$\mathbb{A}$-Differentiability over Associative Algebras

Abstract

The unital associative algebra structure $\mathbb{A}$ on ${\mathbb{R}}^n$ allows for defining elementary functions and functions defined by convergent power series. For these, the usual derivative has a simple form even for higher-order derivatives, which allows us to have the $\mathbb{A}$-calculus. Thus, we introduce $\mathbb{A}$-differentiability. Rules for $\mathbb{A}$-differentiation are obtained: a product rule, left and right quotients, and a chain rule. Convergent power series are $\mathbb{A}$-differentiable, and their $\mathbb{A}$-derivatives are the power series defined by their $\mathbb{A}$-derivatives. Therefore, we use associative algebra structures to calculate the usual derivatives. These calculations are carried out without using partial derivatives, but only by performing operations in the corresponding algebras. For $f\left(x\right)=x^2$, we obtain $d{f}_x\left(v\right)=vx+xv$, and for $f\left(x\right)=x^{-1}$, $d{f}_x\left(v\right)=-x^{-1}v{x}^{-1}$. Taylor approximations of order k and expansion by the Taylor series are performed. The pre-twisted differentiability for the case of non-commutative algebras is introduced and used to solve families of quadratic ordinary differential equations.

1. Introduction

The Irish mathematician, physicist, and astronomer William Hamilton discovered the quaternions in 1843 [1,2]. The complex numbers can be seen as the subalgebra of all matrices in $M_2\left(\mathbb{R}\right)$ of the type

$$\left(\begin{array}{cc}a& -b\\ b& a\end{array}\right).$$

The quaternions can be seen as the set of all the matrices in $M_4\left(\mathbb{R}\right)$ of the type

$$\left(\begin{array}{cccc}a& -b& -c& -d\\ b& a& d& -c\\ c& -d& a& b\\ d& c& -b& a\end{array}\right).$$

(1)

In a more general framework, Clifford algebras, used in mathematics, physics, engineering, and other fields, extend and generalize the real, complex, and quaternionic analysis; see [3]. These are associative algebras.

In this article, we consider finite-dimensional associative algebras $\mathbb{A}$ that as linear spaces are ${\mathbb{R}}^n$ for some $n\in \mathbb{N}$, although many results hold more generally. We consider algebras $\mathbb{A}$ with identity, although some of the results apply to algebras without identity.

Complex analysis extends to real analysis. The objective of the theory of analytic functions on algebras is to extend the results of complex analysis to the case of algebras. For the commutative case, there are a significant number of results; see [4,5,6,7,8]. This theory has had an important impact on the solution of both ordinary and partial differential equations; see [9,10,11,12,13] and the references therein.

The introduction of differentiability over noncommutative associative algebras has been a challenge. For the left and right derivatives defined by the left and right quotients, respectively, not even the function $f\left(x\right)=x^2$ is differentiable. In the quaternions case, for the left derivative, only functions $f\left(x\right)=a+xb$ are differentiable, and for the right derivative, only functions $f\left(x\right)=a+bx$ are differentiable; see [14]. In [15], a definition of differentiability is given over Clifford algebras, for functions from a vector subspace of the algebra to the algebra. In this paper, we consider the structures of the associative algebra $\mathbb{A}$ in ${\mathbb{R}}^n$ and define the $\mathbb{A}$-differentiability of functions $f:U\subset \mathbb{A}\to \mathbb{A}$. We obtain versions for the case of algebras of some differentiation rules, and we calculate the $\mathbb{A}$-derivatives for elementary functions defined over associative algebras, as well as functions defined by convergent power series.

The differentiability introduced in this work is similar to that given by Dzagnidze [16]; see also [17]. The difference between these two definitions is that, in the one given by Dzagnidze, the derivative at a point is an element of the algebra considered, which, in our opinion, loses the approximation character given by the derivative, so the author does not introduce Taylor approximations. With the definition given in this work, Taylor approximations of order k and the expansion by the Taylor series are performed. This approach is applied to differentiable functions defined on associative algebras.

The introduction of differentiability on non-commutative associative algebras facilitates the calculation of the usual (Fréchet) derivative of elementary functions and is defined over associative algebras, as well as functions defined by convergent power series. The rules enable the systematic and efficient differentiation of these functions, without the need to express functions in terms of their components, to then calculate partial derivatives as usually done and finally obtain an elementary function or a function defined by a convergent power series. Additionally, for convergent power series, it may be impossible to perform the usual derivative calculations directly. However, the introduced notion of differentiability allows us to obtain explicit expressions for the Fréchet derivatives of such functions.

Therefore, the significant achievement of the differentiability rules presented in this work is to relate the standard calculus of these functions to the underlying structures of associative algebras. This approach introduces concepts of more advanced calculus, where vector-valued functions are expressed in a concise manner.

Clifford algebras, including quaternions, provide practical applications of the results discussed here. These algebras are fundamental in mathematics, physics, and engineering, among other fields. Additionally, the introduced derivatives can be extended to infinite-dimensional cases by considering Banach algebras, thereby providing systematic methods for calculating Fréchet derivatives of families of functions.

The main contributions studied in this paper are as follows:

• we extend classical results of real and complex analysis to the setting of finite-dimensional, non-commutative, associative algebras,
• we present a simple and systematic framework for computing Fréchet derivatives of functions defined through algebraic operations,
• we develop Taylor approximations for functions defined via structures of associative algebras,
• we introduce differential equations related to derivatives in non-commutative associative algebras and use this to solve a family of quadratic systems of ordinary differential equations, and
• we show how, using the introduced calculus, one can calculate usual the line integrals of an important family of functions.

Section 2 recalls associative algebras with identity and defines polynomials and power series in associative algebras. In Section 3, the definition of $\mathbb{A}$-differentiability is introduced, some differentiability rules are given, and it is proved that the derivative of a convergent power series is given by the corresponding series of derivatives. Section 4 contains $\mathbb{A}$-differentiability of k-order, Taylor polynomials and series, $\mathbb{A}$-analyticity, and the usual line integral in ${\mathbb{R}}^n$. In Section 5, we calculate the derivative of the function $f\left(x\right)=x^3$ with respect to the quaternions. The pre-twisted differentiability for associative algebras is introduced in Section 6 and is used to solve a family of quadratic ordinary differential equations.

2. Associative Algebras

2.1. Introduction

We call an algebra $\mathbb{A}$ the $\mathbb{R}$-linear space ${\mathbb{R}}^n$ endowed with an associative product between its elements. In this work, $\mathbb{A}$ denotes an algebra. $\mathbb{A}$ is said to be unital if there is an identity $e\in \mathbb{A}$ for the $\mathbb{A}$-product. An element $a\in \mathbb{A}$ is said to be regular if there is an element denoted by $a^{-1}\in \mathbb{A}$ such that $a{a}^{-1}=a^{-1}a=e$. $a^{-1}$ is called multiplicative inverse of a. The set of all regular elements of $\mathbb{A}$ is denoted by ${\mathbb{A}}^{\ast }$.

To simplify notation, horizontal and vertical notations of elements of ${\mathbb{R}}^n$ will be used without remembering this.

An algebra $\mathbb{A}$ can be determined by knowing the product between the elements of the canonical basis: $e_i{e}_j={\sum }_{k=1}^n{c}_{ijk}{e}_k$, see [18]. The first fundamental representation of an algebra $\mathbb{A}$ is the isomorphism $R:\mathbb{A}\to {\mathbb{M}}_n\left(\mathbb{R}\right)$ defined on the canonical basis by $R_i=R\left(e_i\right)$, where $R_{ijk}=c_{ikj}$. For a complete classification of three-dimensional algebras, see [19].

On the algebra of the continuous operators of a Banach space into itself, the operators norm is semi-multiplicative. Since the matrices ${\mathbb{M}}_n\left(\mathbb{R}\right)$ can be seen as the set of all the linear operators $L:{\mathbb{R}}^n\to {\mathbb{R}}^n$, this norm can be given to ${\mathbb{M}}_n\left(\mathbb{R}\right)$. Then, using the first fundamental representation $R:\mathbb{A}\to {\mathbb{M}}_n\left(\mathbb{R}\right)$, we can pull back the operators norm of ${\mathbb{M}}_n\left(\mathbb{R}\right)$; see [20] (p. 97). Therefore, $\mathbb{A}$ can be endowed with a semi-multiplicative norm. Since we are working on finite-dimensional Euclidean spaces, these are inherently Banach algebras. Consequently, every absolutely convergent series converges.

In this paper, $\mathbb{A}$ is a unital associative algebra that, as a linear space, is ${\mathbb{R}}^n$, and x denotes the $\mathbb{A}$-variable, which is an n-tuple of real variables. Therefore, $\mathbb{A}$ will be a Banach algebra endowed with the semi-multiplicative norm mentioned above.

2.2. $\mathbb{A}$-Polynomial and $\mathbb{A}$-Power Series

Definition 1.

We define $·:{\mathbb{A}}^{m+1}\times {\mathbb{A}}^m\to \mathbb{A}$ by

$$A·B=\left(\prod _{i=1}^m{a}_i{b}_i\right)a_{m+1}=a_1{b}_2{a}_2{b}_2\dots a_m{b}_m{a}_{m+1},$$

for $A=(a_1,\dots ,a_{m+1})$ and $B=(b_1,\dots ,b_m)$.

An $\mathbb{A}$-monomial of degree 0 is an element $a\in \mathbb{A}$. Hence, an $\mathbb{A}$-polynomial function of degree 0 is given by

$$p_0\left(x\right)=a.$$

(2)

An $\mathbb{A}$-monomial of degree 1 is an expression $axb$, where $a,b\in \mathbb{A}$, and an $\mathbb{A}$-polynomial function of degree 1 is a function defined by a finite sum of $\mathbb{A}$-monomials of degree 1 and an $\mathbb{A}$-polynomial function of degree 0. Thus,

$$p_1\left(x\right)=a+a_{1,1,1}x{a}_{1,1,2}+a_{1,2,1}x{a}_{1,2,2}+\dots +a_{1,k_1,1}x{a}_{1,k_1,2}.$$

(3)

An $\mathbb{A}$-monomial of degree 2 is an expression $axbxc$, where $a,b,c\in \mathbb{A}$, and an $\mathbb{A}$-polynomial function of degree 2 is a function defined by a finite sum of $\mathbb{A}$-monomials of degree 2 and an $\mathbb{A}$-polynomial function of degree 1. Consequently,

$$p\left(x\right)=p_1\left(x\right)+a_{2,1,1}x{a}_{2,1,2}x{a}_{2,1,3}+\dots +a_{3,k_2,1}x{a}_{3,k_2,2}x{a}_{3,k_2,3}.$$

(4)

In this way, $\mathbb{A}$-monomial of degree m is defined by

$$\left(\prod _{i=1}^m{a}_{m,j,i}x\right)a_{m,j,m+1},$$

an homogeneous polynomial function of degree m is defined by

$$p_m\left(x\right)=\sum _{j=1}^{k_m}\left(\prod _{i=1}^m{a}_{m,j,i}x\right)a_{m,j,m+1},$$

(5)

and a $\mathbb{A}$-polynomial function of degree l is defined by

$$p\left(x\right)=a+\sum _{m=1}^l\sum _{j=1}^{k_m}\left(\prod _{i=1}^m{a}_{m,j,i}x\right)a_{m,j,m+1},$$

(6)

where $k_m$ is the number of $\mathbb{A}$-monomials of degree m. By using notations

$$A_{m,j}=\left(a_{m,j,1},\dots ,a_{m,j,m+1}\right)\in {\mathbb{A}}^{m+1},$$

(7)

and

$$X_m=(x,\dots ,x)\in {\mathbb{A}}^m,$$

(8)

we have

$$p\left(x\right)=a+\sum _{m=1}^l\sum _{j=1}^{k_m}{A}_{m,j}·X_m.$$

(9)

Since a product by a constant a in a $\mathbb{A}$-monomial expression is linear, it can be represented by a product by a matrix $A\in M(l,\mathbb{R})$ by the left. Thus, $\mathbb{A}$-monomials of the form $A_{m,j}·X_m$, $m>0$, can be written by $A{x}^m$ for a matrix A, that is, $A{x}^m=A_{m,j}·X_m$, for $a_{m,j,i}\in \mathbb{A}$. Therefore,

$$p\left(x\right)=a+\sum _{m=1}^l{M}_m{x}^m.$$

(10)

A left or right $\mathbb{A}$-rational function is defined by ${\left(Q\left(x\right)\right)}^{-1}P\left(x\right)$ or $P\left(x\right){\left(Q\left(x\right)\right)}^{-1}$, respectively, where P and Q are $\mathbb{A}$-polynomial functions.

If we consider infinite sums in (9), we obtain functions defined by $\mathbb{A}$-power series which have the form

$$s\left(x\right)=a+\sum _{m=1}^{\infty }\sum _{j=1}^{k_m}\left(\prod _{i=1}^m{a}_{m,j,i}x\right)a_{m,j,m+1}=a+\sum _{m=1}^{\infty }\sum _{j=1}^{k_m}{A}_{m,j}·X_m,$$

(11)

defined in the set where they are convergent. By using matrices $M_m\in {\mathbb{M}}_n\left(\mathbb{R}\right)$, the series (11) can be expressed by

$$s\left(x\right)=a+\sum _{m=1}^{\infty }{M}_m{x}^m,$$

(12)

modulus transpositions.

2.3. Convergence of $\mathbb{A}$-Power Series

We say that the series (11) is absolutely convergent if

$$\sum _{m=1}^{\infty }\sum _{j=1}^{k_m}\left(\prod _{i=1}^{m+1}\parallel a_{m,j,i}\parallel \right)x^m,$$

converges. We also say that the series (12) is absolutely convergent if

$$\sum _{m=1}^{\infty }\parallel M_m\parallel x^m,$$

converges. Therefore, the absolute convergence of series (11) implies the absolute convergence of the series (12).

Theorem 1.

Consider $\mathbb{A}$-series (12) and

$$\sum _{m=0}^{\infty }{r}_m{x}^m,r_m=\parallel M_m\parallel .$$

(13)

Let $l={\overline{lim}}_{m\to \infty }{r}_m^{{\displaystyle \frac{1}{k}}}$. Then,

(1) If $l=0$, the series (12) and (13) converge for all $x\in \mathbb{A}$.

(2) If $0<l<\infty$ sets $r=1/l$. Then, the series (12) and (13) converge for $\parallel x\parallel <r$.

Proof. 

The affirmations on the convergence of (13) can be proved in a similar way to the case of series on $\mathbb{C}$. Thus, the convergence of (12) can be obtained immediately. □

For the complex case, r given in Theorem 1 is called the radius of convergence of the series. Let us consider $\mathbb{A}=M_2\left(\mathbb{R}\right)$ and $M_m$ as the matrix with ${\left[M_m\right]}_{11}=m^m$ and zero otherwise. Then $l=\infty$ and $r=0$. However, in this case, for the matrix X with ${\left[X\right]}_{22}=1$ and zero otherwise, we have the convergence of ${\sum }_{m=0}^{\infty }{M}_m{X}^m$. On the other hand, if $M_m$ is the matrix with ${\left[M_m\right]}_{11}=1$ and zero otherwise, then $l=1$ and $r=1$. However, in this case, for the matrix X with ${\left[X\right]}_{22}=2$ and zero otherwise, we have the convergence of ${\sum }_{m=0}^{\infty }{M}_m{X}^m$. In this case, $\parallel M_m\parallel =2$. Therefore, in general, r can not be a convergence radius for associative algebras.

3. $\mathbb{A}$-Differentiability on Associative Algebras

3.1. $\mathbb{A}$-Differentiability

The Fréchet derivative of a function $f:U\subset {\mathbb{R}}^m\to {\mathbb{R}}^n$ at a point x is denoted by $d{f}_x$. We will refer to the Fréchet derivative as the usual derivative. Let us remember that $d{f}_x:{\mathbb{R}}^m\to {\mathbb{R}}^n$ is a linear transformation that satisfies

$$\underset{h\to 0}{lim}{\displaystyle \frac{\parallel f(x+h)-f\left(x\right)-d{f}_x\left(h\right)\parallel }{\parallel h\parallel }}=0,$$

(14)

where we denote norms on ${\mathbb{R}}^m$ and on ${\mathbb{R}}^n$ in the same way.

With respect to the canonical basis of ${\mathbb{R}}^m$, the linear transformation $d{f}_x$ is given by the Jacobain matrix $J{f}_x$. If the k-order derivative $d^k{f}_x$ exists in x, it is a k-tensor $d^k{f}_x:{\left({\mathbb{R}}^m\right)}^k\to {\mathbb{R}}^n$. In this work, we will generally consider functions $f:U\subset {\mathbb{R}}^n\to {\mathbb{R}}^n$ and their derivatives $d{f}_x$, $d^2{f}_x$, etc. We use the notation $C^k\left(U\right)$ for the set of all functions $f:U\subset {\mathbb{R}}^n\to {\mathbb{R}}^n$ that have a continuous derivative of k-order for all x in an open set U, where $C^0\left(U\right)$ denotes the set of continuous functions on U.

Definition 2.

Let $f:U\to \mathbb{A}$ be a differentiable function in the usual sense on an open set $U\subset \mathbb{A}$. We say f is $\mathbb{A}$-differentiable on U if there is an index set I, where $I=\{1,2,\dots ,m\}$ if I is finite, $I=\mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers, if I is infinite, and there are functions $a_i,b_i:U\to {\mathbb{R}}^n$ for $i\in I$ such that

$$d{f}_x\left(v\right)=\sum _{i\in I}{a}_i\left(x\right)v{b}_i\left(x\right).$$

(15)

for all $v\in \mathbb{A}$ and all $x\in U$. The $\mathbb{A}$-derivative of f is the usual derivative $df$.

Linear combinations of $\mathbb{A}$-differentiable functions are $\mathbb{A}$-differentiable functions.

Let us denote by $f_{x_i}$ the directional derivative $f_{x_i}$ in the direction $e_i$ of a differentiable function in the usual sense $f:U\to \mathbb{A}$ on an open set $U\subset \mathbb{A}$. We have the following trivial result.

Corollary 1.

Let $f:U\to \mathbb{A}$ be a differentiable function in the usual sense on an open set $U\subset \mathbb{A}$. Then, f is $\mathbb{A}$-differentiable if and only if there is an index set I, as well as functions $a_i,b_i:U\to {\mathbb{R}}^n$ for $i\in I$ such that

$$f_{x_i}=\sum _{i\in I}{a}_i\left(x\right)e_i{b}_i\left(x\right),i\in \{1,2,\dots ,n\},$$

(16)

where $\{e_1,\dots ,e_n\}$ is the standard basis of ${\mathbb{R}}^n$.

Proof. 

If f is $\mathbb{A}$-differentiable, then there are functions $a_i,b_i:U\to {\mathbb{R}}^n$ such that (15) is satisfied. Consequently, (16) is satisfied.

Now, suppose that (16) is satisfied. Then, there are functions $a_i,b_i:U\to {\mathbb{R}}^n$ such that (16) is satisfied. Therefore,

$$d{f}_x\left(v\right)=\sum _{i=1}^n{v}_{i}d{f}_x\left(e_i\right)=\sum _{i=1}^n{v}_i{f}_{x_i}=\sum _{i=1}^n{v}_i\sum _{i\in I}{a}_i\left(x\right)e_i{b}_i\left(x\right)=\sum _{i\in I}{a}_i\left(x\right)v{b}_i\left(x\right),$$

for all $v\in {\mathbb{R}}^n$ and $x\in U$. Therefore, f is $\mathbb{A}$-differentiable. □

3.2. $\mathbb{A}$-Product Rule

For $f\left(x\right)=c$, where $c\in \mathbb{A}$ is a constant, we set $a_1\left(x\right)=b_1\left(x\right)=0$. Then

$$d{f}_x\left(v\right)=a_1\left(x\right)v{b}_1\left(x\right).$$

(17)

Consequently, f is $\mathbb{A}$-differentiable and $d{f}_x\left(v\right)=0$ for all $v\in \mathbb{A}$ and all $x\in \mathbb{A}$.

For $f\left(x\right)=cx$, where $c\in \mathbb{A}$ is a constant, we set $a_1\left(x\right)=c$, $b_1\left(x\right)=e$. Then

$$d{f}_x\left(v\right)=a_1\left(x\right)v{b}_1\left(x\right).$$

(18)

Hence, f is $\mathbb{A}$-differentiable and $d{f}_x\left(v\right)=cv$ for all $v\in \mathbb{A}$ and all $x\in \mathbb{A}$.

In the following, we have the $\mathbb{A}$-product rule of differentiation.

Proposition 1.

Let f and g be $\mathbb{A}$-differentiable functions on an open set $U\subset \mathbb{A}$. Then, the $\mathbb{A}$-product $fg$ of f and g is $\mathbb{A}$-differentiable, and it satisfies the $\mathbb{A}$-product rule of differentiation

$$d{\left(fg\right)}_x\left(v\right)=d{f}_x\left(v\right)g\left(x\right)+f\left(x\right)d{g}_x\left(v\right),v\in \mathbb{A},x\in U.$$

(19)

Proof. 

Using the properties of a semi-multiplicative norm, the proof of equality (19) can be done in a similar way to the case of a single variable. Since f and g are $\mathbb{A}$-differentiable functions,

$$d{f}_x\left(v\right)=\sum _{i\in I}{a}_{1i}\left(x\right)v{b}_{1i}\left(x\right),d{g}_x\left(v\right)=\sum _{j\in j}{a}_{2i}\left(x\right)v{b}_{2i}\left(x\right).$$

We define $a_i\left(x\right)=a_{1i}\left(x\right)$ and $b_i\left(x\right)=b_{1i}\left(x\right)g\left(x\right)$ for $i\in I$. Let $m_1$ be the cardinality of I, and let $m_2$ the cardinality of J. We define $a_{m+j}\left(x\right)=f\left(x\right)a_{2j}\left(x\right)$ and $b_{m+j}\left(x\right)=b_{2j}\left(x\right)g\left(x\right)$ for $j\in J$. If $K=\{1,2,\dots ,m_1+m_2\}$, then

$$d{\left(fg\right)}_x\left(v\right)=\sum _{i\in K}{a}_i\left(x\right)v{b}_i\left(x\right).$$

Therefore, $fg$ is $\mathbb{A}$-differentiable. □

Proposition 2.

The function $f\left(x\right)=c{x}^2$ is $\mathbb{A}$-differentiable and $d{f}_x\left(v\right)=cxv+cvx$.

Proof. 

Consider $h\left(x\right)=cx$, $k\left(x\right)=x$, $a_1\left(x\right)=c$, $b_1\left(x\right)=x$, $a_2\left(x\right)=cx$, and $b_2\left(x\right)=e$. Applying Proposition 1

$$\begin{array}{cc}\hfill d{f}_x\left(v\right)& =d{h}_x\left(v\right)k\left(x\right)+h\left(x\right)d{k}_x\left(v\right)\hfill \\ \hfill & =cvx+cxv\hfill \\ \hfill & =a_1\left(x\right)v{b}_1\left(x\right)+a_2\left(x\right)v{b}_2\left(x\right).\hfill \end{array}$$

Therefore, the proof is finished. □

Let ${\sigma }_{m,k}:{\left({\mathbb{R}}^n\right)}^m\to {\left({\mathbb{R}}^n\right)}^m$ be the transformation

$${\sigma }_{m,k}(x,\dots ,x,v_1,\dots ,v_k)=\sum _{P\in \mathcal{P}(m,k)}P(x,\dots ,x,v_1,\dots ,v_k),$$

(20)

where $\mathcal{P}(m,k)$ is the set of all the permutations of $(x,\dots ,x,v_1,\dots ,v_k)$, where the variable x appears $(m-k)$ times. Let $P_{m,k}:{\left({\mathbb{R}}^n\right)}^m\to {\mathbb{R}}^n$ be the transformation

$$P_{m,k}(x,\dots ,x,v_1,\dots ,v_k)=\pi \left({\sigma }_{m,k}(x,\dots ,x,v_1,\dots ,v_k)\right),$$

where $\pi :{\left({\mathbb{R}}^n\right)}^m\to {\mathbb{R}}^n$ is defined by the $\mathbb{A}$ product $\pi (x_1,x_2,\dots ,x_m)=x_1{x}_2\dots x_m$. The function $P_{m,k}$ is k linear for $k<m-1$ and m linear for $k=m-1$, $k=m$. $P_{m,m}$ does not depend on x. For simplicity, we use

$${\sigma }_{m,k}(x,v_1,\dots ,v_k)={\sigma }_{m,k}(x,\dots ,x,v_1,\dots ,v_k),$$

and

$$P_{m,k}(x,v_1,\dots ,v_k)=P_{m,k}(x,\dots ,x,v_1,\dots ,v_k).$$

Thus, the polynomial $P_{m,k}(x,v_1,\dots ,v_k)$ has ${\displaystyle \frac{m!}{(m-k)!}}$ monomials. Therefore, $P_{2,1}(x,v)=vx+xv$, and

$$P_{3,2}(x,v_1,v_2)=v_1{v}_{2}x+v_{1}x{v}_2+x{v}_1{v}_2+v_2{v}_{1}x+v_{2}x{v}_1+x{v}_2{v}_1.$$

(21)

We also use

$$P_{m,k}(x,v)=P_{m,k}(x,v,\dots ,v).$$

Proposition 3.

The function $f\left(x\right)=x^m$ is $\mathbb{A}$-differentiable and $d{f}_x\left(v\right)=P_{m,1}(x,v)$.

Proof. 

We have that $d{h}_x\left(v\right)=P_{1,1}(x,v)$ for $h\left(x\right)=x$. By Proposition 2, $d{h}_x\left(v\right)=P_{2,1}(x,v)$ for $h\left(x\right)=x^2$. We assume that $d{h}_x\left(v\right)=P_{k,1}(x,v)$ for $h\left(x\right)=x^k$. Let us consider $h\left(x\right)=x^{k+1}$ and $g\left(x\right)=x^k$, so $h\left(x\right)=xg\left(x\right)$ and $d{h}_x\left(v\right)=vh\left(x\right)+xd{g}_x\left(v\right)=v{x}^k+x{P}_{k,1}(x,v)$. Therefore, $d{h}_x\left(v\right)=P_{k+1,1}(x,v)$. □

If $f\left(x\right)=x^m$, by Proposition 3, we have

$$D{f}_x\left(e\right)=m{x}^{m-1}.$$

(22)

In the following example, we give linear and quadratic local approximations of function $f\left(x\right)=x^3$ in $\mathbb{A}$.

Example 1.

The best affine approximation of $f\left(x\right)=x^3$ around the point $x_0$ is

$$L\left(v\right)=x_0^3+P_{3,1}(x_0,v)=x_0^3+(v{x}_0^2+x_{0}v{x}_0+x_0^{2}v).$$

Consequently,

$${(x_0^3+h)}^3\simeq L\left(h\right)=x_0^3+(h{x}_0^2+x_{0}h{x}_0+x_0^{2}h).$$

The best quadratic approximation of $f\left(x\right)=x^3$ around the point $x_0$ is

$$\begin{array}{cc}\hfill Q(v,v)& =x_0^3+P_{3,1}(x_0,v)+{\displaystyle \frac{P_{3,2}(x_0,v)}{2}}\hfill \\ \hfill & =x_0^3+(v{x}_0^2+x_{0}v{x}_0+x_0^{2}v)+(v^2{x}_0+v{x}_{0}v+x_0{v}^2).\hfill \end{array}$$

Therefore,

$${(x_0+h)}^3\simeq Q(h,h)=x_0^3+(h{x}_0^2+x_{0}h{x}_0+x_0^{2}h)+(h^2{x}_0+h{x}_{0}h+x_0{h}^2).$$

3.3. $\mathbb{A}$-Quotient Rules of Differentiation

The function $f:{\mathbb{A}}^{\ast }\to \mathbb{A}$, $f\left(x\right)=x^{-m}$, for $m\in \mathbb{N}$, is differentiable in the usual sense since its components are rational functions. Its derivative is given in the following result.

Proposition 4.

The function $f\left(x\right)=x^{-m}$ is $\mathbb{A}$-differentiable over the regular set ${\mathbb{A}}^{\ast }$ of $\mathbb{A}$, and its derivative $d{f}_x$ is given by

$$d{f}_x\left(v\right)=-x^{-m}{P}_{m,1}(x,v)x^{-m}.$$

(23)

Proof. 

Let $f\left(x\right)=x^{-m}$ and $g\left(x\right)=x^m$. Then $\left(fg\right)\left(x\right)=e$, by applying the $\mathbb{A}$-product rule

$$\begin{array}{cc}\hfill 0=d{\left(fg\right)}_x\left(v\right)& =d{f}_x\left(v\right)g\left(x\right)+f\left(x\right)d{g}_x\left(v\right)\hfill \\ \hfill & =d{f}_x\left(v\right)x^m+x^{-m}{P}_{m,1}(x,v).\hfill \end{array}$$

From this, we obtain (23). Therefore, f is $\mathbb{A}$-differentiable and satisfies (23). □

If $f\left(x\right)=x^{-m}$, by Proposition 4, we have

$$d{f}_x\left(e\right)=-m{x}^{-(m+1)}.$$

(24)

If $p\left(x\right)$ and $q\left(x\right)$ are polynomials, we call both the left rational function defined by p and q to the function

$$f\left(x\right)=p\left(x\right){\left(q\left(x\right)\right)}^{-1},$$

and the right rational function defined by p and q to the function

$$g\left(x\right)={\left(q\left(x\right)\right)}^{-1}p\left(x\right),$$

functions defined on the set $q^{-1}\left({\mathbb{A}}^{\ast }\right)$.

The left and right $\mathbb{A}$-quotient rules are given in the following result.

Proposition 5.

Let $p,q:U\to \mathbb{A}$ be differentiable functions defined on an open set $U\subset \mathbb{A}$, where q has an image in ${\mathbb{A}}^{\ast }$. Then the left and right quotients $p\left(x\right)q{\left(x\right)}^{-1}$, $q{\left(x\right)}^{-1}p\left(x\right)$ are differentiable functions and satisfy the corresponding left and right $\mathbb{A}$-quotient rules:

(1) If $f\left(x\right)=p\left(x\right)q{\left(x\right)}^{-1}$, then

$$d{f}_x\left(v\right)=q{\left(x\right)}^{-1}[q\left(x\right)d{p}_x\left(v\right)-q\left(x\right)p\left(x\right)q{\left(x\right)}^{-1}d{q}_x\left(v\right)]q{\left(x\right)}^{-1}.$$

(25)

(2) If $g\left(x\right)=q{\left(x\right)}^{-1}p\left(x\right)$, then

$$d{g}_x\left(v\right)=q{\left(x\right)}^{-1}[d{p}_x\left(v\right)q\left(x\right)-d{q}_x\left(v\right)q{\left(x\right)}^{-1}p\left(x\right)q\left(x\right)]q{\left(x\right)}^{-1}.$$

(26)

Proof. 

The chain rule is valid for the usual differentiability. We apply the $\mathbb{A}$-product rule, Proposition 4 for $m=1$, and the chain rule to obtain (25) and (26). The $\mathbb{A}$-differentiability of f is similar to that of $fg$ in Proposition 1. Therefore, a proof can be obtained. □

3.4. $\mathbb{A}$-Chain Rule

The chain rule is valid for the usual differentiability. In the following result, we give the $\mathbb{A}$-chain rule.

Proposition 6.

Let $f,g$ be $\mathbb{A}$-differentiable functions for which $f\left(x\right)$ belongs to the domain of g for all x in the domain of f. If

$$d{f}_x\left(v\right)=\sum _{i\in I}{a}_{1i}\left(x\right)v{b}_{1i}\left(x\right),d{g}_y\left(v\right)=\sum _{j\in J}{a}_{2j}\left(y\right)v{b}_{2j}\left(y\right)$$

then $g\circ f$ is an $\mathbb{A}$-differentiable function ($g\circ f$ denotes g composed with f, as long as it can be defined), and

$$d{(g\circ f)}_x\left(v\right)=\sum _{(i,j)\in I\times J}{a}_{2j}\left(f\left(x\right)\right)a_{1i}\left(x\right)v{b}_{1i}\left(x\right)b_{2j}\left(f\left(x\right)\right).$$

(27)

Proof. 

We have

$$\begin{array}{cc}\hfill d{g}_{f\left(x\right)}\circ d{f}_x\left(v\right)& =\sum _{j\in J}{a}_{2j}\left(f\left(x\right)\right)\left(\sum _{i\in I}{a}_{1i}\left(x\right)v{b}_{1i}\left(x\right)\right)b_{2j}\left(f\left(x\right)\right)\end{array}$$

(28)

$$\begin{array}{cc}\hfill & =\sum _{(i,j)\in I\times J}{a}_{2j}\left(f\left(x\right)\right)a_{1i}\left(x\right)v{b}_{1i}\left(x\right)b_{2j}\left(f\left(x\right)\right).\hfill \end{array}$$

(29)

Take $a_{3ij}\left(x\right)=a_{2j}\left(f\left(x\right)\right)a_{1i}\left(x\right)$ and $b_{3ij}\left(x\right)=b_{1i}\left(x\right)b_{2j}\left(f\left(x\right)\right)$. We obtain

$$d{g}_{f\left(x\right)}\circ d{f}_x\left(v\right)=\sum _{(i,j)\in I\times J}{a}_{3ij}\left(x\right)v{b}_{3ij}\left(x\right).$$

By the usual chain rule, $d{(g\circ f)}_x=d{g}_{f\left(x\right)}\circ d{f}_x$. Therefore, we obtain (27). □

Example 2.

Consider the function

$$h\left(x\right)={(axbxc+kxl+m)}^{-1},$$

defined on the set

$$\{x\in \mathbb{A}:axbxc+kxl+m\in {\mathbb{A}}^{\ast }\}.$$

Let $f\left(x\right)=axbxc+kxl+m$ and $g\left(y\right)=y^{-1}$ be defined on ${\mathbb{A}}^{\ast }$. Hence,

$$d{f}_x\left(v\right)=avbxc+axbvc+kvl,d{g}_y\left(w\right)=-y^{-1}w{y}^{-1}.$$

Then, $h=g\circ f$, and according to the chain rule,

$$d{h}_x\left(v\right)=d{g}_{f\left(x\right)}d{f}_x\left(v\right)=-{\left(f\left(x\right)\right)}^{-1}(avbxc+axbvc+kvl){\left(f\left(x\right)\right)}^{-1}.$$

3.5. $\mathbb{A}$-Polynomials and $\mathbb{A}$-Power Series

In this section, we follow some results (and their proofs) of Chapter 2 of [21]. We show that polynomials and series are differentiable. If the series (11)

$$s\left(x\right)=a+\sum _{m=1}^{\infty }\sum _{j=1}^{k_m}\left(\prod _{i=1}^m{a}_{m,j,i}x\right)a_{m,j,m+1}=a+\sum _{m=1}^{\infty }\sum _{j=1}^{k_m}{A}_{m,j}·X_m$$

converges in some ball $B_R\left(0\right)\subset \mathbb{A}$, then the series

$$\sum _{m=1}^{\infty }\sum _{j=1}^{k_m}{A}_{m,j}·{\sigma }_{m,1}(x,v)$$

converges in $B_R\left(0\right)\subset \mathbb{A}$, since

$$\underset{m\to \infty }{lim\; sup}{\left(m{r}_m\right)}^{{\displaystyle \frac{1}{m-1}}}=\underset{m\to \infty }{lim\; sup}{\left({\left(m{r}_m\right)}^{{\displaystyle \frac{1}{m}}}\right)}^{{\displaystyle \frac{m}{m-1}}}=\underset{m\to \infty }{lim\; sup}{\left(r_m\right)}^{{\displaystyle \frac{1}{m}}},$$

(30)

where $r_m={\sum }_{j=1}^{k_m}{\prod }_{i=1}^{m+1}\parallel a_{m,j,i}\parallel$.

We consider polynomial functions of the form (6) that can be written as (9).

Proposition 7.

The $\mathbb{A}$-polynomial function (9),

$$p\left(x\right)=a+\sum _{m=1}^l\sum _{j=1}^{k_m}{A}_{m,j}·X_m,$$

where $A_{m,j}$ and $X_m$ are defined by (7) and (8), has a k-order $\mathbb{A}$-derivative $d^k{p}_x$ given by

$$d^k{p}_x(v_1,\dots ,v_k)=\sum _{m=k}^l\sum _{j=1}^{k_m}{A}_{m,j}·{\sigma }_{m,k}(x,v_1,\dots ,v_k),$$

(31)

where ${\sigma }_{m,k}$ is defined in (20).

Proof. 

By the $\mathbb{A}$-product rule (1) we have that

$$d{(A_{m,j}·X_m)}_x\left(v\right)=A_{m,j}·{\sigma }_{m,k}(x,v).$$

By the linearity of the operator d, $f\mapsto df$, the proof is obtained for $k=1$. Again, by using the $\mathbb{A}$-product rule applied to function

$$d^k{(A_{m,j}·X_m)}_x(v_1,\dots ,v_k)=A_{m,j}·{\sigma }_{m,k}(x,v_1,\dots ,v_k),$$

we obtain

$$\begin{array}{ccc}\hfill d^{k+1}{(A_{m,j}·X_m)}_x(v_1,\dots ,v_{k+1})& =& d{\left[d^k{(A_{m,j}·X_m)}_x(v_1,\dots ,v_k)\right]}_x\left(v_{k+1}\right)\hfill \\ & =& A_{m,j}·{\sigma }_{m,k+1}(x,v_1,\dots ,v_{k+1}).\hfill \end{array}$$

Therefore, by induction and linearity of the derivative d, the proof is obtained. □

The following result can be used to calculate higher-order derivatives of the function $f\left(x\right)=x^m$. It is a generalization of the power rule stated in Proposition 3.

Corollary 2.

The k-th derivative of $x^m$ is given by

$$d^k{\left(x^m\right)}_x(v_1,\dots ,v_k)=P_{m,k}(x,v_1,\dots ,v_k),$$

and

$$P_{m,k}(x,v)=k!P_{m,m-k}(v,x),$$

for $k=1,\dots ,m$.

Example 3.

By Corollary 2, $P_{3,2}(x_0,v)=P_{3,1}(v,x_0)$. Hence, the quadratic approximation $Q(v,v)$ of $f\left(x\right)=x^3$ around $x_0)$ given in Example 1 can be written as

$$Q(v,v)=x_0^3+P_{3,1}(x_0,v)+P_{3,1}(v,x_0).$$

Corollary 3.

Let $p\left(x\right)$ be a polynomial function of the m degree of the form

$$p\left(x\right)=a+a_{1}x+\dots +a_m{x}^m.$$

For $k\in \mathbb{N}$, p has a k-order $\mathbb{A}$-derivative, and $d^k{p}_x:{\mathbb{A}}^k\to \mathbb{A}$ is given by

$$\begin{array}{c}{d}^k{p}_x(v_1,\dots ,v_k)=a_k{P}_{k,k}(x,v_1,\dots ,v_k)+a_{k+1}{P}_{k+1,k}(x,v_1,\dots ,v_k)\hfill \\ \hfill +\dots +a_m{P}_{m,k}(x,v_1,\dots ,v_k).\end{array}$$

(32)

Then, the first-order derivative $d{p}_x:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is given by

$$d{p}_x\left(v\right)=a_1{P}_{1,1}\left(v\right)+a_2{P}_{2,1}(x,v)+\dots +a_m{P}_{m,1}(x,v).$$

(33)

The absolutely convergent $\mathbb{A}$-series are $\mathbb{A}$-differentiable.

Theorem 2.

Consider ${\sum }_{k=0}^{\infty }{M}_k{x}^k$ defined in $\mathbb{A}$, for $M_k\in {\mathbb{M}}_n\left(\mathbb{R}\right)$, such that $l={\overline{lim}}_{m\to \infty }{\parallel M_k\parallel }^{{\displaystyle \frac{1}{k}}}$, $l<\infty$. Then, $d{f}_x$ exists and is given by

$$d{f}_x\left(v\right)=\sum _{k=1}^{\infty }{M}_k{P}_{k,1}(x,v)$$

(34)

throughout $\parallel x\parallel <r$, where $r=1/l$ if $l>0$, and throughout all x if $l=0$.

Proof. 

We use the operators norm for matrices in ${\mathbb{M}}_n\left(\mathbb{R}\right)$; thus, all of the following properties used are fulfilled. By Theorem 1, we have absolute convergence for $\parallel x\parallel <r$, where $r=1/l$ for $l\ne 0$, and for all $x\in \mathbb{A}$ if $l=0$.

Assume ${\sum }_{k=0}^{\infty }{M}_k{x}^k$ converges absolutely for all $x\in \mathbb{A}$. Then

$${\displaystyle \frac{f(x+h)-f\left(x\right)}{\parallel h\parallel }}=\sum _{k=1}^{\infty }{M}_k{\displaystyle \frac{{(x+h)}^k-x^k}{\parallel h\parallel }},$$

so

$${\displaystyle \frac{f(x+h)-f\left(x\right)}{\parallel h\parallel }}-\sum _{k=1}^{\infty }{M}_k{\displaystyle \frac{P_{k,1}(x,h)}{\parallel h\parallel }}=\sum _{k=1}^{\infty }{M}_k{b}_k,$$

where

$$b_k={(x+h)}^k-\left(x^k+P_{k,1}(x,h)\right)=\sum _{l=2}^k{P}_{k,l}(x,h),$$

(35)

where the last previous equality is satisfied since

$${(x+h)}^k=x^k+\sum _{l=1}^k{P}_{k,l}(x,h).$$

Consequently,

$$\parallel b_k\parallel \le \sum _{l=2}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{l}}\right){\parallel h\parallel }^{l-1}{\parallel x\parallel }^{k-l}\le \parallel h\parallel \sum _{l=2}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{l}}\right){\parallel x\parallel }^{k-l}={\parallel h\parallel (\parallel x\parallel +1)}^k,$$

for $\parallel h\parallel \le 1$. Hence, for $\parallel h\parallel \le 1$,

$${\displaystyle \frac{f(x+h)-f\left(x\right)-{\sum }_{k=1}^{\infty }{M}_k{P}_{k,1}(x,h)}{\parallel h\parallel }}\le \parallel h\parallel \sum _{k=2}^{\infty }\parallel M_k{\parallel (\parallel x\parallel +1)}^k\le A\parallel h\parallel ,$$

since ${\sum }_{k=0}^{\infty }{M}_k{x}^k$ is absolutely convergent for all x. Leaving $h\to 0$, we conclude that (34) is verified.

Suppose ${\sum }_{k=0}^{\infty }{M}_k{x}^k$ converges for $0\le \parallel x\parallel <r$. Let $\parallel x\parallel =r-2\delta$, with $\delta >0$, and assume that $\parallel h\parallel <\delta$. Then $\parallel x+h\parallel <r$. Similarly to the previous case

$${\displaystyle \frac{f(x+h)-f\left(x\right)}{\parallel h\parallel }}-\sum _{k=1}^{\infty }{M}_k{\displaystyle \frac{P_{k,1}(x,h)}{\parallel h\parallel }}=\sum _{k=1}^{\infty }{M}_k{b}_k,$$

where $b_k$ is given by (35). If $x=0$, $b_k=h^{k-1}$, and the proof follows easily. Otherwise, note that $P_{k,l}$ has $\left({\displaystyle \genfrac{}{}{0pt}{}{k}{l}}\right)$ monomials with x appearing exactly $(k-l)$ times and h appearing exactly $(l-1)$ times. Note then for $l\ge 2$ that

$$\left({\displaystyle \genfrac{}{}{0pt}{}{k}{l}}\right)={\displaystyle \frac{k(k-1)\dots (k-l+1)}{l!}}\le k^2\left({\displaystyle \genfrac{}{}{0pt}{}{k}{l-2}}\right).$$

Hence, for $x\ne 0$,

$$\begin{array}{cc}\hfill \parallel b_k\parallel & \le {\displaystyle \frac{k^2\parallel h\parallel }{{\parallel x\parallel }^2}}\sum _{l=2}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{l-2}}\right){\parallel h\parallel }^{l-2}{\parallel x\parallel }^{k-(l-2)}\le {\displaystyle \frac{k^2\parallel h\parallel }{{\parallel x\parallel }^2}}\sum _{l=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right){\parallel h\parallel }^j{\parallel x\parallel }^{k-j}\hfill \\ \hfill & ={\displaystyle \frac{k^2\parallel h\parallel }{{\parallel x\parallel }^2}}(\parallel x\parallel +{\parallel h\parallel )}^k\le {\displaystyle \frac{k^2\parallel h\parallel }{{\parallel x\parallel }^2}}{(R-\delta )}^k,\hfill \end{array}$$

and

$${\displaystyle \frac{\parallel f(x+h)-f\left(x\right)-{\sum }_{k=1}^{\infty }{M}_k{P}_{k,1}(x,h)\parallel }{\parallel h\parallel }}\le {\displaystyle \frac{\parallel h\parallel }{{\parallel x\parallel }^2}}\sum _{k=0}^{\infty }{k}^2\parallel M_k{\parallel (r-\delta )}^k\le A\parallel h\parallel ,$$

since $x\ne 0$ is fixed, and by (30) applied to $r_k=\parallel M_k\parallel$, the series

$$\sum _{k=0}^{\infty }{k}^2\parallel M_k\parallel x^k,$$

also converges for $\parallel x\parallel <r$. Leaving $h\to 0$, again we conclude that (34) is verified. □

4. Higher Order $\mathbb{A}$-Differentiability and Usual Line Integrals

In this work, an index set I is in the finite case $I=\{1,2,\dots ,n\}$ for some $n\in \mathbb{N}$, and in the infinite case $I=\mathbb{N}$.

4.1. $C^k(U,\mathbb{A})$-Differentiability

Definition 3.

Let $f:U\subset \mathbb{A}\to \mathbb{A}$ be $C^1\left(U\right)$ in the usual sense on an open set U. If f is $\mathbb{A}$-differentiable on U with

$$d{f}_x\left(v\right)=\sum _{i\in I}{a}_{i1}\left(x\right)v{a}_{i2}\left(x\right),$$

(36)

for all $v\in \mathbb{A}$, where $a_{i1},a_{i2}:U\to \mathbb{A}$ are $C^0\left(U\right)$, then we say f is $C^1(U,\mathbb{A})$-differentiable.

We define $C^k(U,\mathbb{A})$-differentiability inductively.

Definition 4.

If $f:U\subset \mathbb{A}\to \mathbb{A}$ is $C^k\left(U\right)$ in the usual sense on an open set U and $C^l(U,\mathbb{A})$ for $1\le l\le k$ with

$$d{f}_x^k(v_1,\dots ,v_k)=\sum _{i\in I}\left(\prod _{l=1}^k{a}_{il}\left(x\right)v_l\right)a_{ik+1}\left(x\right),$$

(37)

for all $(v_1,\dots ,v_k)\in {\mathbb{A}}^k$, where $a_{il}:U\to \mathbb{A}$ for $1\le l\le k+1$ are $C^1(U,\mathbb{A})$ and over U, then we say f is $C^{k+1}(U,\mathbb{A})$-differentiable.

$C^k(U,\mathbb{A})$ denotes the set of all $C^k(U,\mathbb{A})$-differentiable functions.

The following result aims to calculate the derivative $k+1$ of f when we know the derivative k of f applied to the elements $(v_1,\dots ,v_k)\in {\mathbb{A}}^k$. It does not consider that $d^{k+1}{f}_x$ is a symmetric (k+1)-multilinear map.

Proposition 8.

If $f:U\subset \mathbb{A}\to \mathbb{A}$ is $C^{k+1}(U,\mathbb{A})$-differentiable with

$$d^k{f}_x(v_1,\dots ,v_k)=\sum _{i\in I}\left(\prod _{l=1}^k{a}_{il}\left(x\right)v_l\right)a_{ik+1}\left(x\right),$$

(38)

for all $(v_1,\dots ,v_k)\in {\mathbb{A}}^k$, where $a_{il}:U\to \mathbb{A}$ for $1\le l\le k+1$ are $\mathbb{A}$-differentiable functions over U

$$d{a}_{ipx}\left(v_{k+1}\right)=\sum _{i\in J_{a_{ip}}}{a}_{ip,j,1}\left(x\right)v_{k+1}{a}_{ip,j,2}\left(x\right),p=1,2,\dots ,k$$

(39)

then $d^{k+1}{f}_x$ is given by

$$\begin{array}{cc}\hfill d^{k+1}{f}_x(v_1,\dots ,v_{k+1})& =\sum _{i\in I}\left(\sum _{j\in J_{a_{i1}}}{a}_{i1,j,1}\left(x\right)v_{k+1}{a}_{i1.j,2}\left(x\right)\right)\left(\prod _{l=2}^k{a}_{il}\left(x\right)v_l\right)a_{ik+1}\left(x\right)\hfill \\ \hfill & +\sum _{i\in I}{a}_{i1}\left(x\right)v_1\left(\sum _{j\in J_{a_{i2}}}{a}_{i2,j,1}\left(x\right)v_{k+1}{a}_{i2.j,2}\left(x\right)\right)\left(\prod _{l=3}^k{a}_{il}\left(x\right)v_l\right)a_{ik+1}\left(x\right)\hfill \\ \hfill & +\dots +\sum _{i\in I}\left(\prod _{l=1}^k{a}_{il}\left(x\right)v_l\right)\left(\sum _{j\in J_{a_{i(k+1)}}}{a}_{i(k+1),j,1}\left(x\right)v_{k+1}{a}_{i(k+1).j,2}\left(x\right)\right).\hfill \end{array}$$

Proof. 

Since $d^k{f}_x$ applied to $(v_1,\dots ,v_k)$ is a product of the symmetric tensor $d^k{f}_x$ by $(v_1,\dots ,v_k)\in {\mathbb{A}}^k$, we have that

$$d{\left(d^k{f}_x(v_1,\dots ,v_k)\right)}_x\left(v_{k+1}\right)=d^{k+1}{f}_x(v_1,\dots ,v_k,v_{k+1}).$$

The proof can be obtained using the $\mathbb{A}$-product rule given in Proposition 1. □

Definition 5.

If $f:U\subset \mathbb{A}\to \mathbb{A}$ is $C^k(U,\mathbb{A})$ for all $k\in \mathbb{N}$, we say f is infinitely $\mathbb{A}$-differentiable on U.

$C^{\infty }(U,\mathbb{A})$ denotes the set of all infinitely $\mathbb{A}$-differentiable functions on U.

Example 4.

Consider the exponential function $E\left(x\right)={\sum }_{k=0}^{\infty }{\displaystyle \frac{x^k}{k!}}$. Thus, by Proposition 3,

$$d{E}_x\left(v\right)=\sum _{k=1}^{\infty }{\displaystyle \frac{P_{k,1}(x,v)}{k!}}.$$

For this series, the index set I given in Definition 3 is $I=\mathbb{N}$, and the functions $a_{ij}\left(x\right)$ have the form $x^l$ for some $l\in \mathbb{N}$. Therefore, E is $C^2(\mathbb{A},\mathbb{A})$. By Corollary 2, the second derivative $d^2{E}_x$ is given by

$$d^2{E}_x(v,w)=\sum _{k=2}^{\infty }{\displaystyle \frac{P_{k,2}(x,v,w)}{k!}},$$

and the l-derivative $d^l{E}_x$ is given by

$$d^l{E}_x(v_1,\dots ,v_k)=\sum _{k=l}^{\infty }{\displaystyle \frac{P_{k,l}(x,v_1,\dots ,v_k)}{k!}}.$$

By using a semi-multiplicative norm, the convergence of these series can be established.

4.2. $\mathbb{A}$-Derivative of Analytical Functions

If f is differentiable in the usual sense at $x_0$, then

$$T\left(x\right)=f\left(x_0\right)+d{f}_{x_0}(x-x_0)$$

is the best affine approximation of f around $x_0$.

If $f:U\subset \mathbb{A}\to \mathbb{A}$ is $C^k(U,\mathbb{A})$, the series

$$T_{f,x_0}^k\left(x\right)=\sum _{k=0}^k{\displaystyle \frac{1}{k!}}{d}^k{f}_{x_0}(x-x_0,\dots ,x-x_0),$$

is called a Taylor approximation of order k of f around $x_0$ or a Taylor polynomial of order k of f around $x_0$, and if $f:U\subset \mathbb{A}\to \mathbb{A}$ is $C^{\infty }(U,\mathbb{A})$, the series

$$T_{f,x_0}\left(x\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{1}{k!}}{d}^k{f}_{x_0}(x-x_0,\dots ,x-x_0),$$

(40)

is called Taylor series of f around $x_0$.

Since we work with the usual derivative, Taylor approximations are approximations of the functions that define them.

Definition 6.

An infinitely $\mathbb{A}$-differentiable function $f:U\subset \mathbb{A}\to \mathbb{A}$ is called $\mathbb{A}$-analytical on an open set U if for any $x_0\in U$ the Taylor series (40) converges to $f\left(x\right)$ for x in a neighborhood $N_{x_0}$ of $x_0$, pointwise.

Therefore, the $\mathbb{A}$-Taylor expansion of F around x is

$$T_{f,x}(x+h)=\sum _{k=0}^{\infty }{\displaystyle \frac{1}{k!}}{d}^k{f}_x(h,\dots ,h)$$

if $x,x+h\in U$.

By Theorems 1 and 2, the series which converge absolutely for $\parallel x\parallel <r$ are analytical for $\parallel x\parallel <r$.

Consider the exponential function

$$E\left(x\right)=Exp\left(x\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{x^k}{k!}}.$$

(41)

By Theorems 2 and 3, the derivative of the exponential function (41) applied to v is given by the series

$$d{E}_x\left(v\right)=\sum _{k=1}^{\infty }{\displaystyle \frac{P_{k,1}(x,v)}{k!}}.$$

(42)

In particular, if we evaluate $d{E}_x$ at the identity e of $\mathbb{A}$, we obtain

$$d{E}_x\left(e\right)=\sum _{k=1}^{\infty }{\displaystyle \frac{P_{k,1}(x,e)}{k!}}=\sum _{k=1}^{\infty }{\displaystyle \frac{k{x}^{k-1}}{k!}}=E\left(x\right).$$

Example 5.

The best $\mathbb{A}$-affine approximation of $E\left(x\right)=Exp\left(x\right)$ around x is given by

$$E(x+h)\simeq E\left(x\right)+\sum _{k=1}^{\infty }{\displaystyle \frac{P_{k,1}(x,h)}{k!}}.$$

The second order derivative $d^2{E}_x$

$$d^2{E}_x(v,w)=\sum _{k=2}^{\infty }{\displaystyle \frac{P_{k,2}(x,v,w)}{k!}}.$$

Consequently,

$$d^2{E}_x(e,e)=\sum _{k=2}^{\infty }{\displaystyle \frac{P_{k,2}(x,e,e)}{k!}}=\sum _{k=2}^{\infty }{\displaystyle \frac{k(k-1)x^{k-2}}{k!}}=E\left(x\right).$$

4.3. The Line Integral

The line integral

$${\int }_{t_0}^t\alpha \left(s\right)ds,$$

of a function $\alpha :[a,b]\to \mathbb{A}$ is defined in the usual way as the line integral in ${\mathbb{R}}^n$. Using the $\mathbb{A}$-variable, we can calculate the line integral for families of functions.

The following proposition corresponds to the fundamental theorem of calculus.

Proposition 9.

If $\gamma :[a,b]\to \mathbb{A}$ is an integrable curve, and f is $\mathbb{A}$-differentiable, then

$${\int }_{t_0}^{t}d{f}_{\gamma \left(s\right)}\left({\gamma }^{\prime }\left(s\right)\right)ds=f\left(\gamma \left(t\right)\right)-f\left(\gamma \left(t_0\right)\right).$$

Proof. 

We have ${\displaystyle \frac{d}{dt}}f\left(\gamma \left(t\right)\right)=d{f}_{\gamma \left(t\right)}\left({\gamma }^{\prime }\left(t\right)\right)$. We apply the fundamental theorem of calculus to the component functions to obtain the proof. □

Example 6.

If $\gamma :[a,b]\to \mathbb{A}$ is an integrable curve, then

$${\int }_{t_0}^t{P}_{m,1}(\gamma \left(s\right),{\gamma }^{\prime }\left(s\right))ds={\left(\gamma \left(t\right)\right)}^m-{\left(\gamma \left(t_0\right)\right)}^m.$$

Example 7.

If $\gamma :[a,b]\to {\mathbb{A}}^{\ast }$ is an integrable curve, then

$${\int }_{t_0}^t-{\left(\gamma \left(s\right)\right)}^{-m}{P}_{m,1}(\gamma \left(s\right),{\gamma }^{\prime }\left(s\right)){\left(\gamma \left(s\right)\right)}^{-m}ds={\left(\gamma \left(t\right)\right)}^{-m}-{\left(\gamma \left(t_0\right)\right)}^{-m}.$$

We also integrate functions defined by power series.

Example 8.

Let $E:\mathbb{A}\to \mathbb{A}$ be the exponential function. If $\gamma :[a,b]\to \mathbb{A}$ is an integrable curve, then

$${\int }_{t_0}^t\sum _{k=1}^{\infty }{\displaystyle \frac{P_{k,1}(\gamma \left(s\right),{\gamma }^{\prime }\left(s\right))}{k!}}ds=E\left(\gamma \left(t\right)\right)-E\left(\gamma \left(t_0\right)\right),$$

(43)

since (42) is satisfied.

5. Quaternions $\mathbb{H}$

5.1. Clifford Algebras

We consider a Clifford algebra structure $\mathbb{A}$ on the linear space ${\mathbb{R}}^n$, where $n=2^m$ for $m\in \mathbb{N}$, with a product defined over the basis $\{e_1,\dots ,e_n\}$ for which is orthonormal for a symmetric bilinear form $h:\mathbb{A}\times \mathbb{A}\to \mathbb{R}$, having the signature $(p,q)$; $h(e_i,e_i)=1$ for $1\le i\le p$, $h(e_i,e_i)=-1$ for $p+1\le i\le n$, and $h(e_i,e_j)=0$ for $i\ne j$. We consider $e=e_1$ and use $e=1$. Then, $\mathbb{A}$ has the product

$$e_i{e}_j=-e_j{e}_i+2{\delta }_{ij},{\delta }_{ij}=h(e_i,e_j).$$

(44)

We consider the quaternion algebra $\mathbb{H}$, which is the linear space ${\mathbb{R}}^4$ with respect to the product

$$\begin{array}{ccccc}·& e_1& e_2& e_3& e_4\\ e_1& e_1& e_2& e_3& e_4\\ e_2& e_2& -e_1& e_4& -e_3\\ e_3& e_3& -e_4& -e_1& e_2\\ e_4& e_4& e_3& -e_2& -e_1\end{array}.$$

(45)

$\mathbb{H}$ is a Clifford algebra, where $h(e_1,e_1)=1$, $h(e_i,e_i)=-1$ for $i=2,\dots ,4$, and $h(e_i,e_j)=0$ for $i\ne j$. Hence, h has the signature $(1,3)$, and $\{e_1,\dots ,e_4\}$ is orthonormal with respect to h. The image of $(a,b,c,d)$ under the first fundamental representation is the matrix (1).

The quaternions as a system of numbers is usually denoted by

$$\mathbb{H}=\{w+xi+yj+zk:w,x,y,z\in \mathbb{R}\},$$

(46)

endowed with the product obtained in Table (45) by identifying 1, i, j, and k with $e_1$,..., $e_4$, respectively.

5.2. The Function $F\left(x\right)=x^3$ in $\mathbb{H}$

In this section, we consider the function $F:{\mathbb{R}}^4\to {\mathbb{R}}^4$, $F\left(x\right)=x^3$, expressed in terms of the component variables $x_1$,..., $x_4$ of the $\mathbb{A}$-variable $x=(x_1,x_2,x_3,x_4)$. This function is given by

$$\begin{array}{cc}\hfill F\left(x\right)& =(x_1^3-3x_1{x}_2^2-3x_1{x}_3^2-3x_1{x}_4^2)e_2+(-x_2^3+3x_1^2{x}_2-x_2{x}_3^2-x_2{x}_4^2)e_2\hfill \\ \hfill & +(-x_3^3+3x_1^2{x}_3-x_2^2{x}_3-x_3{x}_4^2)e_3+(-x_4^3+3x_1^2{x}_4-x_2^2{x}_4-x_3^2{x}_4)e_4.\hfill \end{array}$$

The Jacobian matrix at $x=(x_1,x_2,x_3,x_4)$ is

$$J{F}_x=3x_1^{2}I+\left(\begin{array}{cccc}-3x_2^2-3x_3^2-3x_4^2& -6x_1{x}_2& -6x_1{x}_3& -6x_1{x}_4\\ 6x_1{x}_2& -3x_2^2-x_3^2-x_4^2& -2x_2{x}_3& -2x_2{x}_4\\ 6x_1{x}_3& -2x_2{x}_3& -3x_3^2-x_2^2-x_4^2& -2x_3{x}_4\\ 6x_1{x}_4& -2x_2{x}_4& -2x_{3}2x_4& -3x_4^2-x_2^2-x_3^2\end{array}\right)$$

where $I\in M(4,\mathbb{R})$ is the identity matrix. Thus, the derivative of F is given by $d{F}_x\left(v\right)=J{F}_{x}v$. If we use Theorem 3, we have

$$\begin{array}{cc}\hfill d{F}_x\left(v\right)& =v{x}^2+xvx+x^{2}v\hfill \\ \hfill & =\left(\begin{array}{c}3v_1{x}_1^2-3v_1{x}_2^2-6v_2{x}_1{x}_2-3v_1{x}_3^2-6v_3{x}_1{x}_3-3v_1{x}_4^2-6v_4{x}_1{x}_4\\ 3v_2{x}_1^2-3v_2{x}_2^2+6v_1{x}_1{x}_2-v_2{x}_3^2-2v_3{x}_2{x}_3-v_2{x}_4^2-2v_4{x}_2{x}_4\\ 3v_3{x}_1^2-v_3{x}_2^2-3v_3{x}_3^2+6v_1{x}_1{x}_3-2v_2{x}_2{x}_3-v_3{x}_4^2-2v_4{x}_3{x}_4\\ 3v_4{x}_1^2-v_4{x}_2^2-v_4{x}_3^2-3v_4{x}_4^2+6v_1{x}_1{x}_4-2v_2{x}_2{x}_4-2v_3{x}_3{x}_4\end{array}\right),\hfill \end{array}$$

where the last equality follows from applying the first fundamental representation and performing matrix operations in $M(4,\mathbb{R})$.

6. Pre-Twisted Differentiability

Recently, pre-twisted calculus has been introduced for the case of commutative algebras; see [6] and references therein. We now present a definition of pre-twisted differentiability in the non-commutative setting.

6.1. Definition of $\phi \left(\mathbb{A}\right)$-Differentiability

Recall that $\mathbb{A}$ is the linear space ${\mathbb{R}}^n$ equipped with an associative algebra structure. We consider functions $f,\phi :U\subset {\mathbb{R}}^k\to \subset \mathbb{A}$, where U is an open set.

Definition 7.

Let $f,\phi :U\to \mathbb{A}$ be differentiable functions in the usual sense on an open set $U\subset {\mathbb{R}}^k$. We say f is $\phi \left(\mathbb{A}\right)$-differentiable on U if there is an index set I, where $I=\{1,2,\dots ,m\}$ if I is finite, $I=\mathbb{N}$, where $\mathbb{N}$ is the set of the natural numbers, if I is infinite, and there are functions ${\alpha }_i,{\beta }_i:U\to \mathbb{A}$ for $i\in I$ such that

$$d{f}_x\left(v\right)=\sum _{i\in I}{\alpha }_i\left(x\right)d{\phi }_x\left(v\right){\beta }_i\left(x\right),$$

(47)

for all $v\in {\mathbb{R}}^k$ and all $x\in U$. The $\phi \left(\mathbb{A}\right)$-derivative of f is the usual derivative $d{f}_x$.

In the following example, we will see that the function ${\left(\phi \left(x\right)\right)}^m$ is $\phi \left(\mathbb{A}\right)$-differentiable if $\phi$ is differentiable in the usual sense.

We have the following result.

Proposition 10.

Let g be a $\mathbb{A}$-differentiable function with

$$d{g}_y\left(w\right)=\sum _{i\in I}{a}_i\left(x\right)w{b}_i\left(x\right).$$

If the domain of g contains the image of a differentiable function in the usual sense $\phi :U\subset {\mathbb{R}}^k\to \mathbb{A}$, where U is an open set, then the function $f=g\circ \phi$ is a $\phi \left(\mathbb{A}\right)$-differentiable function, and

$$d{f}_x\left(v\right)=\sum _{i\in I}{a}_i\left(\phi \left(x\right)\right)d{\phi }_x\left(v\right)b_i\left(\phi \left(x\right)\right),$$

for all x in the domain of φ and $v\in {\mathbb{R}}^k$.

Proof. 

We apply the chain rule for the usual differentiability and the definition of $\phi \left(A\right)$-differentiability. Therefore, we obtain the proof. □

We have the following examples.

Example 9.

Let $\phi :U\subset {\mathbb{R}}^k\to \mathbb{A}$ be a differentiable function in the usual sense on an open set U, and $f\left(x\right)={\left(\phi \left(x\right)\right)}^m$ for a positive integer m. Then, $f=p\circ \phi$, where $p\left(x\right)=x^m$, and by the chain rule

$$d{f}_x\left(v\right)=d{p}_{\phi \left(x\right)}d{\phi }_x\left(v\right)=P_{m,1}(\phi \left(x\right),d{\phi }_x\left(v\right)).$$

Example 10.

Let $\phi :U\subset {\mathbb{R}}^k\to \mathbb{A}$ be a differentiable function in the usual sense on an open set U with image in ${\mathbb{A}}^{\ast }$, and $f\left(x\right)={\left(\phi \left(x\right)\right)}^{-m}$ for a positive integer m. Then, $f=p\circ \phi$, where $p\left(x\right)=x^{-m}$, and by the chain rule

$$d{f}_x\left(v\right)=d{p}_{\phi \left(x\right)}d{\phi }_x\left(v\right)=-{\left(\phi \left(x\right)\right)}^{-m}{P}_{m,1}(\phi \left(x\right),d{\phi }_x\left(v\right)){\left(\phi \left(x\right)\right)}^{-m}.$$

6.2. On Solutions of Quadratic ODE Systems

In the following proposition for each n-dimensional algebra $\mathbb{A}$, we use a pre-twisted $\mathbb{A}$-variable for solving a quadratic n-dimensional family of ordinary differential equations (ODEs). Therefore, we show that $\phi \left(\mathbb{A}\right)$-differentiability can be used to solve ODE systems.

Proposition 11.

Consider the quadratic ODE

$${\displaystyle \frac{dx}{dt}}=xax,$$

(48)

where $xax$ denotes the $\mathbb{A}$-product, $a\in \mathbb{A}$. If $c\in \mathbb{A}$ is a regular element, then

$$x\left(t\right)=-{\left(c+ta\right)}^{-1}$$

(49)

is a solution defined in an interval $I\subset \mathbb{R}$ containing 0 with $x\left(0\right)=-c^{-1}$.

Proof. 

Let $\phi \left(t\right)=c+ta$ and $f\left(y\right)=-y^{-1}$ defined in ${\mathbb{A}}^{\ast }$. Then, $d{\phi }_t=a$, and by Proposition 4, $d{f}_y\left(w\right)=y^{-1}w{y}^{-1}$. By the chain rule,

$$d{(f\circ \phi )}_t=d{f}_{\phi \left(t\right)}d{\phi }_t={(c+ta)}^{-1}a{(c+ta)}^{-1}.$$

Therefore, if $x\left(t\right)=-{(c+ta)}^{-1}$, we have

$${\displaystyle \frac{dx}{dt}}=(-{(c+ta)}^{-1})a(-{(c+ta)}^{-1})={(c+ta)}^{-1}a{(c+ta)}^{-1}.$$

□

Example 11.

Consider the ODE (11) with respect to the quaternion variable. Then,

$$\begin{array}{cc}\hfill f(x_1,x_2,x_3,x_4)& =(a_1{x}_1^2-a_1{x}_2^2-2a_2{x}_1{x}_2-a_1{x}_3^2-2a_3{x}_1{x}_3-a_1{x}_4^2-2a_4{x}_1{x}_4)e_1\hfill \\ \hfill & +(a_2{x}_1^2-a_2{x}_2^2+2a_1{x}_1{x}_2+a_2{x}_3^2-2a_3{x}_2{x}_3+a_2{x}_4^2-2a_4{x}_2{x}_4)e_2\hfill \\ \hfill & +(a_3{x}_1^2+a_3{x}_2^2-a_3{x}_3^2+2a_1{x}_1{x}_3-2a_2{x}_2{x}_3+a_3{x}_4^2-2a_4{x}_3{x}_4)e_3\hfill \\ \hfill & +(a_4{x}_1^2+a_4{x}_2^2+a_4{x}_3^2-a_4{x}_4^2+2a_1{x}_1{x}_4-2a_2{x}_2{x}_4-2a_3{x}_3{x}_4)e_4.\hfill \end{array}$$

By Proposition 11, $x\left(t\right)=-{(c+ta)}^{-1}$ is a solution. Hence,

$$x\left(t\right)={\displaystyle \frac{1}{q\left(t\right)}}(-t{a}_1-c_1,t{a}_2+c_2,t{a}_3+c_3,t{a}_4+c_4),$$

where

$$q\left(t\right)=(a_1^2+a_2^2+a_3^2+a_4^2)t^2+(2a_1{c}_1+2a_2{c}_2+2a_3{c}_3+2a_4{c}_4)t+c_1^2+c_2^2+c_3^2+c_4^2,$$

and

$$x\left(0\right)={\displaystyle \frac{1}{c_1^2+c_2^2+c_3^2+c_4^2}}\left(-c_1,c_2,c_3,c_4\right).$$

Example 12.

Consider the ODE (11) with respect to the $M_2\left(\mathbb{R}\right)$-variable. Then,

$$\begin{array}{cc}\hfill f(x_1,x_2,x_3,x_4)& =(a_1{x}_1^2+a_3{x}_1{x}_2+a_2{x}_1{x}_3+a_4{x}_2{x}_3)e_1\hfill \\ \hfill & +(a_3{x}_2^2+a_1{x}_1{x}_2+a_2{x}_1{x}_4+a_4{x}_2{x}_4)e_2\hfill \\ \hfill & +(a_2{x}_3^2+a_1{x}_1{x}_3+a_3{x}_1{x}_4+a_4{x}_3{x}_4)e_3\hfill \\ \hfill & +(a_1{x}_2{x}_3+a_4{x}_4^2+a_3{x}_2{x}_4+a_2{x}_3{x}_4)e_4.\hfill \end{array}$$

By Proposition 11, $x\left(t\right)=-{(c+ta)}^{-1}$ is a solution. Hence,

$$x\left(t\right)={\displaystyle \frac{1}{q\left(t\right)}}(a_{4}t+c_4,-a_{2}t-c_2,-a_{3}t-c_3,a_{1}t+c_1),$$

where

$$q\left(t\right)=(a_2{a}_3-a_1{a}_4)t^2+(-a_1{c}_4+a_2{c}_3+a_3{c}_2-a_4{c}_1)t+c_2{c}_3-c_1{c}_4,$$

and

$$x\left(0\right)={\displaystyle \frac{1}{c_2{c}_3-c_1{c}_4}}\left(c_4,-c_2,-c_3,c_1\right).$$

Example 13.

The best $\mathbb{A}$-quadratic approximation of $E\left(x\right)=Exp\left(x\right)$ around x is given by

$$E(x+h)\simeq E\left(x\right)+\sum _{k=1}^{\infty }{\displaystyle \frac{P_{k,1}(x,h)}{k!}}+\sum _{k=2}^{\infty }{\displaystyle \frac{P_{k,2}(x,h)}{k!}}.$$

In the same way, one can calculate higher-order derivatives for the exponential function and other analytical functions.

7. Discussion and Conclusions

This work has introduced a concept of $\mathbb{A}$-differentiability over associative algebras, establishing fundamental rules similar to the usual calculus and proving that the derivative of a convergent power series coincides with its term-wise differentiation. We have extended the analysis to $C^k(U,\mathbb{A})$-differentiability, Taylor polynomials and series, and $\mathbb{A}$-analyticity, providing a framework for differentiable functions.

This tool allows us to calculate the usual derivative for elementary functions and functions defined by convergent power series over associative algebra. As an example, the derivative of the function $f\left(x\right)=x^3$ was explicitly computed in the quaternionic setting. Additionally, line integrals were calculated for specific function families. Finally, the notion of pre-twisted differentiability was introduced, providing a new skill for solving a family of quadratic ordinary differential equations.

These results contribute to the broader understanding of usual differentiation and line integration in non-commutative settings, expanding usual skills for calculus over associative algebras.