right)}{\partial z}}{\widehat{c}}_{3,xz}(D,\mathbf{x})dzdx=$$

$${\int }_{\Omega }{\displaystyle \frac{1}{c_1({\alpha }_3,z)}}{\displaystyle \frac{\partial \left({\widehat{c}}_1({\alpha }_3,z)F_x\left(\mathbf{x}\right)\right)}{\partial z}}{\widehat{c}}_{3,xz}(D,\mathbf{x})dzdx=$$

$${\int }_{\Omega }\left({\widehat{D}}_z^{{\alpha }_3}{F}_x\left(\mathbf{x}\right)\right){\widehat{c}}_{3,xz}(D,\mathbf{x})dzdx.$$

(821)

The second integral on the right side of Equation (820) is

$${\int }_{\Omega }{\displaystyle \frac{\partial \left({\widehat{c}}_{3,x}(D,\mathbf{x})F_x\left(\mathbf{x}\right)\right)}{\partial y}}dxdy=$$

$${\int }_{\Omega }{\displaystyle \frac{1}{{\widehat{c}}_{3,xy}(D,\mathbf{x})}}{\displaystyle \frac{\partial \left({\widehat{c}}_{3,x}(D,\mathbf{x})F_x\left(\mathbf{x}\right)\right)}{\partial y}}{\widehat{c}}_{3,xy}(D,\mathbf{x})dxdy=$$

$${\int }_{\Omega }{\displaystyle \frac{1}{(c_1({\alpha }_1,x)c_1({\alpha }_2,y){\widehat{c}}_1({\alpha }_3,z)}}{\displaystyle \frac{\partial \left(c_1({\alpha }_1,x){\widehat{c}}_1({\alpha }_2,y){\widehat{c}}_1({\alpha }_3,z)F_x\left(\mathbf{x}\right)\right)}{\partial y}}{\widehat{c}}_{3,xy}(D,\mathbf{x})dxdy=$$

$${\int }_{\Omega }{\displaystyle \frac{1}{c_1({\alpha }_2,y)}}{\displaystyle \frac{\partial \left({\widehat{c}}_1({\alpha }_2,y)F_x\left(\mathbf{x}\right)\right)}{\partial y}}{\widehat{c}}_{3,xy}(D,\mathbf{x})dxdy=$$

$${\int }_{\Omega }\left({\widehat{D}}_y^{{\alpha }_2}{F}_x\left(\mathbf{x}\right)\right){\widehat{c}}_{3,xy}(D,\mathbf{x})dxdy.$$

(822)

As a result, substituting Equations (821) and (822) into Equation (820), we get

$${\int }_L{\widehat{c}}_{3,x}(D,\mathbf{x})F_x\left(\mathbf{x}\right)dx={\int }_{\Omega }\left(\left({\widehat{D}}_z^{{\alpha }_3}{F}_x\left(\mathbf{x}\right)\right){\widehat{c}}_{3,xz}(D,\mathbf{x})dzdx-\left({\widehat{D}}_y^{{\alpha }_2}{F}_x\left(\mathbf{x}\right)\right){\widehat{c}}_{3,xy}(D,\mathbf{x})dxdy\right).$$

(823)

The equations with $F_2\left(\mathbf{x}\right)$ and $F_3\left(\mathbf{x}\right)$ are proved similarly. This proves Equation (807).

(2) In the metric approaach, the NIDS Stokes theorem is proved similarly. In fact, to prove Equation (810), it is sufficient to use

$${\widehat{c}}_1({\alpha }_j,x^j)=1$$

(824)

for all $j=1,2,3$ and equations in the proof of this theorem in the product measure approach. In the case (824), we get

$${\widehat{c}}_{3,x}(D,\mathbf{x})=c_1({\alpha }_1,x),$$

(825)

$${\widehat{c}}_{3,y}(D,\mathbf{x})=c_1({\alpha }_2,y),$$

(826)

$${\widehat{c}}_{3,z}(D,\mathbf{x})=c_1({\alpha }_3,z),$$

(827)

and

$${\widehat{c}}_{3,xz}(D,\mathbf{x})=c_2(d_{xz},\mathbf{x}),$$

(828)

$${\widehat{c}}_{3,yz}(D,\mathbf{x})=c_2(d_{yz},\mathbf{x}),$$

(829)

$${\widehat{c}}_{3,xy}(D,\mathbf{x})=c_2(d_{xy},\mathbf{x}),$$

(830)

where $k=1,2,3$, and $x_1=x$, $x_2=y$, $x_3=z$. This proves Equation (810). □

Remark 61. 

In the metric approach, the Stokes theorem for NIDS can be considered as a special case of the Stokes theorem for NIDS in the product measure approach, when all functions ${\widehat{c}}_1({\alpha }_k,x^k)$ are equal to one, and the NIDS derivatives are

$${\widehat{D}}_{x^k}^{{\alpha }_k}f\left(\mathbf{x}\right):={\displaystyle \frac{1}{c_1({\alpha }_k,x^k)}}{\displaystyle \frac{\partial f\left(\mathbf{x}\right)}{\partial x^k}}=D_{x^k}^{{\alpha }_k}f\left(\mathbf{x}\right),$$

(831)

for all $k=1,2,3$.

Remark 62. 

In accordance with the principle of the self-consistency of NIDS vector calculus (Principle 9), the NIDS integral calculus and the NIDS differential calculus must be mutually consistent. This means that the form of differential operators ${\widehat{Curl}}_{\left(\alpha \right)}$ and ${Curl}_{\left(\alpha \right)}$ must be determined by the fact that the Stokes theorem must be satisfied for the product measure approach and the metric approach, respectively. It can be said that the explicit expression of the NIDS Curl operator is derived in the proof of Theorem 34. Therefore, we can give the following corollary and definition.

Corollary 5. 

In order to the Stokes theorem for NIDS (Theorem 34) in the form (807) and (810) is satisfied, the NIDS curl operators ${\widehat{Curl}}_{\left(\alpha \right)}$ and ${Curl}_{\left(\alpha \right)}$ should be defined by the following definition.

Definition 31. 

Let $W=W_1\times W_2\times W_3$ be a D-dimensional space with $D={\alpha }_1+{\alpha }_2+{\alpha }_3\in (0,3]$. Let L be a contour bounding region (803) on the surface, described by the equations $x^k=x^k(u\left(t\right),v\left(t\right))$ with $t\in [a,b]$ ($k=1,2,3$) in the region G of the space W, and $\Omega \subset G$.

(1) In the product measure approach, the NIDS curl operator is defined as

$${\widehat{Curl}}_{\left(\alpha \right)}\mathbf{F}:=$$

$$\left({\widehat{D}}_y^{{\alpha }_2}{F}_z\left(\mathbf{x}\right)-{\widehat{D}}_z^{{\alpha }_3}{F}_y\left(\mathbf{x}\right)\right)\mathbf{i}+$$

$$\left({\widehat{D}}_z^{{\alpha }_3}{F}_x\left(\mathbf{x}\right)-{\widehat{D}}_x^{{\alpha }_1}{F}_z\left(\mathbf{x}\right)\right)\mathbf{j}+$$

$$\left({\widehat{D}}_x^{{\alpha }_1}{F}_y\left(\mathbf{x}\right)-{\widehat{D}}_y^{{\alpha }_2}{F}_x\left(\mathbf{x}\right)\right)\mathbf{k},$$

(832)

where

$$F_k\left(\mathbf{x}\right),{\displaystyle \frac{\partial \left({\widehat{c}}_{3,j}(D,\mathbf{x})F_k\left(\mathbf{x}\right)\right)}{\partial x^j}}\in C(\Omega )(j\ne k),$$

(833)

for all $k,l=1,2,3$,

$${\widehat{D}}_{x^k}^{{\alpha }_k}f\left(\mathbf{x}\right):={\displaystyle \frac{1}{c_1({\alpha }_k,x^k)}}{\displaystyle \frac{\partial \left({\widehat{c}}_1({\alpha }_k,x^k)f\left(\mathbf{x}\right)\right)}{\partial x^k}}$$

(834)

with $k=1,2,3$, $x_1=x$, $x_2=y$, and $x_3=z$, and $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are unit coordinate vectors.

(2) In the metrics approach, the NIDS curl operator is defined as

$${Curl}_{\left(\alpha \right)}\mathbf{F}:=$$

$$\left(D_y^{{\alpha }_2}{F}_z\left(\mathbf{x}\right)-D_z^{{\alpha }_3}{F}_y\left(\mathbf{x}\right)\right)\mathbf{i}+$$

$$\left(D_z^{{\alpha }_3}{F}_x\left(\mathbf{x}\right)-D_x^{{\alpha }_1}{F}_z\left(\mathbf{x}\right)\right)\mathbf{j}+$$

$$\left(D_x^{{\alpha }_1}{F}_y\left(\mathbf{x}\right)-D_y^{{\alpha }_2}{F}_x\left(\mathbf{x}\right)\right)\mathbf{k},$$

(835)

where

$$F_k\left(\mathbf{x}\right),{\displaystyle \frac{\partial F_k\left(\mathbf{x}\right)}{\partial x^j}}\in C(\Omega )(j\ne k)$$

(836)

for all $k,l=1,2,3$,

$$D_{x^k}^{{\alpha }_k}f\left(\mathbf{x}\right):={\displaystyle \frac{1}{c_1({\alpha }_k,x^k)}}{\displaystyle \frac{\partial f\left(\mathbf{x}\right)}{\partial x^k}}$$

(837)

with $k=1,2,3$, $x_1=x$, $x_2=y$, and $x_3=z$.

Using the definition of the line NIDS integrals (${\mathbf{I}}_L^D$ and ${\mathbf{I}}_L^{D,\ast }$), the surface NIDS integrals (${\mathbf{I}}_{\Omega }^D$ and ${\mathbf{I}}_{\Omega }^{D,\ast }$), and the NIDS curl operators (832) and (835), the Stokes theorem for NIDS can be formulated in the following corollary.

Corollary 6. 

The Stokes Theorem for NIDS (Theorem 34) can be represented by the following equations.

(1) The Product Measure Approach: for the vector field $\mathbf{F}=\mathbf{F}\left(\mathbf{x}\right)$, the circulation of $\mathbf{F}\left(\mathbf{x}\right)$ along the contour L is equal to the flow of the NIDS curl operator ${\widehat{Curl}}_{\left(\alpha \right)}$ of this vector field through the surface Ω bounded by the contour $L=\partial \Omega$:

$$\left({\mathbf{I}}_{\partial \Omega }^D,\mathbf{F}\right)=\left({\mathbf{I}}_{\Omega }^D,{\widehat{Curl}}_{\left(\alpha \right)}\mathbf{F}\right),$$

(838)

where the NIDS curl operator ${\widehat{Curl}}_{\left(\alpha \right)}$ is defined by Equation (832).

(2) The Metric Approach: for the vector field $\mathbf{F}=\mathbf{F}\left(\mathbf{x}\right)$, the circulation of $\mathbf{F}\left(\mathbf{x}\right)$ along the contour L is equal to the flow of the NIDS curl ${Curl}_{\left(\alpha \right)}$ of this vector field through the surface Ω bounded by the contour $L=\partial \Omega$:

$$\left({\mathbf{I}}_{\partial \Omega }^{D,\ast },\mathbf{F}\right)=\left({\mathbf{I}}_{\Omega }^D,{Curl}_{\left(\alpha \right)}\mathbf{F}\right),$$

(839)

where the NIDS curl operator ${Curl}_{\left(\alpha \right)}$ is defined by Equation (835).

Remark 63. 

Let us note the interrelations of various operators, which form self-consistent calculus in the D-dimensional space.

1. 

We can see that the NIDS curl operators are defined so that the Stokes theorem is satisfied for the line and surface NIDS integrals of vector fields.

2. 

These NIDS integrals of vector fields are defined through the line and surface NIDS integrals of scalar fields according to the principle of the self-consistency of NIDS vector calculus (Principle 9), which requires the consistency of these NIDS integrals.

3. 

The NIDS integrals of scalar fields satisfy the correspondence principle for volume, surface, and line NIDS integrals (Principle 4).

These mutual connections of integral and differential NIDS operators ensure the self-consistency of the NIDS calculus that is proposed in this paper.

4.4.4. NIDS Gauss–Ostrogradsky Theorem and NIDS Divergences

Let us define the simple region in NIDS.

Definition 32. 

Let Ω be a region of the D-dimensional space $W=W_1\times W_2\times W_3$ with $D={\alpha }_1+{\alpha }_2+{\alpha }_3\in (0,3]$ that is bounded by two piecewise smooth surfaces

$${\Sigma }_1=\{\mathbf{x}=(x,y,z):z=z_1(x,y)(x,y)\in U\},$$

(840)

$${\Sigma }_2=\{\mathbf{x}=(x,y,z):z=z_2(x,y)(x,y)\in U\},$$

(841)

where $z_1(x,y)⩽z_2(x,y)$ for all $(x,y)\in U$, and the surface

$${\Sigma }_3=\{\mathbf{x}=(x,y,z):z_1(x,y)⩽z⩽z_2(x,y)(x,y)\in \partial U\}$$

(842)

consists of vertical lines that are the generatrices parallel to the Z-axis. Then, this region Ω is called simple along the Z-axis or Z-simple. The union of these surfaces forms the entire boundary $\Sigma =\partial \Omega$ of the region Ω, such that

$$\Sigma ={\Sigma }_1\bigcup {\Sigma }_3\bigcup {\Sigma }_3.$$

(843)

Similarly, the X-simple region is a region bounded by piecewise smooth surfaces $x=x_1(y,z)$, $x=x_2(y,z)$, and a surface consisting of lines that are the generatrices parallel to the X-axis. The Y-simple region is defined in the same way. The region Ω is called simple, if it can be divided into a finite number of Z-simple, X-simple, and Y-regions.

Under the assumptions, which are given in Definition 32, the following NIDS generalization of the Gauss–Ostrogradsky Theorem holds.

Theorem 35 

(Gauss-Ostrogradsky Theorem for NIDS). Let Ω be a region of the D-dimensional space $W=W_1\times W_2\times W_3$ with $D={\alpha }_1+{\alpha }_2+{\alpha }_3\in (0,3]$.

(1) The Product Measure Approach: Let $F_k\left(\mathbf{x}\right)$ ($k=1,2,3$) be functions defined and continuous together with the partial derivatives $\partial \left({\widehat{c}}_1(\alpha ,x^k)F_k\left(\mathbf{x}\right)\right)/\partial x^k$ ($k=1,2,3$) in the domain Ω including its boundary $\Sigma =\partial \Omega$. Then, the following NIDS generalizations of the Gauss–Ostrogradsky equation are satisfied in the coordinate form

$${\int }_{\Sigma }\left(F_x\left(\mathbf{x}\right){\widehat{c}}_{3,yz}(D,\mathbf{x})dydz+F_y\left(\mathbf{x}\right){\widehat{c}}_{3,zx}(D,\mathbf{x})dzdx+F_z\left(\mathbf{x}\right){\widehat{c}}_{3,xy}(D,\mathbf{x})dxdy\right)=$$

$${\int }_{\Omega }\left({\widehat{D}}_x^{{\alpha }_1}{F}_x\left(\mathbf{x}\right))+{\widehat{D}}_y^{{\alpha }_2}{F}_y\left(\mathbf{x}\right))+{\widehat{D}}_z^{{\alpha }_3}{F}_z\left(\mathbf{x}\right))+\right)c_3(D,\mathbf{x})dxdydz,$$

(844)

where $x_1=x$, $x_2=y$, $x_3=z$, and the NIDS operators ${\widehat{D}}_{x^j}^{{\alpha }_j}$ are

$${\widehat{D}}_{x^j}^{{\alpha }_j}f\left(\mathbf{x}\right):={\displaystyle \frac{1}{c_1({\alpha }_j,x^j)}}{\displaystyle \frac{\partial }{\partial x^j}}\left({\widehat{c}}_1({\alpha }_j,x^j)f\left(\mathbf{x}\right)\right)(j=1,2,3),$$

(845)

and

$${\widehat{c}}_{3,yz}(D,\mathbf{x})={\widehat{c}}_1({\alpha }_1,x)c_1({\alpha }_2,y)c_1({\alpha }_3,z),$$

(846)

$${\widehat{c}}_{3,zx}(D,\mathbf{x})=c_1({\alpha }_1,x){\widehat{c}}_1({\alpha }_2,y)c_1({\alpha }_3,z),$$

(847)

$${\widehat{c}}_{3,xy}(D,\mathbf{x})=c_1({\alpha }_1,x)c_1({\alpha }_2,y){\widehat{c}}_1({\alpha }_3,z),$$

(848)

and

$$c_3(D,\mathbf{x})=c_1({\alpha }_1,x)c_1({\alpha }_2,y)c_1({\alpha }_3,z),$$

(849)

with all ${\alpha }_j\in (0,1]$.

(2) The Metric Approach: Let $F_k\left(\mathbf{x}\right)$ ($k=1,2,3$) be functions defined and continuous together with the partial derivatives $\partial F_k\left(\mathbf{x}\right))/\partial x^k$ ($k=1,2,3$) in the domain Ω including its boundary $\Sigma =\partial \Omega$. Then, the following NIDS generalizations of the Gauss–Ostrogradsky equation are satisfied in the coordinate form

$${\int }_{\Sigma }\left(F_x\left(\mathbf{x}\right)c_2(d_{yz},\mathbf{x})dydz+F_y\left(\mathbf{x}\right)c_2(d_{zx},\mathbf{x})dzdx+F_z\left(\mathbf{x}\right)c_2(d_{xy},\mathbf{x})dxdy\right)=$$

$${\int }_{\Omega }\left({\displaystyle \frac{1}{c_1({\alpha }_1,x)}}{\displaystyle \frac{\partial F_x\left(\mathbf{x}\right)}{\partial x}}+{\displaystyle \frac{1}{c_1({\alpha }_2,y)}}{\displaystyle \frac{\partial F_y\left(\mathbf{x}\right)}{\partial y}}+{\displaystyle \frac{1}{c_1({\alpha }_3,z)}}{\displaystyle \frac{\partial F_z\left(\mathbf{x}\right)}{\partial z}}\right)c_3(D,\mathbf{x})dxdydz,$$

(850)

where

$$D_{x^j}^{{\alpha }_j}f\left(\mathbf{x}\right):={\displaystyle \frac{1}{c_1({\alpha }_j,x^j)}}{\displaystyle \frac{\partial f\left(\mathbf{x}\right)}{\partial x^j}},(j=1,2,3),$$

(851)

and

$$c_2(d_{yz},\mathbf{x})=c_1({\alpha }_2,y)c_1({\alpha }_3,z),$$

(852)

$$c_2(d_{xz},\mathbf{x})=c_1({\alpha }_1,x)c_1({\alpha }_3,z),$$

(853)

$$c_2(d_{xy},\mathbf{x})=c_1({\alpha }_1,x)c_1({\alpha }_2,y),$$

(854)

with all ${\alpha }_j\in (0,1]$.

Proof. 

(1) In the product measure approach, let us define the functions

$$P(x,y,z)={\widehat{c}}_{3,yz}(D,\mathbf{x})F_1\left(\mathbf{x}\right)={\widehat{c}}_1({\alpha }_1,x)c_1({\alpha }_2,y)c_1({\alpha }_3,z)F_1\left(\mathbf{x}\right),$$

(855)

$$Q(x,y,z)={\widehat{c}}_{3,zx}(D,\mathbf{x})F_2\left(\mathbf{x}\right)=c_1({\alpha }_1,x){\widehat{c}}_1({\alpha }_2,y)c_1({\alpha }_3,z)F_2\left(\mathbf{x}\right),$$

(856)

$$R(x,y,z)={\widehat{c}}_{3,xy}(D,\mathbf{x})F_3\left(\mathbf{x}\right)=c_1({\alpha }_1,x)c_1({\alpha }_2,y){\widehat{c}}_1({\alpha }_3,z)F_3\left(\mathbf{x}\right).$$

(857)

Let us prove the equation

$${\int }_{\Sigma }{F}_x\left(\mathbf{x}\right)c_{3,yz}(D,\mathbf{x})dydz={\int }_{\Omega }{\displaystyle \frac{1}{c_1({\alpha }_1,x)}}{\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_1,x)F_x\left(\mathbf{x}\right)}{\partial x}}{c}_3(D,\mathbf{x})dxdydz$$

(858)

corresponds to terms of Equation (844) containing only the function $F_1\left(\mathbf{x}\right)$. The equations with terms containing the functions $F_2\left(\mathbf{x}\right)$ and $F_3\left(\mathbf{x}\right)$ are proved similarly.

Let us take a function $R(x,y,z)$, defined and continuous together with its partial derivative $\partial R\left(\mathbf{x}\right)/\partial z$ in the domain $\Omega$ including its boundary. Let us consider the surface integral of the function $R(x,y,z)$ taken over the surface $\Sigma =\partial \Omega$ in the form

$${\int }_{\Sigma }{\widehat{c}}_{3,xy}(D,\mathbf{x})F_3\left(\mathbf{x}\right)dxdy={\int }_{\Sigma }R(x,y,z)dxdy=$$

$${\int }_{{\Sigma }_1}R(x,y,z)dxdy+{\int }_{{\Sigma }_2}R(x,y,z)dxdy+{\int }_{{\Sigma }_3}R(x,y,z)dxdy=$$

$${\int }_{U}R(x,y,z_2(x,y))dxdy-{\int }_{U}R(x,y,z_1(x,y))dxdy=$$

$${\int }_U\left(R(x,y,z_2(x,y))-R(x,y,z_1(x,y))\right)dxdy,$$

(859)

where we take into account that

$${\int }_{{\Sigma }_3}R(x,y,z)dxdy=0.$$

(860)

Using the fundamental theorem of the standard calculus in the form

$${\int }_{z_1(x,y)}^{z_2(x,y)}{\displaystyle \frac{\partial R(x,y,z)}{\partial z}}dz=R(x,y,z_2(x,y))-R(x,y,z_1(x,y)),$$

(861)

Equation (859) gives

$${\int }_U\left(R(x,y,z_2(x,y))-R(x,y,z_1(x,y))\right)dxdy=$$

$${\int }_{U}dxdy{\int }_{z_1(x,y)}^{z_2(x,y)}{\displaystyle \frac{\partial R(x,y,z)}{\partial z}}dz={\int }_{\Omega }{\displaystyle \frac{\partial R(x,y,z)}{\partial z}}dxdydz.$$

(862)

Using Equation (857), we get

$${\int }_{\Omega }{\displaystyle \frac{\partial R(x,y,z)}{\partial z}}dxdydz={\int }_{\Omega }{\displaystyle \frac{1}{c_3(D,\mathbf{x})}}{\displaystyle \frac{\partial R(x,y,z)}{\partial z}}{c}_3(D,\mathbf{x})dxdydz=$$

$${\int }_{\Omega }{\displaystyle \frac{1}{c_3(D,\mathbf{x})}}{\displaystyle \frac{\partial \left({\widehat{c}}_{3,xy}(D,\mathbf{x})F_3\left(\mathbf{x}\right)\right)}{\partial z}}{c}_3(D,\mathbf{x})dxdydz=$$

$${\int }_{\Omega }{\displaystyle \frac{1}{c_1({\alpha }_1,x)c_1({\alpha }_2,y)c_1({\alpha }_3,z)}}{\displaystyle \frac{\partial \left(c_1({\alpha }_1,x)c_1({\alpha }_2,y){\widehat{c}}_1({\alpha }_3,z)F_3\left(\mathbf{x}\right)\right)}{\partial z}}{c}_3(D,\mathbf{x})dxdydz=$$

$${\int }_{\Omega }{\displaystyle \frac{1}{c_1({\alpha }_3,z)}}{\displaystyle \frac{\partial \left({\widehat{c}}_1({\alpha }_3,z)F_3\left(\mathbf{x}\right)\right)}{\partial z}}{c}_3(D,\mathbf{x})dxdydz=$$

$${\int }_{\Omega }{\widehat{D}}_z^{{\alpha }_3}{F}_3\left(\mathbf{x}\right)d{\mu }_3(D,\mathbf{x}),$$

where

$${\widehat{D}}_z^{{\alpha }_3}f\left(\mathbf{x}\right):={\displaystyle \frac{1}{c_1({\alpha }_3,z)}}{\displaystyle \frac{\partial }{\partial z}}\left({\widehat{c}}_1({\alpha }_3,z)f\left(\mathbf{x}\right)\right),$$

(863)

and $d{\mu }_3(D,\mathbf{x})$ is the measured product

$$d{\mu }_3(D,\mathbf{x})=\prod _{k=1}^{3}d{\mu }_1({\alpha }_k,x^k).$$

(864)

The equations with the terms of Equation (844) containing the functions $F_2\left(\mathbf{x}\right)$, $F_3\left(\mathbf{x}\right)$ are proved similarly. This proves Equation (844).

(2) In the metric approach, the NIDS Gauss-Ostrogradsky Theorem is proved similarly. In fact, to prove Equation (850), it is sufficient to use

$${\widehat{c}}_1({\alpha }_j,x^j)=1$$

(865)

for all $j=1,2,3$ and equations in the proof of Theorem in the product measure approach. In this case (865), we get

$${\widehat{c}}_{3,xz}(D,\mathbf{x})=c_2(d_{xz},\mathbf{x}),$$

(866)

$${\widehat{c}}_{3,yz}(D,\mathbf{x})=c_2(d_{yz},\mathbf{x}),$$

(867)

$${\widehat{c}}_{3,xy}(D,\mathbf{x})=c_2(d_{xy},\mathbf{x}),$$

(868)

where $k=1,2,3$ and $x_1=x$, $x_2=y$, $x_3=z$. This proves Equation (850). □

Remark 64. 

It should be emphasized that the factor ${\widehat{c}}_1({\alpha }_k,x^k)$ in the derivative (863) is not equal to one, in the general case of the product measure approach.

In the product measure approach, if the surface $\Sigma =\partial \Omega$ consists only of domains that are parallel to the coordinate planes $XY$, $XZ$, and $YZ$, then all the equations, integrals, and differential operators in the Gauss–Ostrogradsky Theorem for NIDS and its proofs coincide with the induced metric approach.

Remark 65. 

In accordance with the principle of the self-consistency of NIDS vector calculus (Principle 9), the NIDS integral calculus and the NIDS differential calculus must be mutually consistent. This means that the form of the differential operators ${\widehat{Div}}_{\left(\alpha \right)}$ and ${Div}_{\left(\alpha \right)}$ must be determined by the fact that the NIDS Gauss–Ostrogradsky theorem must be satisfied for the product measure approach and the metric approach, respectively. It can be said that the explicit expression of the NIDS divergence is derived in the proof of Theorem 35.

The NIDS divergence of the vector field must be defined such that the NIDS generalization of the Gauss–Ostrogradsky theorem holds and can be proved for the surface and volume NIDS integrals that satisfy the correspondence principle for volume, surface, and line NIDS integrals (Principle 4) and the principle of the self-consistency of NIDS vector calculus (Principle 9). Theorem 35 allows us to define the NIDS divergence in a correct unambiguous form.

Therefore, we can give the following corollary and definition.

Corollary 7. 

In order for the Gauss–Ostrogradsky theorem for NIDS (Theorem 35) in the form (844) and (850) to be satisfied, the NIDS divergence ${\widehat{Div}}_{\left(\alpha \right)}$ and ${Div}_{\left(\alpha \right)}$ should be defined by the following definition.

Definition 33. 

Let $W=W_1\times \dots \times W_n$ be the D-dimensional space with $D={\alpha }_1+\dots +{\alpha }_n\in (0,n]$, and let Ω be a region of the space W that is bounded by a piecewise smooth surface $\Sigma =\partial \Omega$.

(1) The Product Measure Approach: Let $F_k\left(\mathbf{x}\right)$ ($k=1,\dots ,n$) be functions defined and continuous together with its partial derivatives $\partial \left({\widehat{c}}_1({\alpha }_k,x^k)F_k\left(\mathbf{x}\right)\right)/\partial x^k$ ($k=1,2,3$) in the domain Ω including its boundary $\Sigma =\partial \Omega$. Then, the NIDS divergence is defined as

$${\widehat{Div}}_{\left(\alpha \right)}\mathbf{F}\left(\mathbf{x}\right):=\sum _{k=1}^n{\widehat{D}}_{x^k}^{{\alpha }_k}{F}_k\left(\mathbf{x}\right),$$

(869)

where $\left(\alpha \right)=({\alpha }_1,\dots ,{\alpha }_n)$ and

$${\widehat{D}}_{x^k}^{{\alpha }_k}{F}_k\left(\mathbf{x}\right):={\displaystyle \frac{1}{c_1({\alpha }_k,x^k)}}{\displaystyle \frac{\partial }{\partial x^k}}\left({\widehat{c}}_1({\alpha }_k,x^k)F_k\left(\mathbf{x}\right)\right)$$

(870)

with all ${\alpha }_k\in (0,1]$.

(2) The Metric Approach: Let $F_k\left(\mathbf{x}\right)$ ($k=1,\dots ,n$) be functions defined and continuous together with the partial derivatives $\partial F_k\left(\mathbf{x}\right)/\partial x^k$ ($k=1,2,3$) in the domain Ω including its boundary $\Sigma =\partial \Omega$. Then, the NIDS divergence is defined as

$${Div}_{\left(\alpha \right)}\mathbf{F}\left(\mathbf{x}\right):=\sum _{k=1}^n{D}_{x^k}^{{\alpha }_k}{F}_k\left(\mathbf{x}\right),$$

(871)

where $\left(\alpha \right)=({\alpha }_1,\dots ,{\alpha }_n)$, and

$$D_{x^k}^{{\alpha }_k}f\left(\mathbf{x}\right):={\displaystyle \frac{1}{c_1({\alpha }_k,x^k)}}{\displaystyle \frac{\partial f\left(\mathbf{x}\right)}{\partial x^k}},$$

(872)

with all ${\alpha }_k\in (0,1]$.

Using the definition of the surface NIDS integrals (${\mathbf{I}}_{\Sigma }^D$ and ${\mathbf{I}}_{\Sigma }^{D,\ast }$), the volume NIDS integral (${\mathbf{I}}_{\Omega }^D={\mathbf{I}}_{\Omega }^{D,\ast }$), and the NIDS divergence (869), the Gauss–Ostrogradsky theorem for NIDS can be formulated in the following corollary.

Corollary 8. 

The Gauss-Ostrogradsky theorem for NIDS (Theorem 35) can be represented by the following equations.

(1) The Product Measure Approach: the surface NIDS integral ${\mathbf{I}}_{\Sigma }^D$ of a vector field $\mathbf{F}$ over a closed surface $\Sigma =\partial \Omega$ (the flux through the surface), is equal to the volume NIDS integral $I_{\Omega }^D$ of the NIDS divergence ${\widehat{Div}}_{\left(\alpha \right)}$ over the region Ω enclosed by the surface Σ:

$$\left({\mathbf{I}}_{\Sigma }^D,\mathbf{F}\right)=I_{\Omega }^D\left({\widehat{Div}}_{\left(\alpha \right)}\mathbf{F}\right),$$

(873)

where $\partial \Omega =\Sigma$, and the NIDS divergence ${\widehat{Div}}_{\left(\alpha \right)}$ is defined by Equation (869).

(2) The Metric Approach: The surface NIDS integral ${\mathbf{I}}_{\Sigma }^{D,\ast }$ of a vector field $\mathbf{F}$ over a closed surface $\Sigma =\partial \Omega$ (the flux through the surface) is equal to the volume NIDS integral $I_{\Omega }^D$ of the NIDS divergence $Di{v}_{\left(\alpha \right)}$ over the region Ω enclosed by the surface Σ:

$$\left({\mathbf{I}}_{\Sigma }^{D,\ast },\mathbf{F}\right)=I_{\Omega }^D\left({Div}_{\left(\alpha \right)}\mathbf{F}\right),$$

(874)

where $\partial \Omega =\Sigma$, and the NIDS divergence ${Div}_{\left(\alpha \right)}$ is defined by Equation (871).

Remark 66. 

In the product measure approach, the NIDS Gauss–Ostrogradsky theorem (Theorem 35) can be written as

$$\left({\mathbf{I}}_{\Sigma }^D,\mathbf{F}\right)=I_{\Omega }^D\left(\sum _{k=1}^n{D}_{x^k}^{{\alpha }_k}\left({\widehat{c}}_1({\alpha }_k,x^k)F_k\left(\mathbf{x}\right)\right)\right).$$

(875)

(1) For the ball region ${\Omega }_B$ with the spherical boundary ${\Sigma }_S=\partial {\Omega }_B$, Theorem 35 gives

$$\left({\mathbf{I}}_{{\Sigma }_S}^D,\mathbf{F}\right)=I_{{\Omega }_B}^D\left(\sum _{k=1}^n{D}_{x^k}^{{\alpha }_k}\left(c_1({\alpha }_k,x^k)F_k\left(\mathbf{x}\right)\right)\right),$$

(876)

and the NIDS divergence is

$${\widehat{Div}}_{\left(\alpha \right)}\mathbf{F}\left(\mathbf{x}\right)=\sum _{k=1}^n{D}_{x^k}^{{\alpha }_k}\left(c_1({\alpha }_k,x^k)F_k\left(\mathbf{x}\right)\right),$$

(877)

where the functions $c_1({\alpha }_k,x^k)$ are used instead of ${\widehat{c}}_1({\alpha }_k,x^k)$.

(2) For the parallelepiped region ${\Omega }_P$ with boundary ${\Sigma }_P=\partial {\Omega }_P$, where the edges of the parallelepiped are parallel to the coordinate axes, Theorem 35 gives

$$\left({\mathbf{I}}_{{\Sigma }_P}^D,\mathbf{F}\right)=I_{{\Omega }_P}^D\left(\sum _{k=1}^n{D}_{x^k}^{{\alpha }_k}{F}_k\left(\mathbf{x}\right)\right),$$

(878)

and the NIDS divergence is

$${\widehat{Div}}_{\left(\alpha \right)}\mathbf{F}\left(\mathbf{x}\right)=\sum _{k=1}^n{D}_{x^k}^{{\alpha }_k}{F}_k\left(\mathbf{x}\right)={Div}_{\left(\alpha \right)}\mathbf{F}\left(\mathbf{x}\right),$$

(879)

where we use ${\widehat{c}}_1({\alpha }_k,x^k)=1$.

As a result, in the product measure approach, the NIDS divergence in some sense depends on the type of surface $\Sigma =\partial \Omega$, through dependence on ${\widehat{c}}_1({\alpha }_k,x^k)$.

Remark 67. 

Let us note the interrelations of various operators, which form self-consistent calculus in the D-dimensional space.

1. 

We can see that the NIDS divergences are defined so that the NIDS Gauss–Ostrogradsky theorem is satisfied for the surface and volume NIDS integrals of vector fields.

2. 

These surface NIDS integrals of vector fields are defined through the surface NIDS integrals of scalar fields according the principle of self-consistency of NIDS vector calculus (Principle 9), which requires the consistency of the surface NIDS integrals.

3. 

The NIDS integrals of scalar fields satisfy the correspondence principle for volume, surface, and line NIDS integrals (Principle 4).

These mutual connections of integral and differential NIDS operators ensure the self-consistency of the NIDS calculus that is proposed in this paper.

4.4.5. NIDS Green First and Second Identities

Let us give NIDS generalizations of the well-known first and second Green equations [155,156].

The NIDS identity can be derived from the NIDS Gauss–Ostrogradsky Theorem 35 applied to the vector field

$${\mathbf{F}}_1\left(\mathbf{x}\right)=\varphi \left(\mathbf{x}\right){\widehat{\nabla }}_{\left(\alpha \right)}\psi \left(\mathbf{x}\right),$$

(880)

$${\mathbf{F}}_2\left(\mathbf{x}\right)=\varphi \left(\mathbf{x}\right){\widehat{\nabla }}_{\left(\alpha \right)}\psi \left(\mathbf{x}\right)-\psi \left(\mathbf{x}\right){\widehat{\nabla }}_{\left(\alpha \right)}\varphi \left(\mathbf{x}\right).$$

(881)

Using the vector (880) and the Gauss–Ostrogradsky theorem in the form

$$\left({\mathbf{I}}_{\Sigma }^D,{\mathbf{F}}_1\right)=I_{\Omega }^D\left({\widehat{Div}}_{\left(\alpha \right)}{\mathbf{F}}_1\right),$$

(882)

we get

$$\left({\mathbf{I}}_{\Sigma }^D,\varphi \left(\mathbf{x}\right){\widehat{\nabla }}_{\left(\alpha \right)}\psi \left(\mathbf{x}\right)\right)=I_{\Omega }^D{\widehat{Div}}_{\left(\alpha \right)}\left(\varphi \left(\mathbf{x}\right){\widehat{\nabla }}_{\left(\alpha \right)}\psi \left(\mathbf{x}\right)\right).$$

(883)

Using

$${Div}_{\left(\alpha \right)}\left(\varphi \left(\mathbf{x}\right){\widehat{\nabla }}_{\left(\alpha \right)}\psi \left(\mathbf{x}\right)\right)=\varphi \left(\mathbf{x}\right)\Delta \psi \left(\mathbf{x}\right)+\left({\widehat{\nabla }}_{\left(\alpha \right)}\varphi \left(\mathbf{x}\right),{\widehat{\nabla }}_{\left(\alpha \right)}\psi \left(\mathbf{x}\right)\right),$$

(884)

Equation (883) gives

$$\left({\mathbf{I}}_{\Sigma }^D,\varphi \left(\mathbf{x}\right){\widehat{\nabla }}_{\left(\alpha \right)}\psi \left(\mathbf{x}\right)\right)=I_{\Omega }^D\left(\varphi \left(\mathbf{x}\right)\Delta \psi \left(\mathbf{x}\right)+\left({\widehat{\nabla }}_{\left(\alpha \right)}\varphi \left(\mathbf{x}\right),{\widehat{\nabla }}_{\left(\alpha \right)}\psi \left(\mathbf{x}\right)\right)\right).$$

(885)

As a result, we derive

$$I_{\Omega }^D\left(\varphi \Delta \psi +\left({\widehat{\nabla }}_{\left(\alpha \right)}\varphi ,{\widehat{\nabla }}_{\left(\alpha \right)}\psi \right)\right)=I_{\Sigma }^D\left(\varphi \left(\mathbf{n},{\widehat{\nabla }}_{\left(\alpha \right)}\psi \right)\right),$$

(886)

which is the NIDS Green first identity.

Using the vector (881) and the Gauss–Ostrogradsky theorem, we get

$$I_{\Omega }^D\left(\varphi \Delta \psi -\psi \Delta \varphi \right)=I_{\Sigma }^D\left(\varphi \left(\mathbf{n},{\widehat{\nabla }}_{\left(\alpha \right)}\psi \right)-\psi \left(\mathbf{n},{\widehat{\nabla }}_{\left(\alpha \right)}\varphi )\right)\right),$$

(887)

which is the NIDS Green second identity.

4.5. Properties of NIDS Vector Differential Operators

4.5.1. NIDS Gradient, Divergence, and Curl Operators

Let us give Definitions 29, 31, and 33 of the vector differential operators of the first order in the product measure approach by one definition. For the metric approach, these operators are defined in Definition 23.

Definition 34 

(Vector Operators in Product Measure Approach). Let $W=W_1\times \dots \times W_n$ be a D-dimensional space with $D={\alpha }_1+\dots +{\alpha }_n$, and let Ω be a region of W. Then, in the product measure, the first-order NIDS differential operators of the NIDS vector calculus are defined in the following forms.

The NIDS gradient of scalar function $U=U\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^1(\Omega )$ is defined as the vector field

$${\widehat{Grad}}_{\left(\alpha \right)}U\left(\mathbf{x}\right)=\sum _{j=1}^n{\mathbf{e}}_j{\widehat{D}}_{x^j}^{{\alpha }_j}U\left(\mathbf{x}\right).$$

(888)

The NIDS divergence of the vector field $\mathbf{A}\left(\mathbf{x}\right)={\sum }_{k=1}^n{A}^k\left(\mathbf{x}\right){\mathbf{e}}_k$ with all $A^k\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^1(\Omega )$ is defined as the scalar field

$${\widehat{Div}}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right)=\sum _{k=1}^n{\widehat{D}}_{x^k}^{{\alpha }_k}{A}_k\left(\mathbf{x}\right).$$

(889)

The NIDS curl operator for vector field $\mathbf{A}\left(\mathbf{x}\right)={\sum }_{k=1}^n{A}^k\left(\mathbf{x}\right){\mathbf{e}}_k$ with all $A^k\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^1(\Omega )$ is defined as the vector field

$${\widehat{Curl}}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right)=\sum _{i,j,k=1}^3{\mathbf{e}}_i{\epsilon }_{ijk}{\widehat{D}}_{x^j}^{{\alpha }_j}{A}^k\left(\mathbf{x}\right),$$

(890)

where ${\epsilon }_{ijk}$ is the Levi–Civita symbol.

Using Definition 34 of the NIDS gradient, divergence, and curl operators, we can see that all these operators can be expressed via the NIDS nabla operator. Let $W=W_1\times \dots \times W_n$ be the D-dimensional space with $D={\alpha }_1+\dots +{\alpha }_n\in (0,n]$. In the product measure approach, the NIDS nabla operator is

$${\widehat{\nabla }}_{\left(\alpha \right)}U\left(\mathbf{x}\right):=\sum _{k=1}^n{\mathbf{e}}_k{\widehat{D}}_{x^k}^{{\alpha }_k}U\left(\mathbf{x}\right),$$

(891)

where

$${\widehat{D}}_{x^j}^{{\alpha }_j}U\left(\mathbf{x}\right)={\displaystyle \frac{1}{c_1({\alpha }_j,x^j)}}{\displaystyle \frac{\partial \left({\widehat{c}}_1({\alpha }_j,x^j)U\left(\mathbf{x}\right)\right)}{\partial x^j}}.$$

(892)

Then, in the product measure approach, the first order vector NIDS operators are written as

$${\widehat{Grad}}_{\left(\alpha \right)}U\left(\mathbf{x}\right)={\widehat{\nabla }}_{\left(\alpha \right)}U\left(\mathbf{x}\right),$$

(893)

$${\widehat{Div}}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right)=\left({\widehat{\nabla }}_{\left(\alpha \right)},\mathbf{A}\left(\mathbf{x}\right)\right),$$

(894)

where $\left(\alpha \right)=({\alpha }_1,\dots ,{\alpha }_n)$, and

$${\widehat{Curl}}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right)=\left[{\widehat{\nabla }}_{\left(\alpha \right)}\times \mathbf{A}\left(\mathbf{x}\right)\right],$$

(895)

where $\left(\alpha \right)=({\alpha }_1,{\alpha }_2,{\alpha }_3)$.

In the metric approach, the NIDS nabla operator is

$${\nabla }_{\left(\alpha \right)}U\left(\mathbf{x}\right):=\sum _{k=1}^n{\mathbf{e}}_k{D}_{x^k}^{{\alpha }_k}U\left(\mathbf{x}\right),$$

(896)

where

$$D_{x^j}^{{\alpha }_j}U\left(\mathbf{x}\right)={\displaystyle \frac{1}{c_1({\alpha }_j,x^j)}}{\displaystyle \frac{\partial U\left(\mathbf{x}\right)}{\partial x^j}},$$

(897)

and the first order vector NIDS operators are written as

$${Grad}_{\left(\alpha \right)}U\left(\mathbf{x}\right)={\nabla }_{\left(\alpha \right)}U\left(\mathbf{x}\right),$$

(898)

$${Div}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right)=\left({\nabla }_{\left(\alpha \right)},\mathbf{A}\left(\mathbf{x}\right)\right),$$

(899)

where $\left(\alpha \right)=({\alpha }_1,\dots ,{\alpha }_n)$, and

$${Curl}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right)=\left[{\nabla }_{\left(\alpha \right)}\times \mathbf{A}\left(\mathbf{x}\right)\right],$$

(900)

where $\left(\alpha \right)=({\alpha }_1,{\alpha }_2,{\alpha }_3)$.

Remark 68. 

The use of the same nabla operator (891) to define all three vector operators of the first order in the product measure approach is a necessary requirement for the self-consistency of differential NIDS vector calculus in this approach. In the metric approach, the nabla operator (896) can be described by Equation (891) with all ${\widehat{c}}_1({\alpha }_j,x^j)=1$.

Therefore, the requirement 3c of Principle 9 about the self-consistency of the NIDS vector calculus is satisfied.

Note that not all papers devoted to vector calculus in fractional-dimensional spaces satisfy this requirement. For example, two type del-operators are used in the Balankin and Mena paper [157]. This paper also contains mathematical errors and false statements in Tables 1 and 5 about the Tarasov paper [19].

4.5.2. Properties of First-Order Vector NIDS Operators

Let us first describe some properties of the operator ${\widehat{D}}_{x^j}^{{\alpha }_j}$ that is used in the equations that define the NIDS gradient, the NIDS divergence, and the NIDS curl operator in the product measure approach.

Property 4 

(Linearity Property). Let W be a D-dimensional space $W=W_1\times \dots \times W_n$ with $\left(\alpha \right)=({\alpha }_1,\dots ,{\alpha }_n)$ and $f\left(\mathbf{x}\right),g\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^1(\Omega )$ for domain $\Omega \subset W$, and $a,b\in \mathbb{R}$.

Then, the equation

$${\widehat{D}}_{x^j}^{{\alpha }_j}(af\left(\mathbf{x}\right)+bg\left(\mathbf{x}\right))=a{\widehat{D}}_{x^j}^{{\alpha }_j}f\left(\mathbf{x}\right)+b{\widehat{D}}_{x^j}^{{\alpha }_j}g\left(\mathbf{x}\right)$$

(901)

is satisfied for all $\mathbf{x}\in \Omega$, for all $j=1,\dots ,n$.

Property 5 

(Action on Constant). Let W be a D-dimensional space $W=W_1\times \dots \times W_n$ with $\left(\alpha \right)=({\alpha }_1,\dots ,{\alpha }_n)$.

The equation

$${\widehat{D}}_{x^j}^{{\alpha }_j}1={\displaystyle \frac{1}{c_1({\alpha }_j,x^j)}}{\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_j,x^j)}{\partial x^j}}$$

(902)

is satisfied for all $\mathbf{x}\in \Omega$.

Property 6 

(Product Rule). Let W be a D-dimensional space $W=W_1\times \dots \times W_n$ with $\left(\alpha \right)=({\alpha }_1,\dots ,{\alpha }_n)$ and $f\left(\mathbf{x}\right),g\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^1(\Omega )$ for domain $\Omega \subset W$.

Then, the equation

$${\widehat{D}}_{x^j}^{{\alpha }_j}\left(fg\right)=\left({\widehat{D}}_{x^j}^{{\alpha }_j}f\right)g+f\left({\widehat{D}}_{x^j}^{{\alpha }_j}g\right)-\left(fg\right)\left({\widehat{D}}_{x^j}^{{\alpha }_j}1\right)$$

(903)

is satisfied for all $\mathbf{x}\in \Omega$ and for all $j=1,\dots ,n$.

Proof. 

Using

$${\widehat{D}}_{x^j}^{{\alpha }_j}f\left(\mathbf{x}\right)=D_{x^j}^{{\alpha }_j}\left({\widehat{c}}_1({\alpha }_j,x^j)f\left(\mathbf{x}\right)\right),$$

(904)

where the operator is

$$D_{x^j}^{{\alpha }_j}f\left(\mathbf{x}\right)={\displaystyle \frac{1}{c_1({\alpha }_j,x^j)}}{\displaystyle \frac{\partial }{\partial x^j}}f\left(\mathbf{x}\right),$$

(905)

which satisfies the standard product (Leibniz) rule, we get

$${\widehat{D}}_{x^j}^{{\alpha }_j}\left(f\left(\mathbf{x}\right)g\left(\mathbf{x}\right)\right)=D_{x^j}^{{\alpha }_j}\left({\widehat{c}}_1({\alpha }_j,x^j)f\left(\mathbf{x}\right)g\left(\mathbf{x}\right)\right)=$$

$$g\left(\mathbf{x}\right)D_{x^j}^{{\alpha }_j}\left({\widehat{c}}_1({\alpha }_j,x^j)f\left(\mathbf{x}\right)\right)+{\widehat{c}}_1({\alpha }_j,x^j)f\left(\mathbf{x}\right)\left(D_{x^j}^{{\alpha }_j}g\left(\mathbf{x}\right)\right).$$

(906)

Using

$$D_{x^j}^{{\alpha }_j}\left({\widehat{c}}_1({\alpha }_j,x^j)g\left(\mathbf{x}\right)\right)=g\left(\mathbf{x}\right)D_{x^j}^{{\alpha }_j}{\widehat{c}}_1({\alpha }_j,x^j)+{\widehat{c}}_1({\alpha }_j,x^j)D_{x^j}^{{\alpha }_j}g\left(\mathbf{x}\right),$$

(907)

we get

$${\widehat{c}}_1({\alpha }_j,x^j)D_{x^j}^{{\alpha }_j}g\left(\mathbf{x}\right)=D_{x^j}^{{\alpha }_j}\left({\widehat{c}}_1({\alpha }_j,x^j)g\left(\mathbf{x}\right)\right)-g\left(\mathbf{x}\right)D_{x^j}^{{\alpha }_j}{\widehat{c}}_1({\alpha }_j,x^j).$$

(908)

Substitution of expression (908) into (906) gives

$${\widehat{D}}_{x^j}^{{\alpha }_j}\left(f\left(\mathbf{x}\right)g\left(\mathbf{x}\right)\right)=g\left(\mathbf{x}\right)D_{x^j}^{{\alpha }_j}\left({\widehat{c}}_1({\alpha }_j,x^j)f\left(\mathbf{x}\right)\right)+$$

$$f\left(\mathbf{x}\right)D_{x^j}^{{\alpha }_j}\left({\widehat{c}}_1({\alpha }_j,x^j)g\left(\mathbf{x}\right)\right)-g\left(\mathbf{x}\right)f\left(\mathbf{x}\right)D_{x^j}^{{\alpha }_j}{\widehat{c}}_1({\alpha }_j,x^j)=$$

$$g\left(\mathbf{x}\right){\widehat{D}}_{x^j}^{{\alpha }_j}f\left(\mathbf{x}\right)+f\left(\mathbf{x}\right){\widehat{D}}_{x^j}^{{\alpha }_j}g\left(\mathbf{x}\right)-g\left(\mathbf{x}\right)f\left(\mathbf{x}\right){\widehat{D}}_{x^j}^{{\alpha }_j}1,$$

(909)

where we use (904). This ends the proof. □

Note that Property 6 is the well-known characteristic property of the linear differential operators of the order $⩽1$ [158], p. 38.

Let us describe the properties of the NIDS gradient, the NIDS divergence, and the NIDS curl operator in the product measure approach to the NIDS vector calculus.

Property 7 

(Product Rule for NIDS Gradient). Let W be a D-dimensional space $W=W_1\times \dots \times W_n$ with $\left(\alpha \right)=({\alpha }_1,\dots ,{\alpha }_n)$ and $f\left(\mathbf{x}\right),g\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^1(\Omega )$ for domain $\Omega \subset W$.

Then, the equation

$${\widehat{Grad}}_{\left(\alpha \right)}\left(fg\right)=\left({\widehat{Grad}}_{\left(\alpha \right)}f\right)g+f\left({\widehat{Grad}}_{\left(\alpha \right)}g\right)-\left(fg\right)\left({\widehat{Grad}}_{\left(\alpha \right)}1\right)$$

(910)

is satisfied for all $\mathbf{x}\in \Omega$.

Proof. 

Using Equation (888), we get

$${\widehat{Grad}}_{\left(\alpha \right)}\left(fg\right)=\sum _{j=1}^n{\mathbf{e}}_j{\widehat{D}}_{x^j}^{{\alpha }_j}\left(fg\right).$$

(911)

Using Equation (903) of property 6, Equation (911) is written as

$${\widehat{Grad}}_{\left(\alpha \right)}\left(fg\right)=$$

$$\left(\sum _{j=1}^n{\mathbf{e}}_j{\widehat{D}}_{x^j}^{{\alpha }_j}f\right)g+f\left(\sum _{j=1}^n{\mathbf{e}}_j{\widehat{D}}_{x^j}^{{\alpha }_j}g\right)-\left(fg\right)\left(\sum _{j=1}^n{\mathbf{e}}_j{\widehat{D}}_{x^j}^{{\alpha }_j}1\right)=$$

$$\left({\widehat{Grad}}_{\left(\alpha \right)}f\right)g+f\left({\widehat{Grad}}_{\left(\alpha \right)}g\right)-\left(fg\right)\left({\widehat{Grad}}_{\left(\alpha \right)}1\right).$$

(912)

This ends the proof. □

Property 8 

(Product Rule for NIDS Divergence). Let W be D-dimensional space $W=W_1\times \dots \times W_n$ with $\left(\alpha \right)=({\alpha }_1,\dots ,{\alpha }_n)$ and $U\left(\mathbf{x}\right),\mathbf{F}\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^1(\Omega )$ for domain $\Omega \subset W$.

Then, the equation

$${\widehat{Div}}_{\left(\alpha \right)}\left(U\mathbf{F}\right)=\left({\widehat{Grad}}_{\left(\alpha \right)}U,\mathbf{F}\right)+U\left({\widehat{Div}}_{\left(\alpha \right)}\mathbf{F}\right)-U\left(\mathbf{F},{\widehat{Grad}}_{\left(\alpha \right)}1\right)$$

(913)

is satisfied for all $\mathbf{x}\in \Omega$.

Proof. 

Using Equation (889), we get

$${\widehat{Div}}_{\left(\alpha \right)}\left(U\mathbf{F}\right)=\sum _{j=1}^n{\widehat{D}}_{x^j}^{{\alpha }_j}\left(U{F}_j\right).$$

(914)

Using Equation (903) of Property 6, Equation (914) is written as

$${\widehat{Div}}_{\left(\alpha \right)}\left(U\mathbf{F}\right)=$$

$$\sum _{j=1}^n\left(F_j{\widehat{D}}_{x^j}^{{\alpha }_j}U\right)+U\left(\sum _{j=1}^n{\widehat{D}}_{x^j}^{{\alpha }_j}{F}_j\right)-U\left(\sum _{j=1}^n{F}_j{\widehat{D}}_{x^j}^{{\alpha }_j}1\right)=$$

$$\left(\mathbf{F},{\widehat{Grad}}_{\left(\alpha \right)}U\right)+U\left({\widehat{Div}}_{\left(\alpha \right)}\mathbf{F}\right)-U\left(\mathbf{F},{\widehat{Grad}}_{\left(\alpha \right)}1\right).$$

(915)

This ends the proof. □

Property 9 

(Product Rule for NIDS Curl Operator). Let W be a D-dimensional space $W=W_1\times W_2\times W_3$ with $\left(\alpha \right)=({\alpha }_1,{\alpha }_2,{\alpha }_3)$, and $U\left(\mathbf{x}\right),\mathbf{F}\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^1(\Omega )$ for domain $\Omega \subset W$.

Then, the equation

$${\widehat{Curl}}_{\left(\alpha \right)}\left(U\mathbf{F}\right)=\left[{\widehat{Grad}}_{\left(\alpha \right)}U\times \mathbf{F}\right]+U{\widehat{Curl}}_{\left(\alpha \right)}\mathbf{F}-U\left[{\widehat{Grad}}_{\left(\alpha \right)}1\times \mathbf{F}\right]$$

(916)

is satisfied for all $\mathbf{x}\in \Omega$.

Proof. 

Using Equation (890), we get

$${\widehat{Curl}}_{\left(\alpha \right)}\left(U\mathbf{F}\right)=\sum _{i,j,k=1}^3{\mathbf{e}}_i{\epsilon }_{ijk}{\widehat{D}}_{x^j}^{{\alpha }_j}\left(U{F}^k\right).$$

(917)

Using Equation (903) of Property 6, Equation (917) is written as

$${\widehat{Curl}}_{\left(\alpha \right)}\left(U\mathbf{F}\right)=$$

$$\sum _{i,j,k=1}^3{\mathbf{e}}_i{\epsilon }_{ijk}\left({\widehat{D}}_{x^j}^{{\alpha }_j}U\right)F^k+U\sum _{i,j,k=1}^3{\mathbf{e}}_i{\epsilon }_{ijk}\left({\widehat{D}}_{x^j}^{{\alpha }_j}{F}^k\right)-U\sum _{i,j,k=1}^3{\mathbf{e}}_i{\epsilon }_{ijk}\left({\widehat{D}}_{x^j}^{{\alpha }_j}1\right)F^k=$$

$$\left[{\widehat{Grad}}_{\left(\alpha \right)}U\times \mathbf{F}\right]+U{\widehat{Curl}}_{\left(\alpha \right)}\mathbf{F}-U\left[{\widehat{Grad}}_{\left(\alpha \right)}1\times \mathbf{F}\right],$$

where we use the vector product (the cross product)

$$\left[\mathbf{A}\times \mathbf{B}\right]=\sum _{i,j,k=1}^3{\mathbf{e}}_i{\epsilon }_{ijk}{A}^j{B}^k.$$

(918)

This ends the proof. □

Property 10 

(Vector Product Rule for NIDS Divergence). Let W be a D-dimensional space $W=W_1\times W_2\times W_3$ with $\left(\alpha \right)=({\alpha }_1,{\alpha }_2,{\alpha }_3)$ and $\mathbf{A}\left(\mathbf{x}\right),\mathbf{B}\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^1(\Omega )$ for domain $\Omega \subset W$.

Then, the equation

$${\widehat{Div}}_{\left(\alpha \right)}\left[\mathbf{A}\times \mathbf{B}\right]=\left(\mathbf{B},{\widehat{Curl}}_{\left(\alpha \right)}\mathbf{A}\right)-\left(\mathbf{A},{\widehat{Curl}}_{\left(\alpha \right)}\mathbf{B}\right)-\left(\left[\mathbf{A}\times \mathbf{B}\right],{\widehat{Grad}}_{\left(\alpha \right)}1\right)$$

(919)

is satisfied for all $\mathbf{x}\in \Omega$.

Proof. 

Using Equations (889) and (918), we get

$${\widehat{Div}}_{\left(\alpha \right)}\left[\mathbf{A}\times \mathbf{B}\right]=\sum _{j=1}^3{\widehat{D}}_{x^i}^{{\alpha }_i}{\left(\left[\mathbf{A}\times \mathbf{B}\right]\right)}_i=$$

$$\sum _{i=1}^3{\widehat{D}}_{x^i}^{{\alpha }_i}\left(\sum _{j,k=1}^3{\epsilon }_{ijk}{A}^j{B}^k\right)=\sum _{i,j,k=1}^3{\epsilon }_{ijk}{\widehat{D}}_{x^i}^{{\alpha }_i}\left(A^j{B}^k\right).$$

(920)

Using Equation (903) of Property 6, Equation (920) is written as

$$\sum _{k=1}^3\left(B^k\left(\sum _{i,j=1}^3{\epsilon }_{ijk}{\widehat{D}}_{x^i}^{{\alpha }_i}{A}^j\right)\right)+\sum _j^3\left(A^j\left(\sum _{i,k=1}^3{\epsilon }_{ijk}{\widehat{D}}_{x^i}^{{\alpha }_i}{B}^k\right)\right)-\sum _{i,j,k=1}^3{\epsilon }_{ijk}\left(A^j{B}^k\right){\widehat{D}}_{x^i}^{{\alpha }_i}1=$$

$$\sum _{k=1}^3\left(B^k\left(\sum _{i,j=1}^3{\epsilon }_{kij}{\widehat{D}}_{x^i}^{{\alpha }_i}{A}^j\right)\right)-\sum _j^3\left(A^j\left(\sum _{i,k=1}^3{\epsilon }_{jik}{\widehat{D}}_{x^i}^{{\alpha }_i}{B}^k\right)\right)-\sum _{i,j,k=1}^3{\epsilon }_{jki}\left(A^j{B}^k\right){\widehat{D}}_{x^i}^{{\alpha }_i}1=$$

$$\left(\mathbf{B},{\widehat{Curl}}_{\left(\alpha \right)}\mathbf{A}\right)-\left(\mathbf{A},{\widehat{Curl}}_{\left(\alpha \right)}\mathbf{B}\right)-\left(\left[\mathbf{A}\times \mathbf{B}\right],{\widehat{Grad}}_{\left(\alpha \right)}1\right),$$

(921)

where we use ${\epsilon }_{ijk}=-{\epsilon }_{jik}$. This ends the proof. □

Property 11 

(Vector Product Rule for NIDS Curl Operator). Let W be a D-dimensional space $W=W_1\times W_2\times W_3$ with $\left(\alpha \right)=({\alpha }_1,{\alpha }_2,{\alpha }_3)$, and $\mathbf{A}\left(\mathbf{x}\right),\mathbf{B}\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^1(\Omega )$ for domain $\Omega \subset W$.

Then, the equation

$${\widehat{Curl}}_{\left(\alpha \right)}\left[\mathbf{A}\times \mathbf{B}\right]=\left(\mathbf{B},{\widehat{\nabla }}_{\left(\alpha \right)}\right)\mathbf{A}-\left(\mathbf{A},{\widehat{\nabla }}_{\left(\alpha \right)}\right)\mathbf{B}+$$

$$\mathbf{A}{\widehat{Div}}_{\left(\alpha \right)}\mathbf{B}-\mathbf{B}{\widehat{Div}}_{\left(\alpha \right)}\mathbf{A}+\mathbf{B}\left(\mathbf{A},{\widehat{Grad}}_{\left(\alpha \right)}1\right)-\mathbf{A}\left(\mathbf{B},{\widehat{Grad}}_{\left(\alpha \right)}1\right)$$

(922)

is satisfied for all $\mathbf{x}\in \Omega$.

Proof. 

Using Equations (890) and (918), we get

$${\widehat{Curl}}_{\left(\alpha \right)}\left[\mathbf{A}\times \mathbf{B}\right]=\sum _{i,j,k=1}^3{\mathbf{e}}_i{\epsilon }_{ijk}{\widehat{D}}_{x^j}^{{\alpha }_j}{\left(\left[\mathbf{A}\times \mathbf{B}\right]\right)}^k=$$

$$\sum _{i,j,k=1}^3{\mathbf{e}}_i{\epsilon }_{ijk}{\widehat{D}}_{x^j}^{{\alpha }_j}\left(\sum _{l,m=1}^3{\epsilon }_{klm}{A}^l{B}^m\right)=$$

$$\sum _{i,j,k,l,m=1}^3{\mathbf{e}}_i{\epsilon }_{ijk}{\epsilon }_{klm}{\widehat{D}}_{x^j}^{{\alpha }_j}\left(A^l{B}^m\right).$$

(923)

Using

$${\epsilon }_{ijk}{\epsilon }_{klm}={\epsilon }_{kij}{\epsilon }_{klm},$$

(924)

the property of the Levi–Civita symbol

$$\sum _{i=1}^3{\epsilon }_{kij}{\epsilon }_{klm}={\delta }_{il}{\delta }_{jm}-{\delta }_{im}{\delta }_{jl},$$

(925)

and the product rule in Property 6, Equation (923) gives

$${\widehat{Curl}}_{\left(\alpha \right)}\left[\mathbf{A}\times \mathbf{B}\right]=$$

$$\sum _{i,j,l,m=1}^3{\mathbf{e}}_i{\delta }_{il}{\delta }_{jm}{\widehat{D}}_{x^j}^{{\alpha }_j}\left(A^l{B}^m\right)-\sum _{i,j,l,m=1}^3{\mathbf{e}}_i{\delta }_{im}{\delta }_{jl}{\widehat{D}}_{x^j}^{{\alpha }_j}\left(A^l{B}^m\right)=$$

$$\sum _{l,m=1}^3{\mathbf{e}}_l{\widehat{D}}_{x^m}^{{\alpha }_m}\left(A^l{B}^m\right)-\sum _{l,m=1}^3{\mathbf{e}}_m{\widehat{D}}_{x^l}^{{\alpha }_l}\left(A^l{B}^m\right)=$$

$$\sum _{l,m=1}^3{\mathbf{e}}_l\left({\widehat{D}}_{x^m}^{{\alpha }_m}{A}^l\right)B^m+\sum _{l,m=1}^3{\mathbf{e}}_l\left({\widehat{D}}_{x^m}^{{\alpha }_m}{B}^m\right)A^l-\sum _{l,m=1}^3{\mathbf{e}}_l{A}^l{B}^m\left({\widehat{D}}_{x^m}^{{\alpha }_m}1\right)-$$

$$\sum _{l,m=1}^3{\mathbf{e}}_m\left({\widehat{D}}_{x^l}^{{\alpha }_l}{A}^l\right)B^m-\sum _{l,m=1}^3{\mathbf{e}}_m\left({\widehat{D}}_{x^l}^{{\alpha }_l}{B}^m\right)A^l+\sum _{l,m=1}^3{\mathbf{e}}_m{A}^l{B}^m\left({\widehat{D}}_{x^l}^{{\alpha }_l}1\right)=$$

$$\sum _{m=1}^3{B}^m{\widehat{D}}_{x^m}^{{\alpha }_m}\left(\sum _{l=1}^3{\mathbf{e}}_l{A}^l\right)+\left(\sum _{l=1}^3{\mathbf{e}}_l{A}^l\right)\left(\sum _{m=1}^3{\widehat{D}}_{x^m}^{{\alpha }_m}{B}^m\right)-\left(\sum _{l=1}^3{\mathbf{e}}_l{A}^l\right)\left(\sum _{m=1}^3{B}^m{\widehat{D}}_{x^m}^{{\alpha }_m}1\right)-$$

$$\left(\sum _{m=1}^3{\mathbf{e}}_m{B}^m\right)\left(\sum _{l=1}^3{\widehat{D}}_{x^l}^{{\alpha }_l}{A}^l\right)-\sum _{l=1}^3{A}^l{\widehat{D}}_{x^l}^{{\alpha }_l}\left(\sum _{m=1}^3{\mathbf{e}}_m{B}^m\right)+\left(\sum _{m=1}^3{\mathbf{e}}_m{B}^m\right)\left(\sum _{l=1}^3{A}^l{\widehat{D}}_{x^l}^{{\alpha }_l}1\right)=$$

$$\left(\mathbf{B},{\widehat{\nabla }}_{\left(\alpha \right)}\right)\mathbf{A}+\mathbf{A}{\widehat{Div}}_{\left(\alpha \right)}\mathbf{B}-\mathbf{A}\left(\mathbf{B},{\widehat{Grad}}_{\left(\alpha \right)}1\right)-$$

$$\mathbf{B}{\widehat{Div}}_{\left(\alpha \right)}\mathbf{A}-\left(\mathbf{A},{\widehat{\nabla }}_{\left(\alpha \right)}\right)\mathbf{B}+\mathbf{B}\left(\mathbf{A},{\widehat{Grad}}_{\left(\alpha \right)}1\right).$$

(926)

This ends the proof. □

Property 12 

(Scalar Product Rule for NIDS Gradient). Let W be a D-dimensional space $W=W_1\times \dots \times W_n$ with $\left(\alpha \right)=({\alpha }_1,\dots ,{\alpha }_n)$, and $\mathbf{A}\left(\mathbf{x}\right),\mathbf{B}\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^1(\Omega )$ for domain $\Omega \subset W$.

Then, the equation

$${\widehat{Grad}}_{\left(\alpha \right)}\left(\mathbf{A},\mathbf{B}\right)=$$

$$\sum _{j=1}^n{B}^j\left({\widehat{Grad}}_{\left(\alpha \right)}{A}^j\right)+\sum _{j=1}^n{A}^j\left({\widehat{Grad}}_{\left(\alpha \right)}{B}^j\right)-\left(\mathbf{A},\mathbf{B}\right){\widehat{Grad}}_{\left(\alpha \right)}1$$

(927)

is satisfied for all $\mathbf{x}\in \Omega$.

Proof. 

Using Equation (888), we get

$${\widehat{Grad}}_{\left(\alpha \right)}\left(\mathbf{A},\mathbf{B}\right)=\sum _{k=1}^n{\mathbf{e}}_k{\widehat{D}}_{x^k}^{{\alpha }_k}\left(\mathbf{A},\mathbf{B}\right)=$$

$$\sum _{k=1}^n{\mathbf{e}}_k{\widehat{D}}_{x^k}^{{\alpha }_k}\left(\sum _{j=1}^n{A}^j{B}^j\right)=\sum _{k,j=1}^n{\mathbf{e}}_k{\widehat{D}}_{x^k}^{{\alpha }_k}\left(A^j{B}^j\right)=$$

$$\sum _{k,j=1}^n{\mathbf{e}}_k\left({\widehat{D}}_{x^k}^{{\alpha }_k}{A}^j\right)B^j+\sum _{k,j=1}^n{\mathbf{e}}_k\left({\widehat{D}}_{x^k}^{{\alpha }_k}{B}^j\right)A^j-\sum _{k,j=1}^n{\mathbf{e}}_k\left(A^j{B}^j\right){\widehat{D}}_{x^k}^{{\alpha }_k}1=$$

$$\sum _{j=1}^n{B}^j\left(\sum _{k,j=1}^n{\mathbf{e}}_k{\widehat{D}}_{x^k}^{{\alpha }_k}{A}^j\right)+\sum _{j=1}^n{A}^j\left(\sum _{k,j=1}^n{\mathbf{e}}_k{\widehat{D}}_{x^k}^{{\alpha }_k}{B}^j\right)-\sum _{j=1}^n\left(A^j{B}^j\right)\sum _{k=1}^n{\mathbf{e}}_k{\widehat{D}}_{x^k}^{{\alpha }_k}1=$$

$$\sum _{j=1}^n{B}^j\left({\widehat{Grad}}_{\left(\alpha \right)}{A}^j\right)+\sum _{j=1}^n{A}^j\left({\widehat{Grad}}_{\left(\alpha \right)}{B}^j\right)-\left(\mathbf{A},\mathbf{B}\right){\widehat{Grad}}_{\left(\alpha \right)}1.$$

(928)

This ends the proof. □

Property 13 

(Scalar Product Rule for NIDS Gradient). Let W be a D-dimensional space $W=W_1\times W_2\times W_3$ with $\left(\alpha \right)=({\alpha }_1,{\alpha }_2,{\alpha }_3)$, and $\mathbf{A}\left(\mathbf{x}\right),\mathbf{B}\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^1(\Omega )$ for domain $\Omega \subset W$.

Then, the equation

$${\widehat{Grad}}_{\left(\alpha \right)}\left(\mathbf{A},\mathbf{B}\right)=\left(\mathbf{B},{\widehat{\nabla }}_{\left(\alpha \right)}\right)\mathbf{A}+\left(\mathbf{A},{\widehat{\nabla }}_{\left(\alpha \right)}\right)\mathbf{B}+$$

$$\left[\mathbf{A}\times {\widehat{Curl}}_{\left(\alpha \right)}\mathbf{B}\right]+\left[\mathbf{B}\times {\widehat{Curl}}_{\left(\alpha \right)}\mathbf{A}\right]-\left(\mathbf{A},\mathbf{B}\right){\widehat{Grad}}_{\left(\alpha \right)}1$$

(929)

is satisfied for all $\mathbf{x}\in \Omega$.

Proof. 

The proof is constructed similarly to the proofs given above.

One can use Equation (903) of Property 6 in the form

$${\widehat{\nabla }}_{\left(\alpha \right)}\left(\mathbf{A},\mathbf{B}\right)={\widehat{\nabla }}_{\left(\alpha \right)}\left(\stackrel{\downarrow }{\mathbf{A}},\mathbf{B}\right)+{\widehat{\nabla }}_{\left(\alpha \right)}\left(\mathbf{A},\stackrel{\downarrow }{\mathbf{B}}\right)-\left(\mathbf{A},\mathbf{B}\right){\widehat{Grad}}_{\left(\alpha \right)}1,$$

(930)

where the vertical arrow indicates the factor to which the operator is applied in a given term. Then, we can use the equality

$$\left[\mathbf{A}\times \left[\mathbf{B}\times \mathbf{C}\right]\right]=\mathbf{B}\left(\mathbf{A},\mathbf{C}\right)-\mathbf{C}\left(\mathbf{A},\mathbf{B}\right)$$

(931)

in the form

$$\mathbf{C}\left(\mathbf{A},\mathbf{B}\right)=\left[\mathbf{A}\times \left[\mathbf{C}\times \mathbf{B}\right]\right]+\left(\mathbf{A},\mathbf{C}\right)\mathbf{B}$$

(932)

with $\mathbf{C}={\widehat{\nabla }}_{\left(\alpha \right)}$, such that

$${\widehat{\nabla }}_{\left(\alpha \right)}\left(\mathbf{A},\stackrel{\downarrow }{\mathbf{B}}\right)=\left[\mathbf{A}\times \left[{\widehat{\nabla }}_{\left(\alpha \right)}\times \mathbf{B}\right]\right]+\left(\mathbf{A},{\widehat{\nabla }}_{\left(\alpha \right)}\right)\mathbf{B},$$

(933)

$${\widehat{\nabla }}_{\left(\alpha \right)}\left(\stackrel{\downarrow }{\mathbf{A}},\mathbf{B}\right)={\widehat{\nabla }}_{\left(\alpha \right)}\left(\mathbf{B},\stackrel{\downarrow }{\mathbf{A}}\right)=\left[\mathbf{B}\times \left[{\widehat{\nabla }}_{\left(\alpha \right)}\times \mathbf{A}\right]\right]+\left(\mathbf{B},{\widehat{\nabla }}_{\left(\alpha \right)}\right)\mathbf{A}.$$

(934)

Substitution of Equations (933) and (934) into Equation (930) gives (929). This ends the proof. □

Remark 69. 

In the metric approach, the equations of these properties are satisfied, where we should use the notations ${Grad}_{\left(\alpha \right)}U$, ${Div}_{\left(\alpha \right)}\mathbf{A}$, ${Curl}_{\left(\alpha \right)}\mathbf{A}$, instead of ${\widehat{Grad}}_{\left(\alpha \right)}U$, ${\widehat{Div}}_{\left(\alpha \right)}\mathbf{A}$, and ${\widehat{Curl}}_{\left(\alpha \right)}\mathbf{A}$ and take into account that

$${\widehat{c}}_1({\alpha }_j,x^j)=1,$$

(935)

$${Grad}_{\left(\alpha \right)}1=0.$$

(936)

For the metric approach, the equations of the properties look the same as for standard vector calculus.

Remark 70. 

These properties of the differential vector operators of the first order in the product measure approach is a necessary requirement for the self-consistency of differential NIDS vector calculus. Therefore, the requirement 3a of Principle 9 about the self-consistency of the NIDS vector calculus is satisfied.

4.5.3. Second-Order Vector NIDS Operators

Let us consider differential operators of the second order of the NIDS vector calculus.

Property 14. 

Let W be a D-dimensional space $W=W_1\times \dots \times W_n$ with $\left(\alpha \right)=({\alpha }_1,\dots ,{\alpha }_n)$ and $U\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^2(\Omega )$ for domain $\Omega \subset W$.

Then, the equation

$${\widehat{Div}}_{\left(\alpha \right)}{\widehat{Grad}}_{\left(\alpha \right)}U\left(\mathbf{x}\right)={}^S\widehat{\Delta }_{\left(\alpha \right)}U\left(\mathbf{x}\right)$$

(937)

is satisfied for all $\mathbf{x}\in \Omega$, where ${}^S\widehat{\Delta }$ is the scalar Laplacian

$${}^S\widehat{\Delta }_{\left(\alpha \right)}U\left(\mathbf{x}\right):=\sum _{k=1}^n{\left({\widehat{D}}_{x^k}^{{\alpha }_k}\right)}^{2}U\left(\mathbf{x}\right)$$

(938)

for $U\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^2(\Omega )$ and all $\left(\alpha \right)=({\alpha }_1,{\alpha }_2,{\alpha }_3)$ with ${\alpha }_k\in (0,1]$.

Proof. 

Using Equations (888) and (889), we get

$${\widehat{Div}}_{\left(\alpha \right)}{\widehat{Grad}}_{\left(\alpha \right)}U\left(\mathbf{x}\right)=\sum _{k=1}^n{\widehat{D}}_{x^k}^{{\alpha }_k}{\widehat{D}}_{x^k}^{{\alpha }_k}U\left(\mathbf{x}\right)={}^S\widehat{\Delta }_{\left(\alpha \right)}U\left(\mathbf{x}\right).$$

(939)

This ends the proof. □

Remark 71. 

Note that the scalar Laplacian can be defined by using the nabla operator as

$${}^S\widehat{\Delta }_{\left(\alpha \right)}U\left(\mathbf{x}\right):=\left({\widehat{\nabla }}_{\left(\alpha \right)},{\widehat{\nabla }}_{\left(\alpha \right)}\right)U\left(\mathbf{x}\right)$$

(940)

for $U\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^2(\Omega )$ with the domain $\Omega \subset W$.

Property 14 can also be used to define the scalar Laplacian as

$${}^S\widehat{\Delta }_{\left(\alpha \right)}U\left(\mathbf{x}\right):={\widehat{Div}}_{\left(\alpha \right)}{\widehat{Grad}}_{\left(\alpha \right)}U\left(\mathbf{x}\right),$$

(941)

for $U\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^2(\Omega )$.

Property 15. 

Let W be a D-dimensional space $W=W_1\times W_2\times W_3$ with $\left(\alpha \right)=({\alpha }_1,{\alpha }_2,{\alpha }_3)$ and $U\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^2(\Omega )$ for domain $\Omega \subset W$.

Then, the equation

$${\widehat{Curl}}_{\left(\alpha \right)}{\widehat{Grad}}_{\left(\alpha \right)}U\left(\mathbf{x}\right)=\mathbf{0}$$

(942)

is satisfied for all $\mathbf{x}\in \Omega$ and all $\left(\alpha \right)=({\alpha }_1,{\alpha }_2,{\alpha }_3)$ with ${\alpha }_k\in (0,1]$, where $\mathbf{0}$ is the zero vector.

Proof. 

Using Equations (888) and (890), we get

$${\widehat{Curl}}_{\left(\alpha \right)}{\widehat{Grad}}_{\left(\alpha \right)}U\left(\mathbf{x}\right)=$$

$$\sum _{i,j,k=1}^3{\mathbf{e}}_i{\epsilon }_{ijk}{\widehat{D}}_{x^j}^{{\alpha }_j}{\left({\widehat{Grad}}_{\left(\alpha \right)}U\left(\mathbf{x}\right)\right)}^k=$$

$$\sum _{i,j,k=1}^3{\mathbf{e}}_i{\epsilon }_{ijk}{\widehat{D}}_{x^j}^{{\alpha }_j}{\widehat{D}}_{x^k}^{{\alpha }_k}U\left(\mathbf{x}\right)=$$

$$\sum _{i=1}^3{\mathbf{e}}_i\left(\sum _{j,k=1}^3{\epsilon }_{ijk}{\widehat{D}}_{x^j}^{{\alpha }_j}{\widehat{D}}_{x^k}^{{\alpha }_k}U\left(\mathbf{x}\right)\right).$$

(943)

Using the symmetry of the product

$${\widehat{D}}_{x^j}^{{\alpha }_j}{\widehat{D}}_{x^k}^{{\alpha }_k}={\widehat{D}}_{x^k}^{{\alpha }_k}{\widehat{D}}_{x^j}^{{\alpha }_j}$$

(944)

by the indices j and k and the anti-symmetry of the Levi–Civita symbol

$${\epsilon }_{ijk}=-{\epsilon }_{ikj},$$

(945)

we get

$$\sum _{j,k=1}^3{\epsilon }_{ijk}{\widehat{D}}_{x^j}^{{\alpha }_j}{\widehat{D}}_{x^k}^{{\alpha }_k}U\left(\mathbf{x}\right)=0,$$

(946)

if $U\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^2(\Omega )$. Substitution of expression (946) into Equation (943) gives

$$\sum _{i=1}^3{\mathbf{e}}_i\left(\sum _{j,k=1}^3{\epsilon }_{ijk}{\widehat{D}}_{x^j}^{{\alpha }_j}{\widehat{D}}_{x^k}^{{\alpha }_k}U\left(\mathbf{x}\right)\right)=\sum _{i=1}^3{\mathbf{e}}_{i}0=\mathbf{0}.$$

(947)

This ends the proof. □

Property 16. 

Let W be a D-dimensional space $W=W_1\times W_2\times W_3$ with $\left(\alpha \right)=({\alpha }_1,{\alpha }_2,{\alpha }_3)$ and $\mathbf{A}\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^2(\Omega )$ for domain $\Omega \subset W$.

Then, the equation

$${\widehat{Div}}_{\left(\alpha \right)}{\widehat{Curl}}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right)=0$$

(948)

is satisfied for all $\mathbf{x}\in \Omega$ and all $\left(\alpha \right)=({\alpha }_1,{\alpha }_2,{\alpha }_3)$ with ${\alpha }_k\in (0,1]$.

Proof. 

Using Equations (889) and (890), we get

$${\widehat{Div}}_{\left(\alpha \right)}{\widehat{Curl}}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right)=$$

$$\sum _{i=1}^3{\widehat{D}}_{x^i}^{{\alpha }_i}{\left({\widehat{Curl}}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right)\right)}^i=$$

$$\sum _{i,j,k=1}^3{\epsilon }_{ijk}{\widehat{D}}_{x^i}^{{\alpha }_i}{\widehat{D}}_{x^j}^{{\alpha }_j}{A}^k=$$

$$\sum _{k,i,j=1}^3\left(\sum _{i,j=1}^3{\epsilon }_{kij}{\widehat{D}}_{x^i}^{{\alpha }_i}{\widehat{D}}_{x^j}^{{\alpha }_j}\right)A^k.$$

(949)

Using the symmetry of the operator product

$${\widehat{D}}_{x^i}^{{\alpha }_i}{\widehat{D}}_{x^j}^{{\alpha }_j}={\widehat{D}}_{x^j}^{{\alpha }_j}{\widehat{D}}_{x^i}^{{\alpha }_i}$$

(950)

by the indices i and j and the anti-symmetry of the Levi–Civita symbol

$${\epsilon }_{kij}=-{\epsilon }_{kji},$$

(951)

we get

$$\sum _{i,g=1}^3{\epsilon }_{kij}{\widehat{D}}_{x^i}^{{\alpha }_i}{\widehat{D}}_{x^j}^{{\alpha }_j}f\left(\mathbf{x}\right)=0,$$

(952)

if $f\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^2(\Omega )$. Substitution of Equation (952) into Equation (949) gives

$$\sum _{k,i,j=1}^3\left(\sum _{i,j=1}^3{\epsilon }_{kij}{\widehat{D}}_{x^i}^{{\alpha }_i}{\widehat{D}}_{x^j}^{{\alpha }_j}{A}^k\right)=\sum _{k,i,j=1}^{3}0=0.$$

(953)

This ends the proof. □

Property 17. 

Let W be a D-dimensional space $W=W_1\times W_2\times W_3$ with $\left(\alpha \right)=({\alpha }_1,{\alpha }_2,{\alpha }_3)$, and $\mathbf{A}\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^2(\Omega )$ for domain $\Omega \subset W$.

Then, the equation

$${\widehat{Curl}}_{\left(\alpha \right)}{\widehat{Curl}}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right)={\widehat{Grad}}_{\left(\alpha \right)}{\widehat{Div}}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right)-\left({\widehat{\nabla }}_{\left(\alpha \right)},{\widehat{\nabla }}_{\left(\alpha \right)}\right)\mathbf{A}\left(\mathbf{x}\right)$$

(954)

is satisfied for all $\mathbf{x}\in \Omega$ and all $\left(\alpha \right)=({\alpha }_1,{\alpha }_2,{\alpha }_3)$ with ${\alpha }_k\in (0,1]$.

Proof. 

Using Equation (890), we get

$${\widehat{Curl}}_{\left(\alpha \right)}{\widehat{Curl}}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right)=$$

$$\sum _{i,j,k=1}^3{\mathbf{e}}_i{\epsilon }_{ijk}{\widehat{D}}_{x^j}^{{\alpha }_j}{\left({\widehat{Curl}}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right)\right)}^k=$$

$$\sum _{i,j,k=1}^3{\mathbf{e}}_i{\epsilon }_{ijk}{\widehat{D}}_{x^j}^{{\alpha }_j}\left(\sum _{l,m=1}^3{\epsilon }_{klm}{\widehat{D}}_{x^l}^{{\alpha }_l}{A}^m\right)=$$

$$\sum _{i,j,k,l,m=1}^3{\mathbf{e}}_i{\epsilon }_{ijk}{\epsilon }_{klm}{\widehat{D}}_{x^j}^{{\alpha }_j}{\widehat{D}}_{x^l}^{{\alpha }_l}{A}^m.$$

(955)

Using

$${\epsilon }_{ijk}{\epsilon }_{klm}={\epsilon }_{kij}{\epsilon }_{klm}$$

(956)

and the property of the Levi–Civita symbol

$$\sum _{i=1}^3{\epsilon }_{kij}{\epsilon }_{klm}={\delta }_{il}{\delta }_{jm}-{\delta }_{im}{\delta }_{jl},$$

(957)

Equation (955) gives

$${\widehat{Curl}}_{\left(\alpha \right)}{\widehat{Curl}}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right)=$$

$$\sum _{i,j,l,m=1}^3{\mathbf{e}}_i{\delta }_{il}{\delta }_{jm}{\widehat{D}}_{x^j}^{{\alpha }_j}{\widehat{D}}_{x^l}^{{\alpha }_l}{A}^m-\sum _{i,j,l,m=1}^3{\mathbf{e}}_i{\delta }_{im}{\delta }_{jl}{\widehat{D}}_{x^j}^{{\alpha }_j}{\widehat{D}}_{x^l}^{{\alpha }_l}{A}^m=$$

$$\sum _{l,m=1}^3{\mathbf{e}}_l{\widehat{D}}_{x^m}^{{\alpha }_m}{\widehat{D}}_{x^l}^{{\alpha }_l}{A}^m-\sum _{l,m=1}^3{\mathbf{e}}_m{\widehat{D}}_{x^l}^{{\alpha }_l}{\widehat{D}}_{x^l}^{{\alpha }_l}{A}^m=$$

$$\left(\sum _{l=1}^3{\mathbf{e}}_l{\widehat{D}}_{x^l}^{{\alpha }_l}\right)\left(\sum _{m=1}^3{\widehat{D}}_{x^m}^{{\alpha }_m}{A}^m\right)-\sum _{l=1}^3{\left({\widehat{D}}_{x^l}^{{\alpha }_l}\right)}^2\left(\sum _{m=1}^3{\mathbf{e}}_m{A}^m\right)=$$

$${\widehat{Grad}}_{\left(\alpha \right)}{\widehat{Div}}_{\left(\alpha \right)}\mathbf{A}\left(\mathbf{x}\right)-\left({\widehat{\nabla }}_{\left(\alpha \right)},{\widehat{\nabla }}_{\left(\alpha \right)}\right)\mathbf{A}.$$

(958)

This ends the proof. □

Remark 72. 

In the standard vector calculus, in addition to the scalar NIDS Laplacian, we can use the vector Laplacian [154]. Therefore, in addition to the definition of the scalar NIDS Laplacian ${}^S\Delta _{\left(\alpha \right)}$, we can define the vector NIDS Laplacian ${}^V\widehat{\Delta }_{\left(\alpha \right)}$ by the equation

$${}^V\widehat{\Delta }_{\left(\alpha \right)}\mathbf{A}:={\widehat{Grad}}_{\left(\alpha \right)}{\widehat{Div}}_{\left(\alpha \right)}\mathbf{A}-{\widehat{Curl}}_{\left(\alpha \right)}{\widehat{Curl}}_{\left(\alpha \right)}\mathbf{A},$$

(959)

if $\mathbf{A}\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^2(\Omega )$ for domain $\Omega \subset W=W_1\times W_2\times W_3$.

For the D-dimensional space $W=W_1\times \dots \times W_n$ with arbitrary integer $n\in \mathbb{N}$, the vector NIDS Laplace operator can be defined as

$${}^V\widehat{\Delta }_{\left(\alpha \right)}\mathbf{A}:=\sum _{k=1}^n{\mathbf{e}}_k\left({\widehat{\nabla }}_{\left(\alpha \right)},{\widehat{\nabla }}_{\left(\alpha \right)}\right)A^k\left(\mathbf{x}\right)=\sum _{k=1}^n{\mathbf{e}}_k{\left({\widehat{D}}_{x^k}^{{\alpha }_k}\right)}^2{A}^k\left(\mathbf{x}\right),$$

(960)

if $\mathbf{A}\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^2(\Omega )$ for domain $\Omega \subset W=W_1\times \dots \times W_n$.

Remark 73. 

These properties of the differential vector operators of the second order in the product measure approach are a necessary requirement for the self-consistency of differential NIDS vector calculus. Therefore, the requirement 3c of Principle 9 about the self-consistency of the NIDS vector calculus is satisfied.

4.6. NIDS Vector Operators in Orthogonal Coordinate Systems

A curvilinear coordinate system is called orthogonal, if, at each point, the coordinate curves passing through the point form right angles with each other, i.e., the tangents to the curve lines are mutually orthogonal at the point. Equations of the NIDS operators for the orthogonal coordinates can be obtained from the equations of these operators in the Cartesian coordinates by standard methods. As examples of the NIDS integral operators, we will describe the equations of the volume NIDS integrals in the orthogonal coordinates. As examples of the NIDS differential operators, we will derive explicit expressions for the NIDS gradient in the cylindrical and spherical coordinates and the NIDS divergence in the cylindrical coordinates in the D-dimensional space.

4.6.1. NIDS Integration in Orthogonal Coordinates

Let $W=W_1\times \dots \times W_n$ be a D-dimensional space with $D={\alpha }_1+\dots +{\alpha }_n$, and let function $f\left(\mathbf{x}\right)\in C(\Omega )$ for the domain $\Omega \subset W$. The volume NIDS integration of function $f\left(\mathbf{x}\right)$ over the region $\Omega$ is described as

$${\int }_{\Omega }f\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x})={\int }_{\Omega }f\left(\mathbf{x}\right)\prod _{k=1}^{n}d{\mu }_1({\alpha }_k,x_k)={\int }_{\Omega }f\left(\mathbf{x}\right)\prod _{k=1}^n{c}_1({\alpha }_k,x_k)d{x}_k,$$

(961)

where

$$d{\mu }_1({\alpha }_k,x_k)=c_1({\alpha }_k,x_k)d{x}_k=c_1\left({\alpha }_k\right){|x_k|}^{{\alpha }_k-1}d{x}_k,$$

(962)

$$c_1\left({\alpha }_k\right)={\displaystyle \frac{{\pi }^{{\alpha }_k/2}}{\Gamma ({\alpha }_k/2)}},$$

(963)

and all ${\alpha }_k\in (0,1]$.

(1) Let us consider the case $W=W_1\times W_2\times W_3$, $n=3$, and $D\in (0,3]$. The Cartesian coordinates $(x_1,x_2,x_3)$ can be transformed to the orthogonal coordinates $(q_1,q_2,q_3)$ by the equations

$$x_k=x_k(q_1,q_2,q_3),(k=1,2,3).$$

(964)

The NIDS integration (961) in the orthogonal coordinates $(q_1,q_2,q_3)$ is written as

$${\int }_{\Omega }f\left(\mathbf{x}\right)d{\mu }_3(D,\mathbf{x})=$$

$$C({\alpha }_1,{\alpha }_2,{\alpha }_3){\int }_{\Omega }d{x}_{1}d{x}_{2}d{x}_{3}f\left(\mathbf{x}\right)\left(\prod _{k=1}^3{|x_k|}^{{\alpha }_k-1}\right)=$$

$$C({\alpha }_1,{\alpha }_2,{\alpha }_3){\int }_{\Omega }d{q}_{1}d{q}_{2}d{q}_3{J}_3(q_1,q_2,q_3)f\left(\mathbf{x}\right)\left(\prod _{k=1}^n{|x_k(q_1,q_2,q_3)|}^{{\alpha }_k-1}\right),$$

where $J_3(q_1,q_2,q_3)$ is the Jacobian of the coordinate change

$$J_3(q_1,q_2,q_3)=H_1(q_1,q_2,q_3)H_2(q_1,q_2,q_3)H_3(q_1,q_2,q_3),$$

(965)

and $H_j(q_1,q_2,q_3)$ are the Lame coefficients

$$H_j(q_1,q_2,q_3)=\sqrt{{\left({\displaystyle \frac{\partial x_1}{\partial q_j}}\right)}^2+{\left({\displaystyle \frac{\partial x_2}{\partial q_j}}\right)}^2+{\left({\displaystyle \frac{\partial x_3}{\partial q_j}}\right)}^2},$$

(966)

and

$$C({\alpha }_1,{\alpha }_2,{\alpha }_3)={\displaystyle \frac{{\pi }^{{\alpha }_1/2}}{\Gamma ({\alpha }_1/2)}}{\displaystyle \frac{{\pi }^{{\alpha }_2/2}}{\Gamma ({\alpha }_2/2)}}{\displaystyle \frac{{\pi }^{{\alpha }_3/2}}{\Gamma ({\alpha }_3/2)}},$$

(967)

where all ${\alpha }_k\in (0,1]$.

Example 14. 

For example, in the spherical coordinates $(r,\varphi ,\theta )$, such that

$$x_1=rsin\theta cos\varphi ,$$

(968)

$$x_2=rsin\theta sin\varphi ,$$

(969)

$$x_3=rcos\theta ,$$

(970)

where $r\in [0,\infty )$, $\varphi \in [0,2\pi )$, $\theta \in [0,\pi ]$, the volume NIDS integration is written as

$${\int }_{\Omega }f\left(\mathbf{x}\right)d{\mu }_3(D,\mathbf{x})=$$

$$C({\alpha }_1,{\alpha }_2,{\alpha }_3){\int }_{\Omega }f\left(\mathbf{x}\right)|x_1{|}^{{\alpha }_1-1}|x_2{|}^{{\alpha }_2-1}{|x_3|}^{{\alpha }_3-1}d{x}_{1}d{x}_{2}d{x}_3=$$

$$C({\alpha }_1,{\alpha }_2,{\alpha }_3){\int }_{\Omega }{r}^{{\alpha }_1+{\alpha }_2+{\alpha }_3-1}{|cos\varphi |}^{{\alpha }_1-1}{|sin\varphi |}^{{\alpha }_2-1}·{|sin\theta |}^{{\alpha }_1+{\alpha }_2-1}{|cos\theta |}^{{\alpha }_3-1}drd\varphi d\theta ,$$

where $J_3(r,\varphi ,\theta )=r^{2}sin\theta$ is the Jacobian of the coordinate change.

(2) Let us consider the case $W=W_1\times \dots \times W_n$ and $D\in (0,n]$. The Cartesian coordinates $(x_1,\dots ,x_n)$ can be transformed to the orthogonal coordinates $(q_1,\dots ,q_n)$ by the equation

$$x_k=x_k(q_1,\dots ,q_n),(k=1,\dots ,n).$$

(971)

The NIDS integration (961) in the orthogonal coordinates $(q_1,\dots ,q_n)$ is written as

$${\int }_{\Omega }f\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x})=C({\alpha }_1,\dots ,{\alpha }_n){\int }_{\Omega }f\left(\mathbf{x}\right)\left(\prod _{k=1}^n{|x_k|}^{{\alpha }_k-1}\right)\prod _{k=1}^{n}d{x}_k=$$

$$C({\alpha }_1,\dots ,{\alpha }_n){\int }_{\Omega }f(x_1,\dots ,x_n)\left(\prod _{k=1}^n{|x_k(x_1,\dots ,x_n)|}^{{\alpha }_k-1}\right)J_n(q_1,\dots ,q_n)\prod _{j=1}^{n}d{q}_j,$$

(972)

where $J_n(q_1,\dots ,q_n)$ is the Jacobian

$$J_n(q_1,\dots ,q_n)=\prod _{j=1}^n{H}_j(q_1,\dots ,q_n),$$

(973)

and $H_j(q_1,\dots ,q_n)$ are the Lame coefficients

$$H_j(q_1,\dots ,q_n)=\sqrt{\sum _{k=1}^n{\left({\displaystyle \frac{\partial x_k}{\partial q_j}}\right)}^2}(k=1,\dots ,n),$$

(974)

and

$$C({\alpha }_1,\dots ,{\alpha }_n)=\prod _{k=1}^n{\displaystyle \frac{{\pi }^{{\alpha }_k/2}}{\Gamma ({\alpha }_k/2)}}.$$

(975)

Example 15. 

As an example, one can consider the “hyperspherical” coordinates [159], where r is the radial coordinate, ${\varphi }_k\in [0,\pi ]$ with $k=1,\dots ,n-2$, and ${\varphi }_{n-1}\in [0,2\pi )$ are angles. The coordinates $x_k$ with $k=1,\dots ,n$ and $n>3$ are expressed through the radial coordinate and angles by the equations

$$x_1=rcos\left({\varphi }_1\right),$$

(976)

$$x_k=rcos\left({\varphi }_k\right)\prod _{j=1}^{k-1}sin\left({\varphi }_j\right),(k=2,\dots ,n-1),$$

(977)

$$x_n=r\prod _{j=1}^{n-1}sin\left({\varphi }_j\right).$$

(978)

The determinant of the Jacobian matrix is

$$J_n(r,\varphi )=\parallel det{\displaystyle \frac{\partial \mathbf{x}}{\partial (r,\varphi )}}\parallel =r^{n-1}\prod _{j=1}^{n-2}{sin}^{n-1-j}\left({\varphi }_j\right),$$

(979)

where $\varphi :=({\varphi }_1,\dots ,{\varphi }_{n-1})$, and the measure of n-dimensional space is

$$\prod _{k=1}^{n}d{x}_k=J_n(r,{\varphi }_1,\dots ,{\varphi }_{n-1})dr\prod _{k=1}^{n-1}d{\varphi }_k.$$

(980)

The function

$$c_n(D,\mathbf{x}):=\prod _{k=1}^n{c}_1({\alpha }_k,x_k)=C({\alpha }_1,\dots ,{\alpha }_n)\prod _{k=1}^n{|x_k|}^{{\alpha }_k-1}=$$

$$C({\alpha }_1,\dots ,{\alpha }_n){|r|}^{D-n}\prod _{k=1}^{n-1}|cos\left({\varphi }_k\right){|}^{{\alpha }_k-1}\prod _{j=1}^{n-1}{|sin\left({\varphi }_j\right)|}^{{\sum }_{k=j+1}^n{\alpha }_k-(n-j)},$$

(981)

where ${\sum }_{k=n}^n{\alpha }_k={\alpha }_n$.

Then, the NIDS measure can be represented as

$$d{\mu }_n(D,\mathbf{x})=\prod _{k=1}^{n}d{\mu }_1({\alpha }_k,x_k)=c_n(D,\mathbf{x})J_n(r,\varphi )dr\prod _{k=1}^{n-1}d{\varphi }_k=$$

$$C({\alpha }_1,\dots ,{\alpha }_n)r^{D-1}\prod _{k=1}^{n-1}|cos\left({\varphi }_k\right){|}^{{\alpha }_k-1}\prod _{j=1}^{n-1}{|sin\left({\varphi }_j\right)|}^{{\sum }_{k=j+1}^n{\alpha }_k-1}dr\prod _{k=1}^{n-1}d{\varphi }_k.$$

(982)

As a result, we get

$${\int }_{\Omega }f\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x})=$$

$$C({\alpha }_1,\dots ,{\alpha }_n){\int }_{\Omega }f\left(\mathbf{x}\right)r^{D-1}\prod _{k=1}^{n-1}|cos\left({\varphi }_k\right){|}^{{\alpha }_k-1}\prod _{j=1}^{n-1}{|sin\left({\varphi }_j\right)|}^{{\sum }_{k=j+1}^n{\alpha }_k-1}dr\prod _{k=1}^{n-1}d{\varphi }_k=$$

$$C({\alpha }_1,\dots ,{\alpha }_n){\int }_{\Omega }f\left(\mathbf{x}\right)r^{D-1}\prod _{j=1}^{n-1}|cos\left({\varphi }_j\right){|}^{{\alpha }_j-1}{|sin\left({\varphi }_j\right)|}^{{\sum }_{k=j+1}^n{\alpha }_k-1}dr\prod _{j=1}^{n-1}d{\varphi }_j,$$

(983)

where $C({\alpha }_1,\dots ,{\alpha }_n$) is defined by Equation (975).

4.6.2. NIDS Differentiation in Orthogonal Coordinates

Let us consider the case $W=W_1\times \dots \times W_n$ and $D\in (0,n]$. In the Cartesian coordinates, the NIDS differential operator of the NIDS vector calculus such as the NIDS gradient, the NIDS divergence, the NIDS curl operator can be expressed through the following elements.

1.

The unit basis vectors ${\mathbf{e}}_k$ or ${\mathbf{e}}^k$, where $k=1,\dots ,n$.

2.

The NIDS derivatives with respect to the Cartesian coordinates $x^k$ with $k=1,\dots ,n$.

3.

The scalar fields $U\left(\mathbf{x}\right)$ or the components of vector field $F_k\left(\mathbf{x}\right)$, where $k=1,\dots ,n$.

For all these elements, it is necessary to make a transformation through new coordinates $(q^1,\dots ,q^n)$. The Cartesian coordinates $(x^1,\dots ,x^n)$ are connected with the orthogonal coordinates $(q_1,\dots ,q_n)$ by the equations

$$x^k=x^k(q^1,\dots ,q^n),(k=1,\dots ,n),$$

(984)

$$q^j=q^j(x^1,\dots ,x^n),(j=1,\dots ,n).$$

(985)

The first-order NIDS derivatives with respect to the Cartesian coordinates $x^k$ have $k=1,\dots ,n$. These NIDS derivatives are expressed in terms of derivatives with respect to new orthogonal coordinates $\partial /\partial q^j$ with $j=1,\dots ,n$ by the equations

$${\displaystyle \frac{\partial }{\partial x^k}}=\sum _{j=1}^n{\left({\displaystyle \frac{\partial q^j\left(\mathbf{x}\right)}{\partial x^k}}\right)}_{x^k=x^k(q^1,\dots ,q^n)}{\displaystyle \frac{\partial }{\partial q^j}}(k=1,\dots ,n),$$

(986)

where $q^j\left(\mathbf{x}\right)=q^j(x^1,\dots ,x^n)$. The NIDS derivatives of the first order are expressed through derivatives with respect to new orthogonal coordinates $\partial /\partial q^j$, where $j=1,\dots ,n$ in the form

$${\displaystyle \frac{1}{c_1({\alpha }^k,x^k)}}{\displaystyle \frac{\partial }{\partial x^k}}=\sum _{j=1}^n{\displaystyle \frac{1}{c_1({\alpha }^k,x^k(q^1,\dots ,q^n)}}\sum _{j=1}^n{\left({\displaystyle \frac{\partial q^j\left(\mathbf{x}\right)}{\partial x^k}}\right)}_{x^k=x^k(q^1,\dots ,q^n)}{\displaystyle \frac{\partial }{\partial q^j}},$$

(987)

where $k=1,\dots ,n$.

The unit basis vectors ${\mathbf{e}}_k$ or ${\mathbf{e}}^k$ with $k=1,\dots ,n$ for the Cartesian coordinates $x^k$ and the unit basis vectors ${\mathbf{w}}_j$ or ${\mathbf{w}}^j$ with $j=1,\dots ,n$ of the orthogonal coordinates $q^j$ are connected by the equations

$${\mathbf{e}}^k=\sum _{j=1}^n{\displaystyle \frac{1}{H_j(q^1,\dots ,q^n)}}{\displaystyle \frac{\partial x^k}{\partial q^j}}{\mathbf{w}}^j(k=1,\dots ,n),$$

(988)

where

$$H_j(q^1,\dots ,q^n):=\sqrt{\sum _{k=1}^n{\left({\displaystyle \frac{\partial x^k}{\partial q^j}}\right)}^2}.$$

(989)

Using Equation (988), we can derive the equations

$${\mathbf{w}}^j=\sum _{k=1}^n{V}_k^j{\mathbf{e}}^k(j=1,\dots ,n),$$

(990)

where

$$\sum _{j=1}^n{\displaystyle \frac{1}{H_j(q^1,\dots ,q^n)}}{\displaystyle \frac{\partial x^k}{\partial q^j}}{V}_l^j={\delta }_l^k.$$

(991)

For cylindrical and spherical coordinate systems, Equations (988) and (990) are given in [160], p. 577.

Note that, for the orthogonal coordinate systems, the following equations are satisfied

$${\mathbf{e}}^k{\mathbf{e}}^k=1,{\mathbf{e}}^k{\mathbf{e}}^l=0(k,l=1,\dots ,n,l\ne k),$$

(992)

$${\mathbf{w}}^j{\mathbf{w}}^j=1,{\mathbf{w}}^j{\mathbf{w}}^i=0(j,i=1,\dots ,n,i\ne j).$$

(993)

For the vector field,

$$\mathbf{F}\left(\mathbf{x}\right)=\sum _{k=1}^n{F}_k\left(\mathbf{x}\right){\mathbf{e}}^k=\sum _{j=1}^n{F}_k^q\left(\mathbf{q}\right){\mathbf{w}}^j,$$

(994)

the components $F_k\left(\mathbf{x}\right)$ of the Cartesian coordinates and the components $F_j^q\left(\mathbf{q}\right)$ of new orthogonal coordinates, where $k,j=1,\dots ,n$, are connected as

$$F_k\left(\mathbf{x}\right)=\sum _{j=1}^n{\left({\displaystyle \frac{\partial q^j\left(\mathbf{x}\right)}{\partial x^k}}\right)}_{x^k=x^k(q^1,\dots ,q^n)}{\left(\sum _{k=1}^n{\left({\displaystyle \frac{\partial q^j}{\partial x^k}}\right)}_{x^k=x^k(q^1,\dots ,q^k)}^2\right)}^{-1/2}{F}_k^q\left(\mathbf{q}\right).$$

(995)

In order to derive the NIDS gradient, the NIDS divergence, and the NIDS curl operator in the orthogonal coordinates from equations of these operators in the Cartesian coordinates, we must substitute Equations (987), (988), and (995) into the equations of the operators defined in Cartesian coordinates. Below, we will provide some examples of obtaining such equations.

4.6.3. NIDS Gradient in Cylindrical Coordinates

Let us obtain the equation of the NIDS gradient in the cylindrical coordinates $(r,\varphi ,z)$ from the expression of the NIDS gradient in the Cartesian coordinates

$${\widehat{Grad}}_{\left(\alpha \right)}U={\displaystyle \frac{1}{c_{3,x}(D,\mathbf{x})}}{\displaystyle \frac{\partial \widehat{U}}{\partial x}}{\mathbf{e}}_x+{\displaystyle \frac{1}{c_{3,y}(D,\mathbf{x})}}{\displaystyle \frac{\partial \widehat{U}}{\partial y}}{\mathbf{e}}_y+{\displaystyle \frac{1}{c_{3,z}(D,\mathbf{x})}}{\displaystyle \frac{\partial \widehat{U}}{\partial z}}{\mathbf{e}}_z=$$

$${\displaystyle \frac{1}{c_1({\alpha }_x,x)}}{\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_x,x)U}{\partial x}}{\mathbf{e}}_x+{\displaystyle \frac{1}{c_1({\alpha }_y,y)}}{\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_y,y)U}{\partial y}}{\mathbf{e}}_y+{\displaystyle \frac{1}{c_1({\alpha }_z,z)}}{\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_z,z)U}{\partial z}}{\mathbf{e}}_z,$$

(996)

where we use notations

$$\widehat{U}:={\widehat{c}}_3(D,\mathbf{x})U\left(\mathbf{x}\right),$$

(997)

and $\mathbf{x}=(x,y,z)$.

For this, we use the coordinate transformations

$$x=rcos\varphi ,$$

(998)

$$y=rsin\varphi ,$$

(999)

$$z=z.$$

(1000)

Using the equations

$$r=\sqrt{x^2+y^2},$$

(1001)

$$\varphi =arctan\left({\displaystyle \frac{y}{x}}\right),$$

(1002)

we get

$${\displaystyle \frac{\partial r}{\partial x}}={\displaystyle \frac{x}{\sqrt{x^2+y^2}}}=cos\varphi ,$$

(1003)

$${\displaystyle \frac{\partial \varphi }{\partial x}}={\displaystyle \frac{-y}{x^2+y^2}}=-{\displaystyle \frac{1}{r}}sin\varphi ,$$

(1004)

and

$${\displaystyle \frac{\partial r}{\partial y}}={\displaystyle \frac{y}{\sqrt{x^2+y^2}}}=sin\varphi ,$$

(1005)

$${\displaystyle \frac{\partial \varphi }{\partial y}}={\displaystyle \frac{x}{x^2+y^2}}={\displaystyle \frac{1}{r}}cos\varphi .$$

(1006)

Then, using Equations (1003)–(1006), we obtain

$${\displaystyle \frac{\partial }{\partial x}}={\displaystyle \frac{\partial r}{\partial x}}{\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{\partial \varphi }{\partial x}}{\displaystyle \frac{\partial }{\partial \varphi }}+{\displaystyle \frac{\partial z}{\partial x}}{\displaystyle \frac{\partial }{\partial z}}=cos\varphi {\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{sin\varphi }{r}}{\displaystyle \frac{\partial }{\partial \varphi }},$$

(1007)

$${\displaystyle \frac{\partial }{\partial y}}={\displaystyle \frac{\partial r}{\partial y}}{\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{\partial \varphi }{\partial y}}{\displaystyle \frac{\partial }{\partial \varphi }}+{\displaystyle \frac{\partial z}{\partial y}}{\displaystyle \frac{\partial }{\partial z}}=sin\varphi {\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{cos\varphi }{r}}{\displaystyle \frac{\partial }{\partial \varphi }}.$$

(1008)

Substitution of Equations (1007) and (1008) and equations

$${\mathbf{e}}_x={\mathbf{e}}_{r}cos\varphi -{\mathbf{e}}_{\varphi }sin\varphi ,$$

(1009)

$${\mathbf{e}}_y={\mathbf{e}}_{r}sin\varphi +{\mathbf{e}}_{\varphi }cos\varphi ,$$

(1010)

where the vector ${\mathbf{e}}_z$ does not change, into Equation (996), we derive the equation of the NIDS gradient in the cylindrical coordinates for the product measure and metric approaches.

In the product measure approach, we derive the equation

$${\widehat{Grad}}_{\left(\alpha \right)}U\left(\mathbf{x}\right)=$$

$${\mathbf{e}}_r{\displaystyle \frac{cos\varphi }{c_1({\alpha }_x,rcos\varphi )}}\left(cos\varphi {\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_x,rcos\varphi )U}{\partial r}}-{\displaystyle \frac{sin\varphi }{r}}{\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_x,rcos\varphi )U}{\partial \varphi }}\right)+$$

$${\mathbf{e}}_{\varphi }{\displaystyle \frac{sin\varphi }{c_1({\alpha }_x,rcos\varphi )}}\left({\displaystyle \frac{sin\varphi }{r}}{\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_x,rcos\varphi )U}{\partial \varphi }}-cos\varphi {\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_x,rcos\varphi )U}{\partial r}}\right)+$$

$${\mathbf{e}}_r{\displaystyle \frac{sin\varphi }{c_1({\alpha }_y,rsin\varphi )}}\left(sin\varphi {\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_y,rsin\varphi )U}{\partial r}}+{\displaystyle \frac{cos\varphi }{r}}{\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_y,rsin\varphi )U}{\partial \varphi }}\right)+$$

$${\mathbf{e}}_{\varphi }{\displaystyle \frac{cos\varphi }{c_1({\alpha }_y,rsin\varphi )}}\left(sin\varphi {\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_y,rsin\varphi )U}{\partial r}}+{\displaystyle \frac{cos\varphi }{r}}{\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_y,rsin\varphi )U}{\partial \varphi }}\right)+$$

$${\displaystyle \frac{1}{c_1({\alpha }_z,z)}}{\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_z,z)U}{\partial z}}{\mathbf{e}}_z.$$

(1011)

Equation (1011) can be rewritten in the form

$${\widehat{Grad}}_{\left(\alpha \right)}U\left(\mathbf{x}\right)=$$

$${\mathbf{e}}_r{\displaystyle \frac{{cos}^2\varphi }{c_1({\alpha }_x,rcos\varphi )}}{\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_x,rcos\varphi )U}{\partial r}}-{\mathbf{e}}_r{\displaystyle \frac{sin\varphi cos\varphi }{r{c}_1({\alpha }_x,rcos\varphi )}}{\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_x,rcos\varphi )U}{\partial \varphi }}+$$

$${\mathbf{e}}_r{\displaystyle \frac{{sin}^2\varphi }{c_1({\alpha }_y,rsin\varphi )}}{\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_y,rsin\varphi )U}{\partial r}}+{\mathbf{e}}_r{\displaystyle \frac{sin\varphi cos\varphi }{r{c}_1({\alpha }_y,rsin\varphi )}}{\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_y,rsin\varphi )U}{\partial \varphi }}+$$

$${\mathbf{e}}_{\varphi }{\displaystyle \frac{{sin}^2\varphi }{r{c}_1({\alpha }_x,rcos\varphi )}}{\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_x,rcos\varphi )U}{\partial \varphi }}-{\mathbf{e}}_{\varphi }{\displaystyle \frac{sin\varphi cos\varphi }{c_1({\alpha }_x,rcos\varphi )}}{\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_x,rcos\varphi )U}{\partial r}}+$$

$${\mathbf{e}}_{\varphi }{\displaystyle \frac{sin\varphi cos\varphi }{c_1({\alpha }_y,rsin\varphi )}}{\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_y,rsin\varphi )U}{\partial r}}+{\mathbf{e}}_{\varphi }{\displaystyle \frac{{cos}^2\varphi }{r{c}_1({\alpha }_y,rsin\varphi )}}{\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_y,rsin\varphi )U}{\partial \varphi }}+$$

$${\displaystyle \frac{1}{c_1({\alpha }_z,z)}}{\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_z,z)U}{\partial z}}{\mathbf{e}}_z.$$

(1012)

In the metric approach, the NIDS gradient in the cylindrical coordinate is described as

$${Grad}_{\left(\alpha \right)}U=$$

$$\left({\mathbf{e}}_{r}cos\varphi -{\mathbf{e}}_{\varphi }sin\varphi \right)\left({\displaystyle \frac{cos\varphi }{c_1({\alpha }_x,rcos\varphi )}}{\displaystyle \frac{\partial U}{\partial r}}-{\displaystyle \frac{sin\varphi }{r}}{\displaystyle \frac{1}{c_1({\alpha }_x,rcos\varphi )}}{\displaystyle \frac{\partial U}{\partial \varphi }}\right)+$$

$$\left({\mathbf{e}}_{r}sin\varphi +{\mathbf{e}}_{\varphi }cos\varphi \right)\left({\displaystyle \frac{sin\varphi }{c_1({\alpha }_y,rsin\varphi )}}{\displaystyle \frac{\partial U}{\partial r}}+{\displaystyle \frac{cos\varphi }{r}}{\displaystyle \frac{1}{c_1({\alpha }_y,rsin\varphi )}}{\displaystyle \frac{\partial U}{\partial \varphi }}\right)+{\displaystyle \frac{1}{c_1({\alpha }_z,z)}}{\displaystyle \frac{\partial U}{\partial z}}{\mathbf{e}}_z.$$

(1013)

Equation (1013) can be written in the form

$${Grad}_{\left(\alpha \right)}U=$$

$${\mathbf{e}}_r\left({\displaystyle \frac{sin\varphi cos\varphi }{r{c}_1({\alpha }_y,rsin\varphi )}}-{\displaystyle \frac{sin\varphi cos\varphi }{r{c}_1({\alpha }_x,rcos\varphi )}}\right){\displaystyle \frac{\partial U}{\partial \varphi }}+$$

$${\mathbf{e}}_r\left({\displaystyle \frac{{sin}^2\varphi }{c_1({\alpha }_y,rsin\varphi )}}+{\displaystyle \frac{{cos}^2\varphi }{c_1({\alpha }_x,rcos\varphi )}}\right){\displaystyle \frac{\partial U}{\partial r}}+$$

$${\mathbf{e}}_{\varphi }\left({\displaystyle \frac{sin\varphi cos\varphi }{c_1({\alpha }_y,rsin\varphi )}}-{\displaystyle \frac{sin\varphi cos\varphi }{c_1({\alpha }_x,rcos\varphi )}}\right){\displaystyle \frac{\partial U}{\partial r}}+$$

$${\mathbf{e}}_{\varphi }\left({\displaystyle \frac{{sin}^2\varphi }{r{c}_1({\alpha }_x,rcos\varphi )}}+{\displaystyle \frac{{cos}^2\varphi }{r{c}_1({\alpha }_y,rsin\varphi )}}\right){\displaystyle \frac{\partial U}{\partial \varphi }}+{\displaystyle \frac{1}{c_1({\alpha }_z,z)}}{\displaystyle \frac{\partial U}{\partial z}}{\mathbf{e}}_z.$$

(1014)

For ${\alpha }_x={\alpha }_y=1$ and ${\alpha }_z\in (0,1]$, we get

$${\displaystyle \frac{sin\varphi cos\varphi }{c_1({\alpha }_y,rsin\varphi )}}-{\displaystyle \frac{sin\varphi cos\varphi }{c_1({\alpha }_x,rcos\varphi )}}=0,$$

(1015)

$${\displaystyle \frac{{sin}^2\varphi }{c_1({\alpha }_y,rsin\varphi )}}+{\displaystyle \frac{{cos}^2\varphi }{c_1({\alpha }_x,rcos\varphi )}}=1,$$

(1016)

$${\displaystyle \frac{{sin}^2\varphi }{r{c}_1({\alpha }_x,rcos\varphi )}}+{\displaystyle \frac{{cos}^2\varphi }{r{c}_1({\alpha }_y,rsin\varphi )}}={\displaystyle \frac{1}{r}}.$$

(1017)

As a result, Equation (1011) takes the form

$${Grad}_{\left(\alpha \right)}U={\mathbf{e}}_r{\displaystyle \frac{\partial U}{\partial r}}+{\mathbf{e}}_{\varphi }{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial U}{\partial \varphi }}+{\displaystyle \frac{1}{c_1({\alpha }_z,z)}}{\displaystyle \frac{\partial U}{\partial z}}{\mathbf{e}}_z.$$

(1018)

For the case ${\alpha }_x={\alpha }_y={\alpha }_z=1$, we obtain the standard equation for the gradient in cylindrical coordinates in the three-dimensional space.

4.6.4. NIDS Gradient in Spherical Coordinates

Let us obtain the equation of NIDS gradient in the spherical coordinates $(r,\varphi ,\theta )$ from the equation of the NIDS gradient in the Cartesian coordinate (996).

The Cartesian coordinates $(x,y,z)$ can be transformed into the spherical coordinates $(r,\varphi ,\theta )$ by the equations

$$x=rsin\theta cos\varphi ,$$

(1019)

$$y=rsin\theta sin\varphi ,$$

(1020)

$$z=rcos\theta ,$$

(1021)

where $r\in [0,\infty )$, $\varphi \in [0,2\pi )$, $\theta \in [0,\pi ]$.

Using the equation

$$r=\sqrt{x^2+y^2+z^2},$$

(1022)

$$\theta =arctan\left({\displaystyle \frac{\sqrt{x^2+y^2}}{z}}\right),$$

(1023)

$$\varphi =arctan\left({\displaystyle \frac{y}{x}}\right),$$

(1024)

we get

$${\displaystyle \frac{\partial r}{\partial x}}={\displaystyle \frac{x}{\sqrt{x^2+y^2+z^2}}}=sin\theta cos\varphi ,$$

(1025)

$${\displaystyle \frac{\partial r}{\partial y}}={\displaystyle \frac{y}{\sqrt{x^2+y^2+z^2}}}=sin\theta sin\varphi ,$$

(1026)

$${\displaystyle \frac{\partial r}{\partial z}}={\displaystyle \frac{z}{\sqrt{x^2+y^2+z^2}}}=cos\theta ,$$

(1027)

$${\displaystyle \frac{\partial \varphi }{\partial x}}=-{\displaystyle \frac{y}{x^2+y^2}}=-{\displaystyle \frac{sin\varphi }{rsin\theta }},$$

(1028)

$${\displaystyle \frac{\partial \varphi }{\partial y}}={\displaystyle \frac{x}{x^2+y^2}}={\displaystyle \frac{cos\varphi }{rsin\theta }},$$

(1029)

$${\displaystyle \frac{\partial \varphi }{\partial z}}=0,$$

(1030)

$${\displaystyle \frac{\partial \theta }{\partial x}}={\displaystyle \frac{zx}{\sqrt{x^2+y^2}(x^2+y^2+z^2)}}={\displaystyle \frac{1}{r}}cos\theta cos\varphi ,$$

(1031)

$${\displaystyle \frac{\partial \theta }{\partial y}}={\displaystyle \frac{zy}{\sqrt{x^2+y^2}(x^2+y^2+z^2)}}={\displaystyle \frac{1}{r}}cos\theta sin\varphi ,$$

(1032)

$${\displaystyle \frac{\partial \theta }{\partial z}}={\displaystyle \frac{\sqrt{x^2+y^2}}{x^2+y^2+z^2}}=-{\displaystyle \frac{1}{r}}sin\theta .$$

(1033)

Substitution of Equations (1025)–(1033) into the equations

$${\displaystyle \frac{\partial }{\partial x}}={\displaystyle \frac{\partial r}{\partial x}}{\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{\partial \varphi }{\partial x}}{\displaystyle \frac{\partial }{\partial \varphi }}+{\displaystyle \frac{\partial \theta }{\partial x}}{\displaystyle \frac{\partial }{\partial \theta }},$$

(1034)

$${\displaystyle \frac{\partial }{\partial y}}={\displaystyle \frac{\partial r}{\partial y}}{\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{\partial \varphi }{\partial y}}{\displaystyle \frac{\partial }{\partial \varphi }}+{\displaystyle \frac{\partial \theta }{\partial y}}{\displaystyle \frac{\partial }{\partial \theta }},$$

(1035)

$${\displaystyle \frac{\partial }{\partial z}}={\displaystyle \frac{\partial r}{\partial z}}{\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{\partial \varphi }{\partial z}}{\displaystyle \frac{\partial }{\partial \varphi }}+{\displaystyle \frac{\partial \theta }{\partial z}}{\displaystyle \frac{\partial }{\partial \theta }},$$

(1036)

gives the equations

$${\displaystyle \frac{\partial }{\partial x}}=sin\theta cos\varphi {\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{sin\varphi }{rsin\theta }}{\displaystyle \frac{\partial }{\partial \varphi }}+{\displaystyle \frac{1}{r}}cos\theta cos\varphi {\displaystyle \frac{\partial }{\partial \theta }},$$

(1037)

$${\displaystyle \frac{\partial }{\partial y}}=sin\theta sin\varphi {\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{cos\varphi }{rsin\theta }}{\displaystyle \frac{\partial }{\partial \varphi }}+{\displaystyle \frac{1}{r}}cos\theta sin\varphi {\displaystyle \frac{\partial }{\partial \theta }},$$

(1038)

$${\displaystyle \frac{\partial }{\partial z}}=cos\theta {\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{1}{r}}sin\theta {\displaystyle \frac{\partial }{\partial \theta }}.$$

(1039)

Then, substitution of Equations (1037)–(1039) and the equations [160], p. 577, in the form

$${\mathbf{e}}_x={\mathbf{e}}_{r}sin\theta cos\varphi +{\mathbf{e}}_{\theta }cos\theta cos\varphi -{\mathbf{e}}_{\varphi }sin\varphi ,$$

(1040)

$${\mathbf{e}}_y={\mathbf{e}}_{r}sin\theta sin\varphi +{\mathbf{e}}_{\theta }cos\theta sin\varphi +{\mathbf{e}}_{\varphi }cos\varphi ,$$

(1041)

$${\mathbf{e}}_z={\mathbf{e}}_{r}cos\theta -{\mathbf{e}}_{\theta }sin\theta$$

(1042)

into Equation (996), gives the expression of the NIDS gradient in the spherical coordinate for the product measure and metric approaches.

In the product measure approach, the NIDS gradient is described by the equation

$${Grad}_{\left(\alpha \right)}U=$$

$$\left({\mathbf{e}}_{r}sin\theta cos\varphi +{\mathbf{e}}_{\theta }cos\theta cos\varphi -{\mathbf{e}}_{\varphi }sin\varphi \right)·$$

$${\displaystyle \frac{1}{c_1({\alpha }_x,rsin\theta cos\varphi )}}\left(sin\theta cos\varphi {\displaystyle \frac{\partial {\widehat{A}}^{-1}U}{\partial r}}-{\displaystyle \frac{sin\varphi }{rsin\theta }}{\displaystyle \frac{\partial {\widehat{A}}^{-1}U}{\partial \varphi }}+{\displaystyle \frac{cos\theta cos\varphi }{r}}{\displaystyle \frac{\partial {\widehat{A}}^{-1}U}{\partial \theta }}\right)+$$

$$\left({\mathbf{e}}_{r}sin\theta sin\varphi +{\mathbf{e}}_{\theta }cos\theta sin\varphi +{\mathbf{e}}_{\varphi }cos\varphi \right)·$$

$${\displaystyle \frac{1}{c_1({\alpha }_y,rsin\theta sin\varphi )}}\left(sin\theta sin\varphi {\displaystyle \frac{\partial {\widehat{B}}^{-1}U}{\partial r}}+{\displaystyle \frac{cos\varphi }{rsin\theta }}{\displaystyle \frac{\partial {\widehat{B}}^{-1}U}{\partial \varphi }}+{\displaystyle \frac{cos\theta sin\varphi }{r}}{\displaystyle \frac{\partial {\widehat{B}}^{-1}U}{\partial \theta }}\right)+$$

$$\left({\mathbf{e}}_{r}cos\theta -{\mathbf{e}}_{\theta }sin\theta \right)·{\displaystyle \frac{1}{c_1({\alpha }_z,rcos\theta )}}\left(cos\theta {\displaystyle \frac{\partial {\widehat{C}}^{-1}U}{\partial r}}-{\displaystyle \frac{sin\theta }{r}}{\displaystyle \frac{\partial {\widehat{C}}^{-1}U}{\partial \theta }}\right),$$

(1043)

where we use the notations

$$\widehat{A}=\widehat{A}(r,\varphi ,\theta )={\widehat{c}}_1^{-1}({\alpha }_x,rsin\theta cos\varphi ),$$

(1044)

$$\widehat{B}=\widehat{B}(r,\varphi ,\theta )={\widehat{c}}_1^{-1}({\alpha }_y,rsin\theta sin\varphi ),$$

(1045)

$$\widehat{C}=\widehat{C}(r,\varphi ,\theta )={\widehat{c}}_1^{-1}({\alpha }_z,rcos\theta ).$$

(1046)

In the metric approach, we have Equation (1043), in which we must use $\widehat{A}=\widehat{B}=\widehat{C}=1$.

In the particular case $U\left(\mathbf{r}\right)=U\left(r\right)$, Equation (1043) of the metric approach takes the form

$$Gra{d}_{\left(\alpha \right)}U\left(r\right)=$$

$$\left({\mathbf{e}}_{r}sin\theta cos\varphi +{\mathbf{e}}_{\theta }cos\theta cos\varphi -{\mathbf{e}}_{\varphi }sin\varphi \right)·{\displaystyle \frac{sin\theta cos\varphi }{c_1({\alpha }_x,rsin\theta cos\varphi )}}{\displaystyle \frac{\partial U\left(r\right)}{\partial r}}+$$

$$\left({\mathbf{e}}_{r}sin\theta sin\varphi +{\mathbf{e}}_{\theta }cos\theta sin\varphi +{\mathbf{e}}_{\varphi }cos\varphi \right)·{\displaystyle \frac{sin\theta sin\varphi }{c_1({\alpha }_y,rsin\theta sin\varphi )}}{\displaystyle \frac{\partial U\left(r\right)}{\partial r}}+$$

$$\left({\mathbf{e}}_{r}cos\theta -{\mathbf{e}}_{\theta }sin\theta \right)·{\displaystyle \frac{cos\theta }{c_1^{-1}({\alpha }_z,rcos\theta )}}{\displaystyle \frac{\partial U\left(r\right)}{\partial r}}.$$

(1047)

For the case ${\alpha }_x={\alpha }_y={\alpha }_z=1$, we obtain the standard equation for the gradient in spherical coordinates in the three-dimensional space.

4.6.5. NIDS Divergence in Cylindrical Coordinates

Let us obtain the equation of the NIDS divergence in the cylindrical coordinates

$$x=rcos\varphi ,$$

(1048)

$$y=rsin\varphi ,$$

(1049)

$$z=z$$

(1050)

from the expression for the NIDS divergence in the Cartesian coordinates

$${\widehat{Div}}_{\left(\alpha \right)}\mathbf{F}={\displaystyle \frac{1}{c_1({\alpha }_x,x)}}{\displaystyle \frac{\partial {\widehat{F}}_x}{\partial x}}+{\displaystyle \frac{1}{c_1({\alpha }_y,y)}}{\displaystyle \frac{\partial {\widehat{F}}_y}{\partial y}}+{\displaystyle \frac{1}{c_1({\alpha }_z,z)}}{\displaystyle \frac{\partial {\widehat{F}}_z}{\partial z}},$$

(1051)

where we use the notations

$${\widehat{F}}_x:={\widehat{c}}_1({\alpha }_x,x)F_x,$$

(1052)

$${\widehat{F}}_y:={\widehat{c}}_1({\alpha }_y,y)F_y,$$

(1053)

$${\widehat{F}}_z:={\widehat{c}}_1({\alpha }_z,z)F_z.$$

(1054)

For this, we use the following standard Equations (1007) and (1008) and equations

$$F_x=F_{r}cos\varphi -F_{\varphi }sin\varphi ,$$

(1055)

$$F_y=F_{r}sin\varphi +F_{\varphi }cos\varphi ,$$

(1056)

which are derived from the equations

$${\mathbf{e}}_r={\mathbf{e}}_{x}cos\varphi +{\mathbf{e}}_{y}sin\varphi ,$$

(1057)

$${\mathbf{e}}_{\varphi }=-{\mathbf{e}}_{x}sin\varphi +{\mathbf{e}}_{y}cos\varphi ,$$

(1058)

and the condition

$$\mathbf{F}=F_x{\mathbf{e}}_x+F_y{\mathbf{e}}_y+F_z{\mathbf{e}}_z=F_r{\mathbf{e}}_r+F_{\varphi }{\mathbf{e}}_{\varphi }+F_z{\mathbf{e}}_z.$$

(1059)

Substitution of Equations (1007), (1008), (1055), and (1056) into Equation (1051), we get the following equations.

For the product measure approach, in the general case, we have

$${\widehat{Div}}_{\left(\alpha \right)}\mathbf{F}=A(r,\varphi )\left(cos\varphi {\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{sin\varphi }{r}}{\displaystyle \frac{\partial }{\partial \varphi }}\right)\left({\widehat{A}}^{-1}{F}_{r}cos\varphi -{\widehat{A}}^{-1}{F}_{\varphi }sin\varphi \right)+$$

$$B(r,\varphi )\left(cos\varphi {\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{sin\varphi }{r}}{\displaystyle \frac{\partial }{\partial \varphi }}\right)\left({\widehat{B}}^{-1}{F}_{r}sin\varphi +{\widehat{B}}^{-1}{F}_{\varphi }cos\varphi \right)+{\displaystyle \frac{1}{c_1({\alpha }_z,z)}}{\displaystyle \frac{\partial {\widehat{F}}_z}{\partial z}},$$

(1060)

where we use the notations

$$A(r,\varphi ):=c_1^{-1}({\alpha }_x,x(r,\varphi ))=c^{-1}\left({\alpha }_x\right)r^{1-{\alpha }_x}{|cos\varphi |}^{1-{\alpha }_x}$$

(1061)

$$B(r,\varphi ):=c_1^{-1}({\alpha }_y,y(r,\varphi ))=c^{-1}\left({\alpha }_x\right)r^{1-{\alpha }_y}{|sin\varphi |}^{1-{\alpha }_y}.$$

(1062)

For the product measure approach, in the general case, Equation (1060) can be written as

$${\widehat{Div}}_{\left(\alpha \right)}\mathbf{F}=$$

$$\left(A(r,\varphi ){\displaystyle \frac{\partial {\widehat{A}}^{-1}(r,\varphi )F_r}{\partial r}}{cos}^2\varphi +B(r,\varphi ){\displaystyle \frac{\partial {\widehat{B}}^{-1}(r,\varphi )F_r}{\partial r}}{sin}^2\varphi \right)+$$

$${\displaystyle \frac{1}{r}}\left(B(r,\varphi ){\displaystyle \frac{\partial {\widehat{B}}^{-1}(r,\varphi )F_r}{\partial \varphi }}-A(r,\varphi ){\displaystyle \frac{\partial {\widehat{A}}^{-1}(r,\varphi )F_r}{\partial \varphi }}\right)sin\varphi cos\varphi +$$

$$\left(A(r,\varphi ){\displaystyle \frac{\partial {\widehat{A}}^{-1}(r,\varphi )F_{\varphi }}{\partial \varphi }}{sin}^2\varphi +B(r,\varphi ){\displaystyle \frac{\partial {\widehat{B}}^{-1}(r,\varphi )F_{\varphi }}{\partial \varphi }}{cos}^2\varphi \right)+$$

$${\displaystyle \frac{1}{r}}\left(A(r,\varphi ){\widehat{A}}^{-1}(r,\varphi )F_r-B(r,\varphi ){\widehat{B}}^{-1}(r,\varphi )F_r\right)sin\varphi cos\varphi +$$

$${\displaystyle \frac{1}{r}}\left(A(r,\varphi ){\widehat{A}}^{-1}(r,\varphi )F_r{sin}^2\varphi +B(r,\varphi ){\widehat{B}}^{-1}(r,\varphi )F_r{cos}^2\varphi \right)+{\displaystyle \frac{1}{c_1({\alpha }_z,z)}}{\displaystyle \frac{\partial {\widehat{F}}_z}{\partial z}}.$$

(1063)

For the metric approsch, we have $\widehat{A}=\widehat{B}=1$, and

$${Div}_{\left(\alpha \right)}\mathbf{F}=$$

$$\left(A(r,\varphi ){cos}^2\varphi +B(r,\varphi ){sin}^2\varphi \right){\displaystyle \frac{\partial F_r}{\partial r}}+$$

$$\left(B(r,\varphi )-A(r,\varphi )\right){\displaystyle \frac{sin\varphi cos\varphi }{r}}{\displaystyle \frac{\partial F_r}{\partial \varphi }}+$$

$$\left(A(r,\varphi ){sin}^2\varphi +B(r,\varphi ){cos}^2\varphi \right){\displaystyle \frac{\partial F_{\varphi }}{\partial \varphi }}+$$

$$\left(A(r,\varphi )-B(r,\varphi )\right){\displaystyle \frac{sin\varphi cos\varphi }{r}}{F}_r+$$

$$\left(A(r,\varphi ){sin}^2\varphi +B(r,\varphi ){cos}^2\varphi \right){\displaystyle \frac{1}{r}}{F}_r+{\displaystyle \frac{1}{c_1({\alpha }_z,z)}}{\displaystyle \frac{\partial {\widehat{F}}_z}{\partial z}}.$$

(1064)

Equation (1064) can be represented in the form

$${Div}_{\left(\alpha \right)}\mathbf{F}=$$

$$\left(A(r,\varphi ){cos}^2\varphi +B(r,\varphi ){sin}^2\varphi \right){\displaystyle \frac{\partial F_r}{\partial r}}+$$

$$\left(B(r,\varphi )-A(r,\varphi )\right){\displaystyle \frac{sin\varphi cos\varphi }{r}}\left({\displaystyle \frac{\partial F_{\varphi }}{\partial r}}+{\displaystyle \frac{\partial F_r}{\partial \varphi }}-F_r\right)+$$

$$\left(A(r,\varphi ){sin}^2\varphi +B(r,\varphi ){cos}^2\varphi \right){\displaystyle \frac{1}{r}}\left({\displaystyle \frac{\partial F_{\varphi }}{\partial \varphi }}+F_r\right)+{\displaystyle \frac{1}{c_1({\alpha }_z,z)}}{\displaystyle \frac{\partial {\widehat{F}}_z}{\partial z}}.$$

(1065)

For the case ${\alpha }_x={\alpha }_y=1$ and ${\alpha }_z\in (0,1]$, we get

$$B(r,\varphi )-A(r,\varphi )=0,$$

(1066)

$$A(r,\varphi ){sin}^2\varphi +B(r,\varphi ){cos}^2\varphi =1,$$

(1067)

and Equation (1065) gives

$${\widehat{Div}}_{\left(\alpha \right)}\mathbf{F}={\displaystyle \frac{\partial F_r}{\partial r}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial F_{\varphi }}{\partial \varphi }}+{\displaystyle \frac{1}{r}}{F}_r+{\displaystyle \frac{1}{c_1({\alpha }_z,z)}}{\displaystyle \frac{\partial {\widehat{F}}_z}{\partial z}}.$$

(1068)

For the case ${\alpha }_x={\alpha }_y={\alpha }_z=1$, Equation (1065) gives the classical expression for integer-dimensional space

$$div\mathbf{F}={\displaystyle \frac{\partial F_r}{\partial r}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial F_{\varphi }}{\partial \varphi }}+{\displaystyle \frac{1}{r}}{F}_r+{\displaystyle \frac{\partial F_z}{\partial z}}.$$

(1069)

The use of Equations (1007), (1008), (1055), and (1056) allows one to obtain the NIDS divergence in the cylindrical coordinates.

4.7. NIDS Derivative with Respect to the NIDS Volume

In this subsection, we proposd the NIDS generalization of the derivative of the multiple integral with respect to the volume of its domain of integration [141,142].

4.7.1. Definition of the NIDS Derivative with Respect to the NIDS Volume

Let W be a D-dimensional space $W=W_1\times \dots \times W_n$ with $D={\alpha }_1+\dots +{\alpha }_n$, where all ${\alpha }_k\in (0,1]$. The volume NIDS integral is the multiple NIDS integral

$${\Phi }_n(D,\Omega ):=I_{\Omega }^{D}f\left(\mathbf{x}\right)={\int }_{\Omega }f\left(\mathbf{x}\right)d{\mu }_n(D,\mathbf{x})={\int }_{\Omega }f\left(\mathbf{x}\right)c_n(D,\mathbf{x})d{x}^1\dots d{x}^n,$$

(1070)

where $\mathbf{x}=(x^1,\dots ,x^n)$, and $\Omega \subset W=W_1\times \dots \times W_n$, $D\in (0,n]$. In the NIDS integral (1070), the function $f\left(\mathbf{x}\right)$ is regarded as being fixed, and the domain of integration $\Omega$ is considered as variable. Then, the NIDS integral (1070) becomes a function ${\Phi }_n(D,\Omega )$ of the domain $\Omega$. Using the additivity property of the NIDS integration, the volume NIDS integral and the function (1070) are additive

$${\Phi }_n(D,{\Omega }_1\bigcup {\Omega }_2):={\Phi }_n(D,{\Omega }_1)+{\Phi }_n(D,{\Omega }_2),\mathit{if}{\Omega }_1\bigcap {\Omega }_2=\varnothing .$$

(1071)

As a class of sets, for which the function ${\Phi }_n(D,\Omega )$ is defined, we can take the totality of all squarable figures contained in the domain $\Omega$, on which $f\left(\mathbf{x}\right)$ is defined.

Note that the set function ${\Phi }_n(D,\Omega )$ can be both positive and negative values, in general.

Definition 35. 

Let W be a D-dimensional space $W=W_1\times \dots \times W_n$ with $D={\alpha }_1+\dots +{\alpha }_n$, where all ${\alpha }_k\in (0,1]$. Let ${\Phi }_n(D,\Omega )$ be an additive set function defined for all the squarable domains Ω, and let $A_n(D,\Omega )$ be the ratio

$$A_n(D,\Omega ):={\displaystyle \frac{{\Phi }_n(D,\Omega )}{V_D(\Omega )}},$$

(1072)

where $V_D(\Omega )$ is the D-dimensional volume of the region $\Omega \subset W$.

A number A is called the limit of the ratio $A_n(D,\Omega )$ at the domain Ω contracted to a point ${\mathbf{x}}_0$, if for every $\epsilon >0$, there is $\delta >0$, such that

$$\left|{\displaystyle \frac{{\Phi }_n(D,\Omega )}{V_D(\Omega )}}-A\right|<\epsilon$$

(1073)

for each domain Ω entirely lying in the ball

$${\Omega }_{B,\delta }:=\{\mathbf{x}:|\mathbf{x}-{\mathbf{x}}_0|⩽\delta ,\delta >0\}$$

(1074)

with the center at the point ${\mathbf{x}}_0$ and the radius $\delta >0$. This limit will be denoted by the symbols

$$\underset{\Omega \to {\mathbf{x}}_0}{lim}{\displaystyle \frac{{\Phi }_n(D,\Omega )}{V_D(\Omega )}}={\displaystyle \frac{d}{d{V}_D(\Omega )}}{\Phi }_n(D,\Omega )$$

(1075)

and will be called the NIDS derivative of the additive set function ${\Phi }_n(D,\Omega )$ with respect to the D-dimensional volume $V_D(\Omega )$ or, briefly, the volume NIDS derivative.

The volume NIDS derivative (1075) is not a set function, but it is an ordinary point function, i.e., a variable quantity dependent on a point ${\mathbf{x}}_0$.

Remark 74. 

It should be emphasized that the volume NIDS integrals in the metric approach and the product measure approach coincide:

$$I_{\Omega }^{D}f\left(\mathbf{x}\right)=I_{\Omega }^{D,\ast }f\left(\mathbf{x}\right).$$

(1076)

The NIDS volumes in these approaches also coincide:

$$V_D(\Omega )=V_D^{\ast }(\Omega ).$$

(1077)

As a result, the volume NIDS derivative in the metric and product measure approaches coincides.

4.7.2. Fundamental Theorem for Volume NIDS Derivative

Let us consider the volume NIDS integral (1070) with the fixed function $f\left(\mathbf{x}\right)$ that is supposed to be continuous in the chosen part of the region $\Omega$. Let us show that the additive set function defined by relation (1070) has the volume NIDS derivative that coincides with the integrand $f\left(\mathbf{x}\right)$.

Theorem 36 

(Fundamental Theorem for Volume NIDS Derivative). Let Ω be a region in the D-dimensional space $W=W_1\times \dots \times W_n$. Let ${\Phi }_n(D,\Omega )$ be defined by Equation (1070), where the function $f\left(\mathbf{x}\right)\in C(\Omega )$ is regarded as being fixed, and the domain of integration Ω is considered as variable.

Then, for the additive set function ${\Phi }_n(D,\Omega )$, the equation

$${\displaystyle \frac{d}{d{V}_D(\Omega )}}{\Phi }_n(D,\Omega )={\displaystyle \frac{d}{d{V}_D(\Omega )}}{I}_{\Omega }^{D}f\left(\mathbf{x}\right)=f\left(\mathbf{x}\right)$$

(1078)

is satisfied for squarable figures contained in the domain Ω, on which $f\left(\mathbf{x}\right)$ is defined.

Proof. 

Let ${\mathbf{x}}_0$ be a fixed point and $\Omega$ be a domain lying within the D-dimensional ball (1074) with center at ${\mathbf{x}}_0$. Denote by m and M the greatest lower bound and the least upper bound of the values of the function $f\left(\mathbf{x}\right)$ in the domain $\Omega$. Using the mean value property of the volume NIDS integral

$$V_D(\Omega )m⩽I_{\Omega }^{D}f\left(\mathbf{x}\right)⩽V_D(\Omega )M,$$

(1079)

where $f\left(\mathbf{x}\right)\in C(\Omega )$, we get

$$m⩽{\displaystyle \frac{1}{V_D(\Omega )}}{I}_{\Omega }^{D}f\left(\mathbf{x}\right)⩽M.$$

(1080)

If the domain $\Omega$ is contracted to the point ${\mathbf{x}}_0$, i.e., when the radius of the circle tends to zero, the numbers m and M tend to the same value. This value is equal to the function $f\left(\mathbf{x}\right)$ at the point ${\mathbf{x}}_0$, because of the continuity of $f\left(\mathbf{x}\right)$ at the point ${\mathbf{x}}_0$. Consequently, the ratio whose values lie between m and M tends to the same limit. Hence, we have that Equation (1078) is satisfied. □

Remark 75. 

The fundamental theorem for the derivative with respect to the NIDS volume (Theorem 36) can be proved in another way. To prove Equation (1078), we can also use Equation (1070) that gives

$$I_{\Omega }^{D}f\left(\mathbf{x}\right)={\int }_{\Omega }f\left(\mathbf{x}\right)c_n(D,\mathbf{x})d{x}^1\dots d{x}^n={\int }_{\Omega }{f}_C\left(\mathbf{x}\right)d{x}^1\dots d{x}^n,$$

(1081)

where

$$f_C\left(\mathbf{x}\right):=f\left(\mathbf{x}\right)c_n(D,\mathbf{x}).$$

(1082)

Using the fundamental theorem for the standard derivative with respect to n-dimensional volume [141,142] in the form

$${\displaystyle \frac{d}{d{V}_n(\Omega )}}{\int }_{\Omega }{f}_C\left(\mathbf{x}\right)d{x}^1\dots d{x}^n=f_C\left(\mathbf{x}\right),$$

(1083)

where $d/d{V}_n(\Omega )$ is the standard derivative with respect to the n-dimensional volume

$${\displaystyle \frac{d}{d{V}_n(\Omega )}}{\Phi }_n(n,\Omega )=\underset{\Omega \to {\mathbf{x}}_0}{lim}{\displaystyle \frac{{\Phi }_n(n,\Omega )}{V_n(\Omega )}},$$

(1084)

which is used in the standard calculus [141,142], we get

$${\displaystyle \frac{d}{d{V}_n(\Omega )}}{I}_{\Omega }^{D}f\left(\mathbf{x}\right)=f\left(\mathbf{x}\right)c_n(D,\mathbf{x}).$$

(1085)

Equation (1083) gives the equality

$${\displaystyle \frac{1}{c_n(D,\mathbf{x})}}{\displaystyle \frac{d}{d{V}_n(\Omega )}}{I}_{\Omega }^{D}f\left(\mathbf{x}\right)=f\left(\mathbf{x}\right),$$

(1086)

which coincides with Equation (1078), if

$${\displaystyle \frac{d}{d{V}_D(\Omega )}}={\displaystyle \frac{1}{c_n(D,\mathbf{x})}}{\displaystyle \frac{d}{d{V}_n(\Omega )}}.$$

(1087)

As a result, since Theorem 36 has already been proved, then we prove that

$${\displaystyle \frac{d}{d{V}_D(\Omega )}}{\Phi }_n(D,\Omega )={\displaystyle \frac{1}{c_n(D,\mathbf{x})}}{\displaystyle \frac{d}{d{V}_n(\Omega )}}{\Phi }_n(D,\Omega ),$$

(1088)

where $d/d{V}_n(\Omega )$ is the standard volume derivative (1084) with respect to the n-dimensional volume of the standard calculus [141,142].

Equation (1078) can be interpreted as the first fundamental theorem for the volume NIDS integral.

4.7.3. NIDS Divergence, Gradient, and Curl Operator via the Volume NIDS Derivative

It is known that, using the derivative with respect to the volume of the additive set functions and fundamental theorems of vector calculus, one can define the divergences, curl operators, and the gradient [155,156,161].

Using Equation (1078) and the NIDS Gauss–Ostrogradsky theorem, the NIDS Stoke theorem, and the NIDS Gradient Theorem, we can define the NIDS divergences, NIDS curl operators, and NIDS gradient in the product measure and metric approaches. This possibility is also based on the fact that the NIDS integrals of vector fields were defined through the NIDS integrals of scalar fields.

For every domain $\Omega \subset W=W_1\times \dots \times W_n$ bounded by a smooth or piecewise smooth surface $\Sigma =\partial \Omega$, and lying in the part of space, where the vector field $\mathbf{F}\left(\mathbf{x}\right)$ is defined, we can associate the set function

$${\Psi }_n(D,\Omega )=\left({\mathbf{I}}_{\Sigma }^D,\mathbf{F}\right)=I_{\Sigma }^D{F}_n\left(\mathbf{x}\right)=I_{\Sigma }^D\left(\mathbf{F},\mathbf{n}\right)\left(\mathbf{x}\right),$$

(1089)

where the function $F_n\left(\mathbf{x}\right)=\left(\mathbf{F},\mathbf{n}\right)\left(\mathbf{x}\right)$ is regarded as being fixed, and the domain of integration $\Omega$ is considered as variable. Note that the quantity $\left({\mathbf{I}}_{\Sigma }^D,\mathbf{F}\right)$ describes the NIDS flux of the vector $\mathbf{F}$ across the outer side of the surface $\Sigma$. It can be easily verified that this set function ${\Psi }_n(D,\Omega )$ is additive.

Remark 76. 

Let Ω be a region bounded by a smooth or piecewise smooth surface $\Sigma =\partial \Omega$ in the part of the D-dimensional space $W=W_1\times \dots \times W_n$.

(1) In the product measure approach, let ${\Psi }_n(D,\Omega )$ be defined by equation

$${\Psi }_n(D,\Omega )=\left({\mathbf{I}}_{\Sigma }^D,\mathbf{F}\right)=I_{\Sigma }^D\left(\mathbf{F},\mathbf{n}\right)\left(\mathbf{x}\right),$$

(1090)

where the function $F_n\left(\mathbf{x}\right)=\left(\mathbf{F},\mathbf{n}\right)\left(\mathbf{x}\right)\in C(\Omega )$ is regarded as being fixed, and the domain of integration Ω is considered variable.

Then, the volume NIDS derivative of the set function ${\Psi }_n(D,\Omega )$ with respect to the NIDS volumes

$${\widehat{Div}}_{\left(\alpha \right)}\mathbf{F}:={\displaystyle \frac{d}{d{V}_D(\Omega )}}{\Psi }_n(D,\Omega )=\underset{\Omega \to {\mathbf{x}}_0}{lim}{\displaystyle \frac{{\Psi }_n(D,\Omega )}{V_D(\Omega )}}$$

(1091)

is the NIDS divergence of the vector field $\mathbf{F}$.

(2) In the metric approach, let ${\Psi }_n^{\ast }(D,\Omega )$ be defined by equation

$${\Psi }_n(D,\Omega )=\left({\mathbf{I}}_{\Sigma }^{D,\ast },\mathbf{F}\right)=I_{\Sigma }^{D,\ast }\left(\mathbf{F},\mathbf{n}\right)\left(\mathbf{x}\right),$$

(1092)

where the function $F_n\left(\mathbf{x}\right)=\left(\mathbf{F},\mathbf{n}\right)\left(\mathbf{x}\right)\in C(\Omega )$ is regarded as being fixed, and the domain of integration Ω is considered as variable.

Then, the divergence of the vector field $\mathbf{F}$, defined by the equation

$${Div}_{\left(\alpha \right)}\mathbf{F}:={\displaystyle \frac{d}{d{V}_D(\Omega )}}{\Psi }_n^{\ast }(D,\Omega )=\underset{\Omega \to {\mathbf{x}}_0}{lim}{\displaystyle \frac{{\Psi }_n^{\ast }(D,\Omega )}{V_D(\Omega )}},$$

(1093)

is the NIDS divergence of the vector field $\mathbf{F}$.

Let us present and prove that Equations (1091) and (1093) of Remark 76 of the NIDS divergence coincides with the equations of Definition 33.

Theorem 37. 

Let $\mathbf{F}$ be a vector field, defined in a domain Ω, such that the functions $F_k\left(\mathbf{x}\right)$ ($k=1,\dots ,n$) are continuous, and the first-order partial NIDS derivatives ${\widehat{D}}_{x^j}^{{\alpha }_j}{F}_k\left(\mathbf{x}\right)$ are also continuous in Ω.

Then, in the product measure approach, the divergence ${\widehat{Div}}_{\left(\alpha \right)}\mathbf{F}$ exists at all the points of the domain Ω, the equation

$${\displaystyle \frac{d}{d{V}_D(\Omega )}}{I}_{\Sigma }^D\left(\mathbf{F},\mathbf{n}\right)\left(\mathbf{x}\right)={\widehat{Div}}_{\left(\alpha \right)}\mathbf{F}$$

(1094)

holds for all $\mathbf{x}\in \Omega$, and

$${\widehat{Div}}_{\left(\alpha \right)}\mathbf{F}=\sum _{j=1}^n{\widehat{D}}_{x^j}^{{\alpha }_j}{F}_j\left(\mathbf{x}\right),$$

(1095)

where

$${\widehat{D}}_{x^j}^{{\alpha }_j}f\left(\mathbf{x}\right):={\displaystyle \frac{1}{c_1({\alpha }_j,x^j)}}{\displaystyle \frac{\partial }{\partial x^j}}\left({\widehat{c}}_1({\alpha }_j,x^j)f\left(\mathbf{x}\right)\right)$$

(1096)

is the NIDS derivative with respect to $x^j$.

Proof. 

Let us consider Equation (844) of the Gauss–Ostrogradsky theorem for NIDS in the form

$$\left({\mathbf{I}}_{\Sigma }^D,\mathbf{F}\right)=I_{\Omega }^D\left(\sum _{j=1}^n{\widehat{D}}_{x^j}^{{\alpha }_j}{F}_j\left(\mathbf{x}\right)\right),$$

(1097)

where $\Sigma =\partial \Omega$.

The action of the volume NIDS derivative (1075) with respect to the NIDS volume

$${\displaystyle \frac{d}{d{V}_D(\Omega )}}={\displaystyle \frac{1}{c_n(D,\mathbf{x})}}{\displaystyle \frac{d}{d{V}_n(\Omega )}}$$

(1098)

on Equation (1097) and Remark 76 give

$${\displaystyle \frac{d}{d{V}_D(\Omega )}}{\Psi }_n(D,\Omega )={\displaystyle \frac{d}{d{V}_D(\Omega )}}{I}_{\Sigma }^D\left(\mathbf{F},\mathbf{n}\right)\left(\mathbf{x}\right)=$$

$${\displaystyle \frac{d}{d{V}_D(\Omega )}}\left({\mathbf{I}}_{\Sigma }^D,\mathbf{F}\right)={\displaystyle \frac{d}{d{V}_D(\Omega )}}{I}_{\Omega }^D\left(\sum _{j=1}^n{\widehat{D}}_{x^j}^{{\alpha }_j}{F}_j\left(\mathbf{x}\right))\right).$$

(1099)

Using the fundamental Theorem 36, the volume NIDS derivative of the right-hand side of the Equation (1099) exists, and it is equal to

$${\displaystyle \frac{d}{d{V}_D(\Omega )}}{I}_{\Omega }^D\left(\sum _{j=1}^n{\widehat{D}}_{x^j}^{{\alpha }_j}{F}_j\left(\mathbf{x}\right)\right)=\sum _{j=1}^n{\widehat{D}}_{x^j}^{{\alpha }_j}{F}_j\left(\mathbf{x}\right).$$

(1100)

As a result, the volume NIDS derivative with respect to the NIDS volume of the left-hand side also exists and is equal to the same expression. This proves Theorem 37. □

For the metric approach, a similar theorem is formulated and proved similarly.

Similar to standard vector calculus [155,156,161], one can define the NIDS generalizations of the curl operator and gradient by using the volume NIDS derivative and additive set functions. This fact is formulated by the following theorem.

Theorem 38. 

Let Ω be a region bounded by a smooth or piecewise smooth surface $\Sigma =\partial \Omega$ in the part of the D-dimensional space $W=W_1\times \dots \times W_n$.

(1) In the product measure approach, if $W=W_1\times W_2\times W_3$, the NIDS curl operator of the vector field $\mathbf{F}\left(\mathbf{x}\right)$ can be defined and derived via the volume NIDS derivative of the vector set function with respect to NIDS volume by the equation

$${\widehat{Curl}}_{\left(\alpha \right)}\mathbf{F}={\displaystyle \frac{d}{d{V}_D(\Omega )}}{I}_{\Sigma }^D\left[\mathbf{F}\times \mathbf{n}\right]\left(\mathbf{x}\right)=\underset{\Omega \to {\mathbf{x}}_0}{lim}{\displaystyle \frac{I_{\Sigma }^D\left[\mathbf{F}\times \mathbf{n}\right]\left(\mathbf{x}\right)}{V_D(\Omega )}},$$

(1101)

for all $\mathbf{x}\in \Omega$.

The NIDS gradient of the scalar field $U\left(\mathbf{x}\right)$ can be defined and derived via the volume NIDS derivative of the vector set function with respect to NIDS volume by the equation

$${\widehat{Grad}}_{\left(\alpha \right)}U={\displaystyle \frac{d}{d{V}_D(\Omega )}}{I}_{\Sigma }^{D}U\left(\mathbf{x}\right)\mathbf{n}\left(\mathbf{x}\right)=\underset{\Omega \to {\mathbf{x}}_0}{lim}{\displaystyle \frac{I_{\Sigma }^{D}U\left(\mathbf{x}\right)\mathbf{n}\left(\mathbf{x}\right)}{V_D(\Omega )}},$$

(1102)

for all $\mathbf{x}\in \Omega$.

(2) In the metric approach, if $W=W_1\times W_2\times W_3$, the NIDS curl operator of the vector field $\mathbf{F}\left(\mathbf{x}\right)$ can be defined and derived via the volume NIDS derivative of the vector set function with respect to the NIDS volume by the equation

$${Curl}_{\left(\alpha \right)}\mathbf{F}={\displaystyle \frac{d}{d{V}_D(\Omega )}}{I}_{\Sigma }^{D,\ast }\left[\mathbf{F}\times \mathbf{n}\right]\left(\mathbf{x}\right)=\underset{\Omega \to {\mathbf{x}}_0}{lim}{\displaystyle \frac{I_{\Sigma }^{D,\ast }\left[\mathbf{F}\times \mathbf{n}\right]\left(\mathbf{x}\right)}{V_D(\Omega )}},$$

(1103)

for all $\mathbf{x}\in \Omega$.

The NIDS gradient of the scalar field $U\left(\mathbf{x}\right)$ can be defined and derived via the volume NIDS derivative of the vector set function with respect to the NIDS volume by the equation

$${Grad}_{\left(\alpha \right)}U={\displaystyle \frac{d}{d{V}_D(\Omega )}}{I}_{\Sigma }^{D,\ast }U\left(\mathbf{x}\right)\mathbf{n}\left(\mathbf{x}\right)=\underset{\Omega \to {\mathbf{x}}_0}{lim}{\displaystyle \frac{I_{\Sigma }^{D,\ast }U\left(\mathbf{x}\right)\mathbf{n}\left(\mathbf{x}\right)}{V_D(\Omega )}},$$

(1104)

for all $\mathbf{x}\in \Omega$.

Proof. 

The statements of Theorem 38 are proved similarly to Theorem 37 by using the fundamental theorem for volume NIDS derivative (Theorem 36).

Let us prove only Equation (1102). To do this, we use Equation (6) from [161], p. 149, (see also, [155,156]) in the form

$${\oint }_{\Sigma }\varphi \left(\mathbf{x}\right)\mathbf{n}\left(\mathbf{x}\right)d{S}_{n-1}={\int }_{\Omega }grad\varphi \left(\mathbf{x}\right)d{V}_n,$$

(1105)

which was proved in Section 14 of [161] for standard vector calculus in integer-dimensional space. For $n_1\left(\mathbf{x}\right)$, Equation (1105) has the form

$${\oint }_{\Sigma }\varphi \left(\mathbf{x}\right)n_1\left(\mathbf{x}\right)d{S}_{n-1}={\int }_{\Omega }{\displaystyle \frac{\partial \varphi \left(\mathbf{x}\right)}{\partial x^1}}d{V}_n,$$

(1106)

where $\Sigma =\partial \Omega$. Using the function

$${\widehat{U}}_{n,(k2,\dots ,kn)}\left(\mathbf{x}\left(\mathbf{u}\right)\right):={\widehat{c}}_{n,(k2\dots ,kn)}(D,\mathbf{x}\left(\mathbf{u}\right))U\left(\mathbf{x}\left(\mathbf{u}\right)\right),$$

(1107)

where $u^1=x^1,\dots ,u^{n-1}=x^{n-1}$, $u^n=x^n(x^1,\dots ,x^{n-1})$, and Equation (1106), we get

$$I_{\Sigma }^{D}U\left(\mathbf{x}\right)n_1\left(\mathbf{x}\right)={\int }_{\Sigma }U\left(\mathbf{x}\left(\mathbf{u}\right)\right)n_1\left(\mathbf{x}\left(\mathbf{u}\right)\right){\widehat{c}}_{n,(k2\dots ,kn)}(D,\mathbf{x}\left(\mathbf{u}\right))d{x}^1\dots d{x}^{n-1}=$$

$${\int }_{\Sigma }{\widehat{U}}_{n,(k2,\dots ,kn)}\left(\mathbf{x}\left(\mathbf{u}\right)\right)n_1\left(\mathbf{x}\left(\mathbf{u}\right)\right)d{x}^1\dots d{x}^{n-1}=$$

$${\oint }_{\Sigma }\varphi \left(\mathbf{x}\right)n_1\left(\mathbf{x}\right)d{S}_{n-1}={\int }_{\Omega }{\displaystyle \frac{\partial \varphi \left(\mathbf{x}\right)}{\partial x^1}}d{V}_n=$$

$${\int }_{\Omega }{\displaystyle \frac{1}{c_n(D,\mathbf{x})}}{\displaystyle \frac{\partial \varphi \left(\mathbf{x}\right)}{\partial x^1}}{c}_n(D,\mathbf{x})d{\mu }_n(n,\mathbf{x})=$$

$${\int }_{\Omega }{\displaystyle \frac{1}{c_n(D,\mathbf{x})}}{\displaystyle \frac{\partial {\widehat{U}}_{n,(k2,\dots ,kn)}\left(\mathbf{x}\right)}{\partial x^1}}d{\mu }_n(D,\mathbf{x})=$$

$$I_{\Omega }^D\left({\displaystyle \frac{1}{c_n(D,\mathbf{x})}}{\displaystyle \frac{\partial {\widehat{U}}_{n,(k2,\dots ,kn)}\left(\mathbf{x}\right)}{\partial x^1}}\right)=$$

$$I_{\Omega }^D\left({\displaystyle \frac{1}{c_1({\alpha }_1,x^1)}}{\displaystyle \frac{\partial {\widehat{c}}_1({\alpha }_1,x^1)U\left(\mathbf{x}\right)}{\partial x^1}}\right)=I_{\Omega }^D\left({\widehat{D}}_{x^1}^{{\alpha }_1}U\right)\left(\mathbf{x}\right),$$

where we assume that $\Sigma$ is described by the equations

$$u^j=x^j,(j=2,\dots ,n),u^n=x^n(x^2,\dots ,x^{n-1}),$$

(1108)

$$d{S}_{n-1}:=d{\mu }_{n-1}(n-1,\mathbf{u})=\prod _{k=1}^{n-1}d{x}^k,d{V}_n:=d{\mu }_n(n,\mathbf{x})=\prod _{k=1}^{n}d{x}^k,$$

(1109)

and we use the notation

$$\varphi \left(\mathbf{x}\right)={\widehat{U}}_{n,(k2,\dots ,kn)}\left(\mathbf{x}\left(\mathbf{u}\right)\right).$$

(1110)

Then, we repeat this for all other components of the normal vector $\mathbf{n}$ and get

$$I_{\Sigma }^{D}U\left(\mathbf{x}\right)n_j\left(\mathbf{x}\right)=I_{\Omega }^D\left({\widehat{D}}_{x^j}^{{\alpha }_j}U\right)\left(\mathbf{x}\right),$$

(1111)

for all $j=1,\dots ,n$.

As a result, we proved

$$I_{\Sigma }^{D}U\left(\mathbf{x}\right)\mathbf{n}\left(\mathbf{x}\right)=I_{\Omega }^D\left({\widehat{Grad}}_{\left(\alpha \right)}U\left(\mathbf{x}\right)\right),$$

(1112)

where $\Sigma =\partial \Omega$. Then, using Fundamental Theorem 36, we get

$${\displaystyle \frac{d}{d{V}_D(\Omega )}}{I}_{\Sigma }^{D}U\left(\mathbf{x}\right)\mathbf{n}\left(\mathbf{x}\right)={\displaystyle \frac{d}{d{V}_D(\Omega )}}{I}_{\Omega }^D\left({\widehat{Grad}}_{\left(\alpha \right)}U\left(\mathbf{x}\right)\right)={\widehat{Grad}}_{\left(\alpha \right)}U\left(\mathbf{x}\right).$$

(1113)

This ends the proof of Equation (1102).

Equations (1101), (1103), and (1104) are proved similarly. □

Note that Equations (1101)–(1104) of Theorem 38 can be used as definitions of the NIDS gradients and the NIDS curl operators.

Remark 77. 

As a result, Theorems 37 and 38 prove that the differential NIDS vector operators (the NIDS gradient, the NIDS divergence, the NIDS curl operator) can be derived by using the NIDS derivative of the additive set functions with respect to the D-dimensional volume $V_D(\Omega )$.

Therefore, requirement 2b of Principle 9 about the self-consistency of the NIDS vector calculus is satisfied.

4.8. Differential Forms in D-Dimensional Space

Differential forms give a unified approach to defining the line, surface, and volume integration in the n-dimensional spaces with any integer dimension [162]. In this approach, the integrands are considered as some independent objects that can be studied, transformed, and integrated. In this subsection, we consider a generalization of the differential forms for D-dimensional spaces with any non-integer value $D\in (0,n]$, $n\in \mathbb{N}$. Note that, in the case $D=n$, the proposed calculus of the NIDS differential forms gives the calculus of the differential forms for Euclidean space.

In order to provide a self-consistent calculus of differential forms in the D-dimensional spaces $W=W_1\times \dots \times W_n$, we propose the following principle.

Principle 12 

(Self-Consistency Principle for the Calculus of NIDS Differential Forms). For the self-consistency of the calculus of the differential forms in D-dimensional spaces, the following conditions must be satisfied.

(1) The differential calculus of the differential forms in the D-dimensional spaces must be consistent with the calculus of the NIDS derivetives. The definition and the exterior derivative of the k-form, including $k=0$, must be described identically for all k-forms, including $k=0$.

(2) The integration of the NIDS differential forms in the D-dimensional spaces must be consistent with the calculus of the volume, surface, and line NIDS integrals.

(3) For the standard generalized Stokes theorem of differential forms, a generalization to D-dimensional spaces must exist and hold.

Principle 12 is used in the definitions and equations proposed below. In this subsection, we propose only some basic elements of the NIDS calculus of differential forms. A detailed description of the calculus of differential forms could be provided in a separate work.

4.8.1. Definition and Example of NIDS Differential Forms

Let us give the definition of the differential k-form in the D-dimensional space.

Definition 36 

(Differential Forms in NIDS). Let $W=W_1\times \dots \times W_n$ be the D-dimensional space with $D={\alpha }_1+\dots +{\alpha }_n\in (0,n]$.

(1) In the product measure approach, the k-form ($1⩽k⩽n$ and $k,n\in \mathbb{N}$) in the D-dimensional space W is the integrand

$${\omega }^k=\sum _{j_1,\dots j_k=1}^n{\widehat{f}}_{j_1\dots j_k}\left(\mathbf{x}\right)\tilde{d}{x}^{j_1}\wedge \dots \wedge \tilde{d}{x}^{j_k}=$$

$$\sum _{j_1,\dots j_k=1}^n{f}_{j_1\dots j_k}\left(\mathbf{x}\right){\widehat{c}}_{n,(j_1\dots j_k)}(D,\mathbf{x})d{x}^{j_1}\wedge \dots \wedge d{x}^{j_k},$$

(1114)

where

$$\tilde{d}{x}^j:=c_1({\alpha }_j,x^j)d{x}^j,$$

(1115)

$${\widehat{f}}_{j_1\dots j_k}\left(\mathbf{x}\right):=f_{j_1\dots j_k}\left(\mathbf{x}\right)\prod _{j=1,j\ne j_1\dots j_k}^n{\widehat{c}}_1({\alpha }_j,x^j),$$

(1116)

the symbol ∧ denotes the exterior product (the wedge product) of two differential forms, and $\mathbf{x}=(x^{j_1},\dots ,x^{j_n})$.

In the metric approach, the k-form in the NIDS is defined by Equations (1114) and (1116), where

$${\widehat{c}}_1({\alpha }_j,x^j)=1$$

(1117)

for all $j=1,\dots ,n$, such that ${\widehat{f}}_{j_1\dots j_k}\left(\mathbf{x}\right)=f_{j_1\dots j_k}\left(\mathbf{x}\right)$ and

$${\omega }^k=\sum _{j_1,\dots j_k=1}^n{f}_{j_1\dots j_k}\left(\mathbf{x}\right)\tilde{d}{x}^{j_1}\wedge \dots \wedge \tilde{d}{x}^{j_k}.$$

(1118)

Differential k-forms in NIDS will also be called the NIDS k-forms or the NIDS differential forms.

The NIDS differential k-form (1114) can be integrated over the k-parametric surface (for $2⩽k⩽n-1$), the curved lines (for $k=1$), and in the D-dimensional space $W=W_1\times \dots \times W_n$. In the case $k=n$, we can consider regions $\Omega \subset W$, and the NIDS n-form is called the volume NIDS form.

Let us give the examples of the NIDS differential forms ${\omega }^k$ in the D-dimensional space, where the superscripts k denote the orders k of the NIDS k-forms.

Example 16. 

As an example of a NIDS differential 1-form in the D-dimensional space $W=W_1\times W_2\times W_3$ with $D={\alpha }_1+{\alpha }_2+{\alpha }_3\in (0,3]$, we can consider the expression

$${\omega }^1={\widehat{F}}_1\left(\mathbf{x}\right)\tilde{d}x+{\widehat{F}}_2\left(\mathbf{x}\right)\tilde{d}y+{\widehat{F}}_3\left(\mathbf{x}\right)\tilde{d}z=$$

$$F_1\left(\mathbf{x}\right){\widehat{c}}_{3,x}(D,\mathbf{x})dx+F_2\left(\mathbf{x}\right){\widehat{c}}_{3,y}(D,\mathbf{x})dy+F_3\left(\mathbf{x}\right){\widehat{c}}_{3,z}(D,\mathbf{x})dz,$$

(1119)

where $\mathbf{x}=(x,y,z)$, and we use

$${\widehat{F}}_1\left(\mathbf{x}\right)=F_1\left(\mathbf{x}\right){\widehat{c}}_1({\alpha }_2,y){\widehat{c}}_1({\alpha }_3,z),$$

(1120)

$${\widehat{F}}_2\left(\mathbf{x}\right)=F_2\left(\mathbf{x}\right){\widehat{c}}_1({\alpha }_1,x){\widehat{c}}_1({\alpha }_3,z),$$

(1121)

$${\widehat{F}}_3\left(\mathbf{x}\right)=F_3\left(\mathbf{x}\right){\widehat{c}}_1({\alpha }_1,x){\widehat{c}}_1({\alpha }_2,y),$$

(1122)

and

$$\tilde{d}x=c_1({\alpha }_1,x)dx,$$

(1123)

$$\tilde{d}y=c_1({\alpha }_2,y)dy,$$

(1124)

$$\tilde{d}z=c_1({\alpha }_3,z)dz.$$

(1125)

The NIDS 1-form (1119) can be integrated over the curved line L:

$$I_L^D{\omega }^1={\int }_L\left(F_1\left(\mathbf{x}\right){\widehat{c}}_{3,x}(D,\mathbf{x})dx+F_2\left(\mathbf{x}\right){\widehat{c}}_{3,y}(D,\mathbf{x})dy+F_3\left(\mathbf{x}\right){\widehat{c}}_{3,z}(D,\mathbf{x})dz\right),$$

(1126)

where

$${\widehat{c}}_{3,x}(D,\mathbf{x})=c_1({\alpha }_1,x){\widehat{c}}_1({\alpha }_2,y){\widehat{c}}_1({\alpha }_3,z),$$

(1127)

$${\widehat{c}}_{3,y}(D,\mathbf{x})={\widehat{c}}_1({\alpha }_1,x)c_1({\alpha }_2,y){\widehat{c}}_1({\alpha }_3,z),$$

(1128)

$${\widehat{c}}_{3,z}(D,\mathbf{x})={\widehat{c}}_1({\alpha }_1,x){\widehat{c}}_1({\alpha }_2,y)c_1({\alpha }_3,z).$$

(1129)

All ${\alpha }_1,{\alpha }_2,{\alpha }_3\in (0,1]$. The symbol ∧ denotes the exterior product (the wedge product) of two differential forms.

Example 17. 

As an example of a NIDS differential 2-form in the D-dimensional space $W=W_1\times W_2\times W_3$ with $D={\alpha }_1+{\alpha }_2+{\alpha }_3\in (0,3]$, we have

$${\omega }^2={\widehat{F}}_1\left(\mathbf{x}\right)\tilde{d}y\wedge \tilde{d}z+{\widehat{F}}_2\left(\mathbf{x}\right)\tilde{d}z\wedge \tilde{d}x+{\widehat{F}}_3\left(\mathbf{x}\right)\tilde{d}x\wedge \tilde{d}y=$$

$$F_1\left(\mathbf{x}\right){\widehat{c}}_{3,yz}(D,\mathbf{x})dy\wedge dz+F_2\left(\mathbf{x}\right){\widehat{c}}_{3,zx}(D,\mathbf{x})dz\wedge dx+F_3\left(\mathbf{x}\right){\widehat{c}}_{3,xy}(D,\mathbf{x})dx\wedge dy,$$

(1130)

$$\widehat{F_j}\left(\mathbf{x}\right)=F_j\left(\mathbf{x}\right){\widehat{c}}_1({\alpha }_j,x^j),$$

(1131)

and $\mathbf{x}=(x,y,z)$. The NIDS 2-form (1130) can be integrated over surface Σ in the NIDS as

$$I_{\Sigma }^D{\omega }^2={\int }_{\Sigma }(F_1\left(\mathbf{x}\right){\widehat{c}}_{3,yz}(D,\mathbf{x})dy\wedge dz+$$

$$F_2\left(\mathbf{x}\right){\widehat{c}}_{3,zx}(D,\mathbf{x})dz\wedge dx+F_3\left(\mathbf{x}\right){\widehat{c}}_{3,xy}(D,\mathbf{x})dx\wedge dy),$$

(1132)

where

$${\widehat{c}}_{3,yz}(D,\mathbf{x})={\widehat{c}}_1({\alpha }_1,x)c_1({\alpha }_2,y)c_1({\alpha }_3,z),$$

(1133)

$${\widehat{c}}_{3,zx}(D,\mathbf{x})=c_1({\alpha }_1,x){\widehat{c}}_1({\alpha }_2,y)c_1({\alpha }_3,z),$$

(1134)

$${\widehat{c}}_{3,xy}(D,\mathbf{x})=c_1({\alpha }_1,x)c_1({\alpha }_2,y){\widehat{c}}_1({\alpha }_3,z).$$

(1135)

The symbol ∧ denotes the exterior product (the wedge product) of two differential forms.

Example 18. 

As an example of a NIDS differential 3-form in the D-dimensional space $W=W_1\times W_2\times W_3$ with $D={\alpha }_1+{\alpha }_2+{\alpha }_3\in (0,3]$, we have

$${\omega }^3=\widehat{f}\left(\mathbf{x}\right)\tilde{d}x\wedge \tilde{d}y\wedge \tilde{d}z=f\left(\mathbf{x}\right)c_3(D,\mathbf{x})dx\wedge dy\wedge dz,$$

(1136)

where we use

$$\widehat{f}\left(\mathbf{x}\right)=f\left(\mathbf{x}\right),$$

(1137)

$$c_3(D,\mathbf{x})=c_1({\alpha }_1,x)c_1({\alpha }_2,y)c_1({\alpha }_3,z).$$

(1138)

The NIDS 3-form (1136) can be integrated, and it is represented by the volume NIDS integral over a region Ω in the NIDS

$$I_{\Omega }^D{\omega }^3={\int }_{\Omega }f\left(\mathbf{x}\right)c_3(D,\mathbf{x})dx\wedge dy\wedge dz.$$

(1139)

Example 19. 

As a specal case of the volume form, one can consider the NIDS differential 1-form in the D-dimensional space $W=W_1$ with $D=\alpha \in (0,1]$ that is described as

$${\omega }^1=f\left(x\right)c_1(\alpha ,x)dx.$$

(1140)

The NIDS 1-form (1140) in NIDS can be integrated over an interval $[a,b]$ contained in the domain of the function $f\left(x\right)$:

$$I_{[a,b]}^D{\omega }^1={\int }_a^{b}f\left(x\right)c_1(\alpha ,x)dx,$$

(1141)

where $\alpha \in (0,1]$.

Remark 78. 

Let us consider the NIDS integration of differential forms in D-dimensional space $W=W_1\times \dots \times W_n$. Let U be an open subset of $W=W_1\times \dots \times W_n$. Every smooth NIDS n-form ${\omega }^n$ on U can be represented as

$${\omega }^n=\widehat{f}\left(\mathbf{x}\right)\tilde{d}{x}^1\wedge \dots \wedge \tilde{d}{x}^n=$$

$$f\left(\mathbf{x}\right)\tilde{d}{x}^1\wedge \dots \wedge \tilde{d}{x}^n=$$

$$f\left(\mathbf{x}\right)c_n(D,\mathbf{x})d{x}^1\wedge \dots \wedge d{x}^n,$$

(1142)

where $f\left(\mathbf{x}\right)$ is the smooth function. The smooth functions have the volume NIDS integrals in the product measure approach. This allows us to define the integral of n-form ${\omega }^n$ in the NIDS $W=W_1\times \dots \times W_n$ as

$$I_U^D{\omega }^n:={\int }_U\widehat{f}\left(\mathbf{x}\right)\tilde{d}{x}^1\wedge \dots \wedge \tilde{d}{x}^n={\int }_{U}f\left(\mathbf{x}\right)c_n(D,\mathbf{x})d{x}^1\dots d{x}^n.$$

(1143)

Note that integral (1143) coincides with the volume NIDS integral, which was defined before.

Remark 79. 

For the correct definition of the NIDS integration of the differential forms, the orientation must be fixed. The skew-symmetry of differential forms means that the sign is changed, when two of their arguments are swapped:

$$\tilde{d}{x}^i\wedge \tilde{d}{x}^j=-\tilde{d}{x}^j\wedge \tilde{d}{x}^i.$$

(1144)

This leads to the fact that the NIDS integral of differential form over an oriented surface (manifold) changes sign, when the orientation of the surface (manifold) is reversed. This property is important for the definition of the NIDS integrals of differential forms in D-dimensional space.

4.8.2. Exterior Derivative of NIDS Differential Forms

The exterior derivative d for the differential forms in NIDS is characterized by the properties

$$d({\omega }_1^k+{\omega }_2^k)=d{\omega }_1^k+d{\omega }_2^k,$$

(1145)

$$d({\omega }_1^k\wedge {\omega }_2^l)=d{\omega }_1^k\wedge {\omega }_2^l+{(-1)}^k{\omega }_1^k\wedge d{\omega }_2^l,$$

(1146)

$$d\left(d{\omega }^k\right)=0.$$

(1147)

Properties (1145)–(1147) are similar to the standard properties of the exterior derivative in integer-dimensional spaces.

The exterior derivative is an operation on k-forms ${\omega }^k$ that gives $(k+1)$-form $d{\omega }^k$.

Theorem 39 

(Exterior Derivative of NIDS k-forms). Let $W=W_1\times \dots \times W_n$ be the D-dimensional space with $D={\alpha }_1+\dots +{\alpha }_n\in (0,n]$. Let ${\omega }^k$ be the k-form ($1⩽k⩽n-1$, $k,n\in \mathbb{N}$) in the D-dimensional space $W=W_1\times \dots \times W_n$.

(1) In the product measure approach, the exterior derivative of the NIDS k-form

$${\omega }^k=\sum _{j_1,\dots j_k=1}^n{\widehat{f}}_{j_1\dots j_k}\left(\mathbf{x}\right)\tilde{d}{x}^{j_1}\wedge \dots \wedge \tilde{d}{x}^{j_k}=$$

$$\sum _{j_1,\dots j_k=1}^n{f}_{j_1\dots j_k}\left(\mathbf{x}\right){\widehat{c}}_{n,(j_1\dots j_k)}(D,\mathbf{x})d{x}^{j_1}\wedge \dots \wedge d{x}^{j_k},$$

(1148)

where

$${\widehat{f}}_{j_1\dots j_k}\left(\mathbf{x}\right)=f_{j_1\dots j_k}\left(\mathbf{x}\right)\prod _{j=1,j\ne j_1,\dots ,j_k}^n{\widehat{c}}_1({\alpha }_j,x^j)$$

(1149)

is the $(k+1)$-form in the D-dimensional space W that is described by the equation

$$d{\omega }^k=\sum _{j_1,\dots j_k=1}^n\sum _{i=1}^n\left(\prod _{j=1,j\ne j_1,\dots ,j_k,i}^n{\widehat{c}}_1({\alpha }_j,x^j)\right)\widehat{D_{x^i}^{{\alpha }_i}}{f}_{j_1\dots j_k}\left(\mathbf{x}\right)\tilde{d}{x}^i\wedge \tilde{d}{x}^{j_1}\wedge \dots \wedge \tilde{d}{x}^{j_k}=$$

$$\sum _{j_1,\dots ,j_k,j_{k+1}=1}^n{\widehat{F}}_{j_1\dots j_k{j}_{k+1}}\left(\mathbf{x}\right)\tilde{d}{x}^{j_1}\wedge \dots \wedge \tilde{d}{x}^{j_k}\wedge \tilde{d}{x}^{j_{k+1}},$$

(1150)

where $1⩽k⩽n-1$, $k,n\in \mathbb{N}$, and the operator $\widehat{D_{x^i}^{{\alpha }_i}}$ is the NIDS derivative

$$\widehat{D_{x^i}^{{\alpha }_i}}f\left(\mathbf{x}\right):={\displaystyle \frac{1}{c_1({\alpha }_i,x^i)}}{\displaystyle \frac{\partial }{\partial x^i}}\left({\widehat{c}}_1({\alpha }_i,x^i)f\left(\mathbf{x}\right)\right),$$

(1151)

$${\widehat{F}}_{j_1\dots j_k{j}_{k+1}}\left(\mathbf{x}\right)={(-1)}^k\prod _{p=k+2}^n{\widehat{c}}_1({\alpha }_p,x^p)F_{j_1\dots j_k{j}_{k+1}}\left(\mathbf{x}\right),$$

(1152)

$$F_{j_1\dots j_k{j}_{k+1}}\left(\mathbf{x}\right):=\widehat{D_{x^i}^{{\alpha }_i}}{f}_{j_1\dots j_k}\left(\mathbf{x}\right),$$

(1153)

where $i=j_{k+1}$.

(2) In the metric approach, the exterior derivative of the k-form in the NIDS is defined by the same Equations (1148)–(1153), where

$${\widehat{c}}_1({\alpha }_j,x^j)=1$$

(1154)

for all $j=1,\dots ,n$. Then, functions (1116) are

$${\widehat{f}}_{j_1\dots j_k}\left(\mathbf{x}\right):=f_{j_1\dots j_k}\left(\mathbf{x}\right).$$

(1155)

Therefore, in the metric approach, the exterior derivative of the NIDS k-form

$${\omega }^k=\sum _{j_1,\dots j_k=1}^n{f}_{j_1\dots j_k}\left(\mathbf{x}\right)\tilde{d}{x}^{j_1}\wedge \dots \wedge \tilde{d}{x}^{j_k}=$$

$$\sum _{j_1,\dots j_k=1}^n{f}_{j_1\dots j_k}\left(\mathbf{x}\right)\left(\prod _{p=1}^k{c}_1({\alpha }_{j_p},x^{j_p})\right)d{x}^{j_1}\wedge \dots \wedge d{x}^{j_k}$$

(1156)

is the $(k+1)$-form in the D-dimensional space W that is described by the equation

$$d{\omega }^k=\sum _{j_1,\dots j_k=1}^n\sum _{i=1}^n{D}_{x^i}^{{\alpha }_i}{f}_{j_1\dots j_k}\left(\mathbf{x}\right)\tilde{d}{x}^i\wedge \tilde{d}{x}^{j_1}\wedge \dots \wedge \tilde{d}{x}^{j_k},$$

(1157)

where $1⩽k⩽n-1$, $k,n\in \mathbb{N}$, and the operator $D_{x^i}^{{\alpha }_i}$ is the NIDS derivative

$$D_{x^i}^{{\alpha }_i}f\left(\mathbf{x}\right):={\displaystyle \frac{1}{c_1({\alpha }_i,x^i)}}{\displaystyle \frac{\partial f\left(\mathbf{x}\right)}{\partial x^i}}$$

(1158)

with ${\alpha }_i\in (0,1]$.

Proof. 

(1) The application of the exterior derivative d to the differential forms (1148) in the D-dimensional space gives

$$d{\omega }^k=\sum _{j_1,\dots j_k=1}^{n}d\left({\widehat{f}}_{j_1\dots j_k}\left(\mathbf{x}\right)\tilde{d}{x}^{j_1}\wedge \dots \wedge \tilde{d}{x}^{j_k}\right)=$$

$$\sum _{j_1,\dots j_k=1}^{n}d\left(f_{j_1\dots j_k}\left(\mathbf{x}\right){\widehat{c}}_{n,(j_1\dots j_k)}(D,\mathbf{x})d{x}^{j_1}\wedge \dots \wedge d{x}^{j_k}\right)=$$

$$\sum _{j_1,\dots j_k=1}^{n}d\left(f_{j_1\dots j_k}^C\left(\mathbf{x}\right)d{x}^{j_1}\wedge \dots \wedge d{x}^{j_k}\right)=$$

$$\sum _{j_1,\dots j_k=1}^{n}d{f}_{j_1\dots j_k}^C\wedge d{x}^{j_1}\wedge \dots \wedge d{x}^{j_k}=$$

$$\sum _{j_1,\dots j_k=1}^n\sum _{i=1}^n{\displaystyle \frac{\partial f_{j_1\dots j_k}^C\left(\mathbf{x}\right)}{\partial x^i}}d{x}^i\wedge d{x}^{j_1}\wedge \dots \wedge d{x}^{j_k},$$

(1159)

where we use properties (1145), (1147), and the notation

$$f_{j_1\dots j_k}^C\left(\mathbf{x}\right):=f_{j_1\dots j_k}\left(\mathbf{x}\right){\widehat{c}}_{n,(j_1\dots j_k)}\left(\mathbf{x}\right).$$

(1160)

The partial derivative of the function (1160) can be represented as

$${\displaystyle \frac{\partial f_{j_1\dots j_k}^C\left(\mathbf{x}\right)}{\partial x^i}}={\displaystyle \frac{\partial }{\partial x^i}}\left(f_{j_1\dots j_k}\left(\mathbf{x}\right){\widehat{c}}_{n,(j_1\dots j_k)}(D,\mathbf{x})\right)=$$

$${\displaystyle \frac{\partial }{\partial x^i}}\left(f_{j_1\dots j_k}\left(\mathbf{x}\right)\prod _{p=1}^k{c}_1({\alpha }_{j_p},x^{j_p})\prod _{p=k+1}^n{\widehat{c}}_1({\alpha }_{j_p},x^{j_p})\right)=$$

$$\left(\prod _{p=1}^k{c}_1({\alpha }_{j_p},x^{j_p})\prod _{p=k+2}^n{\widehat{c}}_1({\alpha }_{j_p},x^{j_p})\right){\displaystyle \frac{\partial }{\partial x^i}}\left({\widehat{c}}_1({\alpha }_i,x^i)f_{j_1\dots j_k}\left(\mathbf{x}\right)\right),$$

(1161)

where we use $j_{k+1}=i$ and the fact that $i\ne j_q$ for $q=1,\dots ,k$, since $d{x}^{j_q}\wedge d{x}^{j_q}=0$. Substitution of Equation (1161) into Equation (1159) gives

$$d{\omega }^k=\sum _{j_1,\dots j_k=1}^n\sum _{i=1}^n\left(\prod _{p=k+2}^n{\widehat{c}}_1({\alpha }_{j_p},x^{j_p})\right){\displaystyle \frac{\partial }{\partial x^i}}\left({\widehat{c}}_1({\alpha }_i,x^i)f_{j_1\dots j_k}\left(\mathbf{x}\right)\right)·$$

$$·\left(\prod _{p=1}^k{c}_1({\alpha }_{j_p},x^{j_p})\right)d{x}^i\wedge d{x}^{j_1}\wedge \dots \wedge d{x}^{j_k}.$$

(1162)

Using the equation

$$d{x}^i\wedge d{x}^{j_1}\wedge \dots \wedge d{x}^{j_k}={(-1)}^{k}d{x}^{j_1}\wedge \dots \wedge d{x}^{j_k}\wedge d{x}^i=$$

$${(-1)}^{k}d{x}^{j_1}\wedge \dots \wedge d{x}^{j_k}\wedge d{x}^{j_{k+1}},$$

(1163)

where we use $i=j_{k+1}$ and

$$d{x}^i={\displaystyle \frac{1}{c_1({\alpha }_i,x^i)}}\tilde{d}{x}^i={\displaystyle \frac{1}{c_1({\alpha }_{j_{k+1}},x^{j_{k+1}})}}\tilde{d}{x}^{j_{k+1}},$$

(1164)

we get

$$\left(\prod _{p=1}^k{c}_1({\alpha }_{j_p},x^{j_p})\right)d{x}^i\wedge d{x}^{j_1}\wedge \dots \wedge d{x}^{j_k}=d{x}^i\wedge \tilde{d}{x}^{j_1}\wedge \dots \wedge \tilde{d}{x}^{j_k}=$$

$${(-1)}^k\tilde{d}{x}^{j_1}\wedge \dots \wedge \tilde{d}{x}^{j_k}\wedge d{x}^{j_{k+1}}=$$

$${(-1)}^k{\displaystyle \frac{1}{c_1({\alpha }_i,x^i)}}\tilde{d}{x}^{j_1}\wedge \dots \wedge \tilde{d}{x}^{j_k}\wedge \tilde{d}{x}^{j_{k+1}},$$

(1165)

where we use $i=j_{k+1}$.

As a result, using Equations (1161) and (1165), Equation (1162) takes the form

$$d{\omega }^k=\sum _{j_1,\dots j_k=1}^n\sum _{i=1}^n{(-1)}^k\left(\prod _{p=k+2}^n{\widehat{c}}_1({\alpha }_{j_p},x^{j_p})\right){\displaystyle \frac{1}{c_1({\alpha }_i,x^i)}}{\displaystyle \frac{\partial }{\partial x^i}}\left({\widehat{c}}_1({\alpha }_i,x^i)f_{j_1\dots j_k}\left(\mathbf{x}\right)\right)·$$

$$·\tilde{d}{x}^{j_1}\wedge \dots \wedge \tilde{d}{x}^{j_k}\wedge \tilde{d}{x}^{j_{k+1}}=$$

$$\sum _{j_1,\dots j_k=1}^n\sum _{i=1}^n{(-1)}^k\left(\prod _{p=k+2}^n{\widehat{c}}_1({\alpha }_{j_p},x^{j_p})\right)\widehat{D_{x^i}^i}{f}_{j_1\dots j_k}\left(\mathbf{x}\right)\tilde{d}{x}^{j_1}\wedge \dots \wedge \tilde{d}{x}^{j_k}\wedge \tilde{d}{x}^{j_{k+1}},$$

(1166)

where we use notation (1151) of the NIDS derivative. Using functions (1152) and (1153), Equation (1166) is represented by Equation (1150). This ends the proof.

(2) In the metric approach, the proof is similar to the proof of the product measure approach, where we should use all $c_1({\alpha }_k,x^k)=1$. □

Remark 80. 

Let us note that in Theorem 39 and Definition 36, the exterior derivative of the 0-form is not considered. This is due to the following fact.

In the integer-dimensional space, the exterior derivative generalizes the differential of a function, the functions $f\left(\mathbf{x}\right)$ can be considered as 0-form, and the exterior derivative has the form $df\left(x\right)=f^{\left(1\right)}\left(x\right)dx$.

In the D-dimensional space, this statement is not true in the general case. Next, we will prove this fact, and we will derive the 0-form for the D-dimensional spaces.

To define 0-forms in NIDS, the following principle is proposed.

Principle 13 

(Self-Consistency Principle for NIDS Differential Calculus of k-Form). For the self-consistency of the NIDS differential calculus of differential forms in D-dimensional spaces, the definition and the exterior derivative of k-form including $k=0$ must be described identically for all k-forms, including $k=0$.

This principle means that Equation (1114) of Definition 36, and Equation (1149) of Theorem 39 must be satisfied for the product measure approach.

Theorem 40. 

Let $W=W_1\times \dots \times W_n$ be a D-dimensional space with $D={\alpha }_1+\dots +{\alpha }_n$ and $W_k={\alpha }_k\in (0,1]$, and let $\Omega \subset W$.

(1) Metric approach: functions $f\left(\mathbf{x}\right)\in C_{\left(\alpha \right)}^1(\Omega )$ can be considered as the 0-form only in the metric approach, since the external derivative of this 0-form gives the 1-form that cannot be described by Equation (1150) of Theorem 39 in the product measure approach.

(2) Product measure approach: If $f\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^1(\Omega )$, the 0-form is the function $\widehat{f}\left(\mathbf{x}\right)$ instead of $f\left(\mathbf{x}\right)$; that is,

$${\omega }^0=\widehat{f}\left(\mathbf{x}\right):={\widehat{c}}_n(D,\mathbf{x})f\left(\mathbf{x}\right)=\left(\prod _{p=1}^n{\widehat{c}}_1({\alpha }_p,x^p)\right)f\left(\mathbf{x}\right).$$

(1167)

In this case, the equation of the external derivative of the 0-form (1167) coincides with Equation (1150) of Theorem 39.

Proof. 

(1) The external derivative of the 0-form ${\omega }^0=f\left(\mathbf{x}\right)$ in the D-dimensional space $W=W_1\times \dots \times W_n$ with $D={\alpha }_1+\dots +{\alpha }_n$ has the form

$$d{\omega }^0=\sum _{i=1}^n{\displaystyle \frac{\partial f\left(\mathbf{x}\right)}{\partial x^i}}d{x}^i=\sum _{i=1}^n{\displaystyle \frac{1}{c_1({\alpha }_i,x^i)}}{\displaystyle \frac{\partial f\left(\mathbf{x}\right)}{\partial x^i}}{c}_1({\alpha }_i,x^i)d{x}^i=\sum _{i=1}^n{D}_{x^i}^{{\alpha }_i}f\left(\mathbf{x}\right)\tilde{d}{x}^i,$$

(1168)

where $D_{x^i}^{{\alpha }_i}$ is used instead of the NIDS derivative ${\widehat{D}}_{x^i}^{{\alpha }_i}$ in the product measure approach. Therefore, functions $f\left(\mathbf{x}\right)\in C_{\left(\alpha \right)}^1(\Omega )$ can be considered as the 0-form only in the metric approach.

(2) The external derivative of the 0-form (1167) is

$$d{\omega }^0=d\widehat{f}\left(\mathbf{x}\right):=\sum _{i=1}^n{\displaystyle \frac{\partial \widehat{f}\left(\mathbf{x}\right)}{\partial x^i}}d{x}^i=$$

$$\sum _{i=1}^n{\displaystyle \frac{\partial }{\partial x^i}}\left(f\left(\mathbf{x}\right)\prod _{p=1}^n{\widehat{c}}_1({\alpha }_p,x^p)\right)d{x}^i=$$

$$\sum _{i=1}^n\left(\prod _{p=1,p\ne i}^n{\widehat{c}}_1({\alpha }_p,x^p)\right){\displaystyle \frac{\partial \left({\widehat{c}}_1({\alpha }_i,x^i)f\left(\mathbf{x}\right)\right)}{\partial x^i}}d{x}^i=$$

$$\sum _{i=1}^n\left(\prod _{p=1,p\ne i}^n{\widehat{c}}_1({\alpha }_p,x^p)\right){\displaystyle \frac{1}{c_1({\alpha }_i,x^i)}}{\displaystyle \frac{\partial \left({\widehat{c}}_1({\alpha }_i,x^i)f\left(\mathbf{x}\right)\right)}{\partial x^i}}{c}_1({\alpha }_i,x^i)d{x}^i=$$

$$\sum _{i=1}^n\left(\prod _{p=1,p\ne i}^n{\widehat{c}}_1({\alpha }_p,x^p)\right)\widehat{D_{x^i}^{{\alpha }_i}}f\left(\mathbf{x}\right)\tilde{d}{x}^i.$$

(1169)

As a result, Equation (1169) gives the 1-form

$$d{\omega }^0=\sum _{i=1}^n\widehat{F_i}\left(\mathbf{x}\right)\tilde{d}{x}^i,$$

(1170)

where

$$\widehat{F_i}\left(\mathbf{x}\right)=\left(\prod _{p=1,p\ne i}^n{\widehat{c}}_1({\alpha }_p,x^p)\right)F_i\left(\mathbf{x}\right),$$

(1171)

$$F_i\left(\mathbf{x}\right)=\widehat{D_{x^i}^{{\alpha }_i}}f\left(\mathbf{x}\right),$$

(1172)

and $i=1,\dots ,n$.

Therefore function (1167) with $f\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^1(\Omega )$ can be considered as the 0-form in the product measure approach. □

Let us give some examples of Equation (1150) of Theorem 39.

Example 20. 

Let us consider the case $n=2$ and the NIDS 1-form

$${\omega }^1=\widehat{f_x}(x,y)\tilde{d}x+\widehat{f_y}(x,y)\tilde{d}y$$

(1173)

in the D-dimensional space $W=W_1\times W_2$ with $D={\alpha }_1+{\alpha }_2$, where

$$\widehat{f_x}(x,y)={\widehat{c}}_1({\alpha }_2,y)f_x(x,y),\widehat{f_y}(x,y)={\widehat{c}}_1({\alpha }_1,x)f_y(x,y),$$

(1174)

$$\tilde{d}x=c_1({\alpha }_1,x)dx,\tilde{d}y=c_1({\alpha }_2,y)dy,$$

(1175)

and with $f\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^1(\Omega )$.

The exterior derivative of the differential form (1173) in NIDS has the form

$$d{\omega }^1={\widehat{F}}_{xy}(x,y)\tilde{d}x\wedge \tilde{d}y=\left(\widehat{D_x^{{\alpha }_1}}{f}_y(x,y)-\widehat{D_y^{{\alpha }_2}}{f}_x(x,y)\right)\tilde{d}x\wedge \tilde{d}y,$$

(1176)

where

$${\widehat{F}}_{xy}(x,y)=F_{xy}(x,y)=\widehat{D_x^{{\alpha }_1}}{f}_y(x,y)-\widehat{D_y^{{\alpha }_2}}{f}_x(x,y).$$

(1177)

Let us note that the NIDS differential form (1173) is a closed NIDS differential 1-form, if

$$\widehat{D_x^{{\alpha }_1}}{f}_y(x,y)-\widehat{D_y^{{\alpha }_2}}{f}_x(x,y)=0$$

(1178)

for all $(x,y)\in \Omega$. If ${\alpha }_1={\alpha }_2=1$, then condition (1178) gives the standard condition

$${\displaystyle \frac{\partial f_y(x,y)}{\partial x}}-{\displaystyle \frac{\partial f_x(x,y)}{\partial y}}=0$$

(1179)

for the differential forms in integer-dimensional space.

Example 21. 

Let us consider the case $n=3$ and the 1-form

$${\omega }^1=\widehat{f_x}\left(\mathbf{x}\right)\tilde{d}x+\widehat{f_y}\left(\mathbf{x}\right)\tilde{d}y+\widehat{f_z}\left(\mathbf{x}\right)\tilde{d}z$$

(1180)

in the D-dimensional space $W=W_1\times W_2\times W_3$ with $D={\alpha }_1+{\alpha }_2+{\alpha }_3$, where

$$\widehat{f_x}\left(\mathbf{x}\right)={\widehat{c}}_2(d_{yz},y)f_x\left(\mathbf{x}\right)={\widehat{c}}_1({\alpha }_2,y){\widehat{c}}_1({\alpha }_3,z)f_x\left(\mathbf{x}\right),$$

(1181)

$$\widehat{f_y}\left(\mathbf{x}\right)={\widehat{c}}_2(d_{xz},\mathbf{x})f_y\left(\mathbf{x}\right)={\widehat{c}}_1({\alpha }_1,x){\widehat{c}}_1({\alpha }_3,z)f_y\left(\mathbf{x}\right),$$

(1182)

$$\widehat{f_z}\left(\mathbf{x}\right)={\widehat{c}}_2(d_{xy},\mathbf{x})f_z\left(\mathbf{x}\right)={\widehat{c}}_1({\alpha }_1,x){\widehat{c}}_1({\alpha }_2,y)f_z\left(\mathbf{x}\right),$$

(1183)

and

$$\tilde{d}x=c_1({\alpha }_1,x)dx,\tilde{d}y=c_1({\alpha }_2,y)dy,\tilde{d}z=c_1({\alpha }_3,z)dz,$$

(1184)

with $f\left(\mathbf{x}\right)\in {\widehat{C}}_{\left(\alpha \right)}^1(\Omega )$.

The exterior derivative of the NIDS differential form (1180) is

$$d{\omega }^1=\widehat{F_{yz}}\left(\mathbf{x}\right)\tilde{d}y\wedge \tilde{d}z+\widehat{F_{zx}}\left(\mathbf{x}\right)\tilde{d}z\wedge \tilde{d}x+\widehat{F_{xy}}\left(\mathbf{x}\right)\tilde{d}x\wedge \tilde{d}y,$$

(1185)

where

$$\widehat{F_{yz}}\left(\mathbf{x}\right)={\widehat{c}}_1({\alpha }_1,x)F_{yz}\left(\mathbf{x}\right)={\widehat{c}}_1({\alpha }_1,x)\left(\widehat{D_y^{{\alpha }_2}}{f}_z\left(\mathbf{x}\right)-\widehat{D_z^{{\alpha }_3}}{f}_y\left(\mathbf{x}\right)\right),$$

(1186)

$$\widehat{F_{zx}}\left(\mathbf{x}\right)={\widehat{c}}_1({\alpha }_2,y)F_{zx}\left(\mathbf{x}\right)={\widehat{c}}_1({\alpha }_2,y)\left(\widehat{D_z^{{\alpha }_1}}{f}_x\left(\mathbf{x}\right)-\widehat{D_x^{{\alpha }_1}}{f}_z\left(\mathbf{x}\right)\right),$$

(1187)

$$\widehat{F_{xy}}\left(\mathbf{x}\right)={\widehat{c}}_1({\alpha }_3,z)F_{xy}\left(\mathbf{x}\right)={\widehat{c}}_1({\alpha }_3,z)\left(\widehat{D_x^{{\alpha }_1}}{f}_y\left(\mathbf{x}\right)-\widehat{D_y^{{\alpha }_2}}{f}_x\left(\mathbf{x}\right)\right).$$

(1188)

Let us note that the NIDS 1-form (1180) is a closed NIDS differential form in the region $\Omega \subset W$, if

$${\widehat{Curl}}_{\left(\alpha \right)}\mathbf{f}\left(\mathbf{x}\right)=0$$

(1189)

for all $\mathbf{x}\in \Omega$. If ${\alpha }_1={\alpha }_2={\alpha }_3=1$, then condition (1189) gives the standard condition $curl\mathbf{f}\left(\mathbf{x}\right)=0$. for the differential forms in integer-dimensional space.

Example 22. 

Let us consider the case $n=3$ and the 2-form

$${\omega }^2=\widehat{f_{yz}}\left(\mathbf{x}\right)\tilde{d}y\wedge \tilde{d}z+\widehat{f_{zx}}\left(\mathbf{x}\right)\tilde{d}z\wedge \tilde{d}x+\widehat{f_{xy}}\left(\mathbf{x}\right)\tilde{d}x\wedge \tilde{d}y=$$

$$f_{yz}\left(\mathbf{x}\right){\widehat{c}}_{3,yz}(D,\mathbf{x})dy\wedge dz+f_{zx}\left(\mathbf{x}\right){\widehat{c}}_{3,xz}(D,\mathbf{x})dz\wedge dx+f_{xy}\left(\mathbf{x}\right){\widehat{c}}_{3,xy}(D,\mathbf{x})dx\wedge dy$$

(1190)

in the D-dimensional space $W=W_1\times W_2\times W_3$ with $D={\alpha }_1+{\alpha }_2+{\alpha }_3$, where

$$\widehat{f_{yz}}\left(\mathbf{x}\right)={\widehat{c}}_1({\alpha }_1,x)f_{yz}\left(\mathbf{x}\right),$$

(1191)

$$\widehat{f_{zx}}\left(\mathbf{x}\right)={\widehat{c}}_1({\alpha }_2,y)f_{zx}\left(\mathbf{x}\right),$$

(1192)

$$\widehat{f_{xy}}\left(\mathbf{x}\right)={\widehat{c}}_1({\alpha }_3,z)f_{xy}\left(\mathbf{x}\right),$$

(1193)

and

$$\tilde{d}x=c_1({\alpha }_1,x)dx,\tilde{d}y=c_1({\alpha }_2,y)dy,\tilde{d}z=c_1({\alpha }_3,z)dz.$$

(1194)

The exterior derivative of the NIDS 2-form (1190) is the NIDS 3-form

$$d{\omega }^2={\widehat{F}}_{xyz}\left(\mathbf{x}\right)\tilde{d}x\wedge \tilde{d}y\wedge \tilde{d}z,$$

(1195)

which is the volume NIDS form in space $W=W_1\times W_2\times W_3$, where

$${\widehat{F}}_{xyz}\left(\mathbf{x}\right)=F_{xyz}\left(\mathbf{x}\right)=\widehat{D_x^{{\alpha }_1}}{f}_{yz}\left(\mathbf{x}\right)+\widehat{D_y^{{\alpha }_2}}{f}_{zx}\left(\mathbf{x}\right)+\widehat{D_z^{{\alpha }_3}}{f}_{xy}\left(\mathbf{x}\right).$$

(1196)

Note that the NIDS 2-form (1190) is a closed NIDS differential form in the region $\Omega \subset W$, if

$${\widehat{Div}}_{\left(\alpha \right)}\mathbf{f}\left(\mathbf{x}\right)=0$$

(1197)

for all $\mathbf{x}\in \Omega$.

4.8.3. General NIDS Stokes Theorem for NIDS Differential Forms

External derivatives allow us to express the fundamental theorems of NIDS vector calculus (the divergence theorem for NIDS, the NIDS Green theorem and the NIDS Stokes theorem), as special cases of the general NIDS Stokes theorem.

This general NIDS Stokes theorem states the following.

Theorem 41 

(General NIDS Stokes Theorem for Differential Forms). Let $W=W_1\times \dots \times W_n$ be the D-dimensional space with $D={\alpha }_1+\dots +{\alpha }_n\in (0,n]$, and let Ω be an orientable surface in the region of the space W that is bounded by a piecewise smooth surface $\Sigma =\partial \Omega$.

The NIDS integral of the k-form ${\omega }^k$ over the boundary $\partial \Omega$ of the surface Ω is equal to the NIDS integral of the NIDS exterior derivative $\tilde{d}{\omega }^k$ over this surface

$$I_{\partial \Omega }^D{\omega }^k=I_{\Omega }^{D}d{\omega }^k.$$

(1198)

The symbolic notation of the general NIDS Stokes Equation (1198) means the equality

$${\int }_{\partial \Omega }\left(\sum _{j_1,\dots j_k=1}^n{\widehat{f}}_{j_1\dots j_k}\left(\mathbf{x}\right)\tilde{d}{x}^{j_1}\wedge \dots \wedge \tilde{d}{x}^{j_k}\right)=$$

$${\int }_{\Omega }\sum _{j_1,\dots ,j_k,j_{k+1}=1}^n{\widehat{F}}_{j_1\dots j_k{j}_{k+1}}\left(\mathbf{x}\right)\tilde{d}{x}^{j_1}\wedge \dots \wedge \tilde{d}{x}^{j_k}\wedge \tilde{d}{x}^{j_{k+1}},$$

(1199)

where

$${\widehat{F}}_{j_1\dots j_k{j}_{k+1}}\left(\mathbf{x}\right)=\left({(-1)}^k\prod _{p=k+2}^n{\widehat{c}}_1({\alpha }_p,x^p)\right)\widehat{D_{x^i}^{{\alpha }_i}}{f}_{j_1\dots j_k}\left(\mathbf{x}\right),$$

(1200)

$$F_{j_1\dots j_k{j}_{k+1}}\left(\mathbf{x}\right)={(-1)}^k\widehat{D_{x^i}^{{\alpha }_i}}{f}_{j_1\dots j_k}\left(\mathbf{x}\right),$$

(1201)

where $i=j_{k+1}$.

Proof. 

This general NIDS Stokes theorem for NIDS differential k-forms is proved in a similar way to the proof of the general Stokes theorem for differential forms in the integer-dimensional space, which is given, for example, in Section 26.4 “Proof of the general Stokes formula for the cube” in [147], pp. 263–265, and Equation (1150), proved in this paper. □

The generalized NIDS Stokes theorem gives the fundamental relationship between the NIDS exterior derivative and the NIDS integration.

4.8.4. Maxwell’s Equations in D-Dimensional Space

Differential forms are actively used to describe electrodynamics (for example, see Chapter 4 in [163]).

The electromagnetic fields in NIDS can be described by the 2-form

$${\mathbf{F}}_{\left(\alpha \right)}:={\displaystyle \frac{1}{2}}{\widehat{F}}_{\mu \nu }\tilde{d}{x}^{\mu }\wedge \tilde{d}{x}^{\nu }=$$

$${\widehat{E}}_x\tilde{d}x\wedge \tilde{d}t+{\widehat{E}}_y\tilde{d}y\wedge \tilde{d}t+{\widehat{E}}_z\tilde{d}z\wedge \tilde{d}t+$$

$${\widehat{B}}_x\tilde{d}y\wedge \tilde{d}z+{\widehat{B}}_y\tilde{d}z\wedge \tilde{d}x+{\widehat{B}}_z\tilde{d}x\wedge \tilde{d}y,$$

(1202)

where

$$\tilde{d}t=c_1({\alpha }_0,t)dt,\tilde{d}x=c_1({\alpha }_1,x)dx,\tilde{d}y=c_1({\alpha }_2,y)dy,\tilde{d}z=c_1({\alpha }_3,z)dz,$$

(1203)

$${\widehat{F}}_{\mu \nu }=F_{\mu \nu }\prod _{\delta \ne \mu ,\nu }^3\widehat{c_1}(\delta ,x^{\delta }).$$

(1204)

Note that time can be considered as one-dimensional (with an integer dimension equal to one). In this case, ${\alpha }_0=1$, and $\tilde{d}t=dt$.

The NIDS generalization of dual 2-form $\star \mathbf{F}$ is defined by the the Hodge star operator ⋆ as

$$\star {\mathbf{F}}_{\left(\alpha \right)}:=\widehat{F^{\delta \lambda }}{\epsilon }_{\delta \lambda \mu \nu }\tilde{d}{x}^{\mu }\wedge \tilde{d}{x}^{\nu }=$$

$$-{\widehat{B}}_x\tilde{d}x\wedge \tilde{d}t-{\widehat{B}}_y\tilde{d}y\wedge \tilde{d}t-{\widehat{B}}_z\tilde{d}z\wedge \tilde{d}t+$$

$${\widehat{E}}_x\tilde{d}y\wedge \tilde{d}z+{\widehat{E}}_y\tilde{d}z\wedge \tilde{d}x+{\widehat{E}}_z\tilde{d}x\wedge \tilde{d}y.$$

(1205)

The NIDS current 1-form is defined as

$${\mathbf{J}}_{\left(\alpha \right)}={\widehat{J}}_{\mu }\tilde{d}{x}^{\mu }=-\widehat{\rho }\tilde{d}t+\widehat{J_x}\tilde{d}x+\widehat{J_y}\tilde{d}y+\widehat{J_z}\tilde{d}z,$$

(1206)

where $J_{\mu }$, $\mu =0,1,2,3$, are the components of the 4-current density, and $J_0=\rho$.

The first Maxwell equation in NIDS is described as

$$d{\mathbf{F}}_{\left(\alpha \right)}=0.$$

(1207)

The second Maxwell equation in NIDS is written as

$$d\star {\mathbf{F}}_{\left(\alpha \right)}=4\pi \star {\mathbf{J}}_{\left(\alpha \right)},$$

(1208)

where $\star {\mathbf{J}}_{\left(\alpha \right)}$ is the NIDS current 3-form

$$\star {\mathbf{J}}_{\left(\alpha \right)}={\displaystyle \frac{1}{6}}\widehat{J^{\delta }}{\epsilon }_{\delta \lambda \mu \nu }\tilde{d}{x}^{\delta }\wedge \tilde{d}{x}^{\mu }\wedge \tilde{d}{x}^{\nu }.$$

(1209)

and $J^{\delta }={\eta }^{\delta \mu }{J}_{\mu }$ with $\mu ,\delta =0,1,2,3$.

For the case ${\alpha }_{\mu }=1$, for all $\mu =0,1,2,3$, Equations (1207) and (1208) give the standard Maxwell equations with standard differential form. If ${\alpha }_{\mu }\in (0,1)$ for all $\mu =1,2,3$, Equations (1207) and (1208) describe the electromagnetic fields in non-integer-dimensional space. In this case, these equations can be interpreted as equations of electromagnetic fields in fractal media.

4.9. Some Comments on the Application of NIDS Calculus in Fractal Physics

The first application of NIDS to fractal systems and media was proposed by Le Guillou and Zinn-Justin in the 1977 paper [4], to describe the physics of some crystals that can be macroscopically considered as fractal media.

Let us note some of the shortcomings of using fractal sets to describe fractal media. Firstly, it should be emphasized that fractals are measurable metric sets with non-integer dimensions [124,125] that should be observed at all scales. Secondly, fractal sets are usually not linear (vector) spaces. Therefore, we cannot use the standard derivatives of integer or fractional orders to describe physical processes on fractal sets. Thirdly, fractal media and fractal distribution of particles are not fractals, since the scaling property is satisfied only over a finite range of scales, bounded below by the size of atoms and above by the size of the region filled with the fractal medium.

All these properties show that the use of fractal sets to describe fractal media and distribution of particles is not entirely adequate and is an inconvenient mathematical tool. Therefore, there are many different approaches to the mathematical description of fractal media. The basic types of approaches were described in the review [69] and the references therein.

The most promising approach to describing fractal media, distributions, and fields is the NIDS approach that uses non-integer-dimensional spaces and continuous models of fractal media (CMFM). The continuous models of fractal media are mathematically based on the concept of non-integer-dimensional space (NIDS) and physically based on the concept of the density of states (DOS). In CMFM, fractality is taken into account through the measure of NIDS integration that describes the density of states (DOS). A significant advantage of the CMFM and NIDS approach is that the non-integer-dimensional space is a linear (vector) space. Another important advantage is that it uses integral and differential calculus in linear (vector) spaces.

The CMFM was first proposed and applied in the author’s papers published from 2004 to 2007 for the following branches of physics:

• Continuum mechanics and hydrodynamics [70,71,72].
• Solid mechanics [73,74].
• Electromagnetic fields on fractal media [75,76,77].
• The magnetohydrodynamics of fractal media [78].
• The gravitational field of the fractal distribution of particles [79].
• The Ginzburg–Landau equation for fractal media [80].
• The Fokker–Planck equation for fractal media [81].
• The Chapman–Kolmogorov equation for fractal media [82].
• The statistical mechanics of fractal distributions [16,83,84].
• Physical kinetics [85,86].

Most of these models were later combined in Chapters 1–7 of the book [87]. Note also the continuum homogenization of fractal media, which is described by Ostoja-Starzewski, Li, Joumaa, and Demmie in [97,103], is important for the basis of the NIDS approach.

A drawback of all these works by the author from 2005 to 2007 was the lack of vector calculus in non-integer-dimensional spaces. In some papers, the NIDS integration was linked to the fractional integration of non-integer order, although the NIDS derivatives remained of integer order. In the review [69], this approach, which is used in the 2004–2007 papers, was incorrectly called the “fractional integration approach”. The use of fractional integrals resulted in incorrect normalization for integrals and derivatives in the form of a numerical coefficient in front of the integrals and derivatives. Some of these works from 2004 to 2007 (primarily those devoted to statistical mechanics and physical kinetics) used fractional powers of coordinates and so-called fractional spaces.

In 2007, the baton of using continuous models of fractal media was picked up by Ostoja-Starzewski. Various continuum models of fractal media were actively developed by Ostoja-Starzewski and then by Ostoja-Starzewski, Li, Demmie, and Joumaa in the following fields:

• The thermomechanics of fractal media [88,89].
• The thermoelasticity of fractal media [90].
• Turbulence as fractal medium [91].
• The extremum and variational principles for fractal porous media [92,93].
• Electromagnetism on anisotropic fractals [94].
• Fractal solids [95,96,97].
• Waves in fractal media [96,98].
• The micropolar continuum mechanics of fractal media [100,101].
• The acoustic-elastodynamics of fractal media [102].

From 2018 to 2023, CMFM were studied by Mashayekhi, Stanisauskis, Pahari, Miles, Hussaini, Oates, Mehnert, and Steinmann in the experimental validation of viscoelasticity models of fractal and non-fractal media [104,105], excluded volume effects in viscoelasticity in polymers [106], fractal effects in elastomers [107], and entropy dynamics in fractal and fractional viscoelasticity of elastomers [108].

The first self-consistent differential NIDS vector calculus of differential operators in the NIDS was proposed only in the 2014 and 2015 papers [18,19], where the differential and integral NIDS operators were suggested. However, these articles did not propose an integral NIDS vector calculus consistent with differential NIDS vector calculus and did not prove the fundamental theorems of the NIDS vector calculus. Despite this, the differential NIDS vector calculus allowed us eliminate some shortcomings of the NIDS models of fractal media proposed in the 2004–2007 papers by using differential and integral operators in the NIDS proposed in [18,19]. In 2014–2018, NIDS vector calculus was applied to the following areas of physics:

• Fractal electrodynamics [109,110].
• The elasticity of fractal material [113].
• The heat transfer in fractal materials [114].
• The acoustic waves in fractal media [115].
• The steady flow of fractal fluid [111,112].
• Quantum mechanics and statistical physics [116].

Note that these application are based on the NIDS integral and differential operators of the vector calculus in NIDS and the concept of the density of states [18,19,69,87]. The physical interpretation of the NIDS integral and differential operators is based on the concept of fractal density of states (DOS) proposed in [18,19,69,87] in the framework of continuum models of the fractal distributions of particles, fields, and fractal media.

Based on the use of CMFM with NIDS and the concept of the density of states [18,19,69,87], the physical interpretation of the operators in NIDS was given. To describe fractal distributions of media by the continuum models in NIDS [18,69,70,71,72,87], we should use the concepts of density of states $c_n(D,\mathbf{x})$ and the distribution function $f\left(\mathbf{x}\right)$, where $\mathbf{x}\in {\mathbb{R}}^n$ and $n\in \mathbb{N}$ (for example, $n=3$) [18,69,87]. The density of states describes closely packed permitted states (or places) in the space ${\mathbb{R}}^n$, where the particles are distributed [18,19,69,87]. The expression $c_n(D,\mathbf{x})d{V}_n$ is equal to the number (or volume) of permitted states between $V_n$ and $V_n+d{V}_n$ that is distributed in the space ${\mathbb{R}}^n$ with the physical dimension $D\in (0,n]$. The distribution function $f\left(\mathbf{x}\right)$, describes a distribution of physical values such as mass, electric charge, and the number of particles (point particles) on a set of permitted states. In general, we cannot reduce all properties of the fractal medium to the distribution function only, and we should use the concept of the density of states.

It should be emphasized that the NIDS derivatives and NIDS integrals are operators of integer orders. These NIDS operators cannot in general be interpreted as fractional derivatives and integrals of non-integer orders and as a fractal derivatives and integrals on fractal sets.

As a result, it can be said that the development of the CMFM and NIDS approach occurred in two stages: the first stage from 2004 to 2014 and the second stage from 2014 to 2025. This 2025 work opens a new stage, the third stage, in the development and application of the continuous models of fractal media and the NIDS approaches. The mathematical tool proposed in this article in the form of the self-consistent NIDS calculus and the NIDS vector calculus will allow for an adequate description of fractal media and distributions of particles and fields in subsequent studies in the future.

5. Conclusions

In this paper, we defined integrals and derivatives in non-integer-dimensional space (NIDS) that form calculus, for which the NIDS generalizations of fundamental theorems of the standard calculus are satisfied. It should be emphasized that the NIDS derivatives and NIDS integrals are operators of integer orders. These operators cannot be interpreted as fractional derivatives and integrals of non-integer orders and as a fractal derivatives and integrals on fractal sets. The NIDS generalization of the standard vector differential operators (gradient, divergence, and curl) and integral operators (the line and surface integrals of vector fields) were proposed. The NIDS generalizations of the standard gradient theorem, the divergence theorem (the Gauss–Ostrogradsky theorem), and the Stokes theorem were proved. In fact, this paper proposed the self-consistent calculus of derivatives and integrals in non-integer-dimensional space, as well as the self-consistent integral and differential vector calculus in NIDS.

Let us note some possible directions of the development of the proposed NIDS calculus.

1.

It is important to generalize the NIDS calculus from Cartesian coordinates to orthogonal coordinate systems (OCS) so that the second correspondence principle is satisfied, at least for integration over volume. It should be emphasized that in both the metric and product measure approaches, the second correspondence principle holds for volume NIDS integrals. In the proposed calculus, D-dimensional spaces were characterized by their dimensions along the Cartesian coordinate axes. In the generalization to OCS, it is assumed that the dimensions of space are generally different along the corresponding orthogonal axes. This should not be confused with the coordinate transformation from the Cartesian system to the spherical system, which is described in this article. For example, in the spherical coordinate system, the NIDS calculus with OCS considers the different dimensions for the radius and for each angle. In this case $D={\alpha }_r+{\alpha }_{\varphi }+{\alpha }_{\theta }$, where ${\alpha }_r$, ${\alpha }_{\varphi }$, ${\alpha }_{\theta }$ are the dimensions of space along the corresponding coordinate axes. Note that elements of such a calculus for the spherically symmetric case are described in the article [19].

2.

It is important to generalize the NIDS calculus from the Euclidean space to arbitrary differential manifolds. This direction of the NIDS generalization is connected with the development of the generalization of the proposed NIDS calculus of the differential forms. Note that the NIDS theory of D-dimensional manifolds is practically obvious in the metric approach, since basic elements are described in this paper. In the product measure approach, it is necessary to generalize the concept of the Riemannian manifold and the metric of the manifold in the form described in Section 2.8.3 of this paper.

3.

It is interesting to generalize the differential geometry of line and surfaces to the case of the D-dimensional space in the product measure approach. Note that the NIDS differential geometry of line in surfaces is practically obvious in the metric approach.

The proposed self-consistent NIDS calculus allows us to describe fractal distributions of particles, fields, and media in the framework of continuum models, in which the non-integer dimensions of the space are taken into account by the concept of the fractal density of states [18,19,69,87].