#
The Multivariate Theory of Connections^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Manifold Constraints in One Axis, Only

#### 2.1. Example #1: Surface Subject to Four Function Constraints

#### 2.2. Example #2: Surface Subject to Two Functions and One Derivative Constraint

## 3. Connecting Functions in Two Directions

## 4. Theory of Connections Surface Subject to Dirichlet Constraints

## 5. Multi-Function Constraints at Generic Coordinates

## 6. Constraints on Function and Derivatives

#### 6.1. Constraints: $c(0,y)$ and $c(x,0)$

#### 6.2. Constraints: $c(0,y)$ and ${c}_{y}(x,0)$

_{y}(x, 0)q(y) + p(x)[c (0, y) − c

_{y}(0, 0) q(y)].

#### 6.3. Neumann Constraints: ${c}_{x}(0,y)$, ${c}_{x}(1,y)$, ${c}_{y}(x,0)$, and ${c}_{y}(x,1)$

#### 6.4. Constraints: $c(0,y)$, ${c}_{y}(x,0)$, and ${c}_{y}(x,1)$

#### 6.5. Generic Mixed Constraints

## 7. Extension to $\mathit{n}$-Dimensional Spaces and Arbitrary-Order Derivative Constraints

#### 7.1. The $\mathcal{M}$ Tensor

- The element of $\mathcal{M}$ for all indices equal to 1 is 0 (i.e., ${\mathcal{M}}_{11\dots 1}=0$).
- The first order tensor obtained by keeping the kth dimension’s index and setting all other dimension’s indices to 1 can be written as,$${\mathcal{M}}_{1,\dots ,1,{i}_{k},1,\dots ,1}={\phantom{\rule{0.166667em}{0ex}}}^{k}{c}_{{\mathit{p}}^{k}}^{{\mathit{d}}^{k}},\phantom{\rule{2.em}{0ex}}\mathrm{where}\phantom{\rule{1.em}{0ex}}{i}_{k}\in [2,{\ell}_{k}+1],$$$${}^{7}{c}_{{\mathit{p}}^{7}}^{{\mathit{d}}^{7}}:=\left(\right)open="\{"\; close="\}">{c|}_{{x}_{7}=-0.3},\phantom{\rule{1.em}{0ex}}{\left(\right)}_{{\displaystyle \frac{{\partial}^{4}c}{\partial {x}_{7}^{4}}}}{x}_{7}=0.5$$
- The generic element of the tensor is ${\mathcal{M}}_{{i}_{1}{i}_{2}\dots {i}_{n}}$, where at least two indices are different from 1. Let m be the number of indices different from 1. Note that m is also the number of constraint “intersections”. In this case, the generic element of the $\mathcal{M}$ tensor is provided by,$${\mathcal{M}}_{{i}_{1}{i}_{2}\dots {i}_{n}}={\phantom{\rule{0.166667em}{0ex}}}^{1}{b}_{{\mathit{p}}_{{i}_{1}-1}^{1}}^{{\mathit{d}}_{{i}_{1}-1}^{1}}\left(\right)open="["\; close="]">{\phantom{\rule{0.166667em}{0ex}}}^{2}{b}_{{\mathit{p}}_{{i}_{2}-1}^{2}}^{{\mathit{d}}_{{i}_{2}-1}^{2}}\left(\right)open="["\; close="]">\dots \left(\right)open="["\; close="]">{\phantom{\rule{0.166667em}{0ex}}}^{n}{b}_{{\mathit{p}}_{{i}_{n}-1}^{n}}^{{\mathit{d}}_{{i}_{n}-1}^{n}}\left[c\left(\mathit{x}\right)\right]$$If $c\left(\mathit{x}\right)\in {C}^{s}$, where $s=\sum _{k=1}^{n}{\mathit{d}}_{{i}_{k}-1}^{k}$, then Clairaut’s theorem states that the sequence of boundary constraint operators provided in Equation (24) can be freely permutated. This permutation becomes obvious by multiple applications of the theorem. For example,$${f}_{xyy}={\left({f}_{xy}\right)}_{y}={\left({f}_{yx}\right)}_{y}={\left({f}_{y}\right)}_{xy}={\left({f}_{y}\right)}_{yx}={f}_{yyx}.$$

- From Step 1: ${M}_{111}=0$
- From Step 2:$$\begin{array}{cc}\hfill {M}_{{i}_{1}11}& =\left\{\begin{array}{ccc}0,& c(0,{x}_{2},{x}_{3}),& c(1,{x}_{2},{x}_{3})\end{array}\right\}=\left\{\begin{array}{ccc}0,& {}^{1}{b}_{0}^{0}\left[c\left(\mathit{x}\right)\right],& {}^{1}{b}_{1}^{0}\left[c\left(\mathit{x}\right)\right]\end{array}\right\}\hfill \\ \hfill {M}_{1{i}_{2}1}& =\left\{\begin{array}{ccc}0,& c({x}_{1},0,{x}_{3}),& {\displaystyle \frac{\partial c}{\partial {x}_{2}}}({x}_{1},0,{x}_{3})\end{array}\right\}=\left\{\begin{array}{ccc}0,& {}^{2}{b}_{0}^{0}\left[c\left(\mathit{x}\right)\right],& {}^{2}{b}_{0}^{1}\left[c\left(\mathit{x}\right)\right]\end{array}\right\}\hfill \\ \hfill {M}_{11{i}_{3}}& =\left\{\begin{array}{ccc}0,& c({x}_{1},{x}_{3},0),& {\displaystyle \frac{\partial c}{\partial {x}_{3}}}({x}_{1},{x}_{2},0)\end{array}\right\}=\left\{\begin{array}{ccc}0,& {}^{3}{b}_{0}^{0}\left[c\left(\mathit{x}\right)\right],& {}^{3}{b}_{0}^{1}\left[c\left(\mathit{x}\right)\right]\end{array}\right\}\hfill \end{array}$$
- From Step 3, a single example is provided,$${\mathcal{M}}_{323}={\phantom{\rule{0.166667em}{0ex}}}^{1}{b}_{1}^{0}\left(\right)open="["\; close="]">{\phantom{\rule{4pt}{0ex}}}^{2}{b}_{0}^{0}\left(\right)open="["\; close="]">{\phantom{\rule{4pt}{0ex}}}^{3}{c}_{0}^{1}\left(\mathit{x}\right){(-1)}^{4}={\displaystyle \frac{\partial c\left(\mathit{x}\right)}{\partial {x}_{3}}}{|}_{\begin{array}{c}{x}_{1}=1\\ {x}_{2}=0\\ {x}_{3}=0\end{array}}$$$${\mathcal{M}}_{323}={\phantom{\rule{0.166667em}{0ex}}}^{2}{b}_{0}^{0}\left(\right)open="["\; close="]">{\phantom{\rule{4pt}{0ex}}}^{3}{b}_{0}^{1}\left(\right)open="["\; close="]">{\phantom{\rule{4pt}{0ex}}}^{1}{c}_{1}^{0}{(-1)}^{4}={\phantom{\rule{0.166667em}{0ex}}}^{3}{b}_{0}^{1}\left(\right)open="["\; close="]">{\phantom{\rule{4pt}{0ex}}}^{1}{b}_{1}^{0}\left(\right)open="["\; close="]">{\phantom{\rule{4pt}{0ex}}}^{2}{c}_{0}^{0}$$Three additional examples are given to help further illustrate the procedure,$${M}_{132}=-{\left(\right)}_{{\displaystyle \frac{\partial c\left(\mathit{x}\right)}{\partial {x}_{2}}}}\begin{array}{c}{x}_{2}=0\\ {x}_{3}=0\end{array}$$

#### 7.2. The **v** Vectors

#### 7.3. Proof

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ToC | Theory of Connections |

PDE | Partial Differential Equations |

ODE | Ordinary Differential Equations |

IVP | Initial Value Problems |

BVP | Boundary Value Problems |

## Appendix A. All combinations of Dirichlet and Neumann constraints

${\mathit{c}}_{\mathit{x},\mathbf{0}}$ | ${\mathit{c}}_{\mathbf{0},\mathit{y}}$ | ${\mathit{c}}_{\mathit{x},\mathbf{1}}$ | ${\mathit{c}}_{\mathbf{1},\mathit{y}}$ | ${\mathit{c}}_{\mathbf{0},\mathit{y}}^{\mathit{x}}$ | ${\mathit{c}}_{\mathbf{1},\mathit{y}}^{\mathit{x}}$ | ${\mathit{c}}_{\mathit{x},\mathbf{0}}^{\mathit{y}}$ | ${\mathit{c}}_{\mathit{x},\mathbf{1}}^{\mathit{y}}$ | $\mathit{v}\left(\mathit{x}\right)$ | $\mathit{v}\left(\mathit{y}\right)$ |

✓ | ✓ | $\left\{\begin{array}{c}1\\ 1\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1\end{array}\right\}$ | ||||||

✓ | ✓ | $\left\{\begin{array}{c}1\\ 1\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ y\end{array}\right\}$ | ||||||

✓ | ✓ | $\left\{\begin{array}{c}1\\ x\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ y\end{array}\right\}$ | ||||||

✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1-{y}^{2}\\ {y}^{2}\end{array}\right\}$ | |||||

✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1\\ y\end{array}\right\}$ | |||||

✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ x\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1-y\\ y\end{array}\right\}$ | |||||

✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ x\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ y-{y}^{2}\\ {y}^{2}\end{array}\right\}$ | |||||

✓ | ✓ | $\left\{\begin{array}{c}1\\ 1\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ y-{y}^{2}/2\\ {y}^{2}/2\end{array}\right\}$ | ||||||

✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ x\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ y-{y}^{2}/2\\ {y}^{2}/2\end{array}\right\}$ | |||||

✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1-x\\ x\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1-y\\ y\end{array}\right\}$ | ||||

✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1-x\\ x\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ y-{y}^{2}\\ {y}^{2}\end{array}\right\}$ | ||||

✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1-x\\ x\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ y-{y}^{2}/2\\ {y}^{2}/2\end{array}\right\}$ | ||||

✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ x-{x}^{2}\\ {x}^{2}\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ y-{y}^{2}\\ {y}^{2}\end{array}\right\}$ | ||||

✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ x-{x}^{2}\\ {x}^{2}\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ y-{y}^{2}/2\\ {y}^{2}/2\end{array}\right\}$ | ||||

✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ x-{x}^{2}/2\\ {x}^{2}/2\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ y-{y}^{2}/2\\ {y}^{2}/2\end{array}\right\}$ | ||||

✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1\\ y\end{array}\right\}$ | |||||

✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ x\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1\\ y\end{array}\right\}$ | |||||

✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1-x\\ x\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1\\ y\end{array}\right\}$ | ||||

✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1-{y}^{2}\\ y-{y}^{2}\\ {y}^{2}\end{array}\right\}$ | ||||

✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ x\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1-{y}^{2}\\ y-{y}^{2}\\ {y}^{2}\end{array}\right\}$ | ||||

✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1\\ y-{y}^{2}/2\\ {y}^{2}/2\end{array}\right\}$ | ||||

✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ x\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1\\ y-{y}^{2}/2\\ {y}^{2}/2\end{array}\right\}$ | ||||

✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1\\ x\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1\\ y\end{array}\right\}$ | ||||

✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ x-{x}^{2}/2\\ {x}^{2}/2\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1\\ y\end{array}\right\}$ | ||||

✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1\\ x\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1\\ y\end{array}\right\}$ | ||||

✓ | ✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1\\ x\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1-{y}^{2}\\ y-{y}^{2}\\ {y}^{2}\end{array}\right\}$ | |||

✓ | ✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1\\ x\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1\\ y-{y}^{2}/2\\ {y}^{2}/2\end{array}\right\}$ | |||

✓ | ✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1-x\\ x\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1-{y}^{2}\\ y-{y}^{2}\\ {y}^{2}\end{array}\right\}$ | |||

✓ | ✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ x-{x}^{2}\\ {x}^{2}\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1-{y}^{2}\\ y-{y}^{2}\\ {y}^{2}\end{array}\right\}$ | |||

✓ | ✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1-x\\ x\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1\\ y-{y}^{2}/2\\ {y}^{2}/2\end{array}\right\}$ | |||

✓ | ✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ x-{x}^{2}\\ {x}^{2}\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1\\ y-{y}^{2}/2\\ {y}^{2}/2\end{array}\right\}$ | |||

✓ | ✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ x-{x}^{2}/2\\ {x}^{2}/2\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1\\ y-{y}^{2}/2\\ {y}^{2}/2\end{array}\right\}$ | |||

✓ | ✓ | ✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1-{x}^{2}\\ x-{x}^{2}\\ {x}^{2}\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1-{y}^{2}\\ y-{y}^{2}\\ {y}^{2}\end{array}\right\}$ | ||

✓ | ✓ | ✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1-{x}^{2}\\ x-{x}^{2}\\ {x}^{2}\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1\\ y-{y}^{2}/2\\ {y}^{2}/2\end{array}\right\}$ | ||

✓ | ✓ | ✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1\\ x-{x}^{2}/2\\ {x}^{2}/2\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1\\ y-{y}^{2}/2\\ {y}^{2}/2\end{array}\right\}$ | ||

✓ | ✓ | ✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1-x\\ x\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1-3{y}^{2}+2{y}^{3}\\ y-2{y}^{2}+{y}^{3}\\ 3{y}^{2}-2{y}^{3}\\ -{y}^{2}+{y}^{3}\end{array}\right\}$ | ||

✓ | ✓ | ✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ -1+x\\ 1\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1-3{y}^{2}+2{y}^{3}\\ y-2{y}^{2}+{y}^{3}\\ 3{y}^{2}-2{y}^{3}\\ -{y}^{2}+{y}^{3}\end{array}\right\}$ | ||

✓ | ✓ | ✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ x-{x}^{2}/2\\ {x}^{2}/2\end{array}\right\}$ | $\left\{\begin{array}{c}1\\ 1-3{y}^{2}+2{y}^{3}\\ y-2{y}^{2}+{y}^{3}\\ 3{y}^{2}-2{y}^{3}\\ -{y}^{2}+{y}^{3}\end{array}\right\}$ | ||

✓ | ✓ | ✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1\\ x\end{array}\right\}$ | |||

✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1-{x}^{2}\\ x-{x}^{2}\\ {x}^{2}\end{array}\right\}$ | ||

✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1\\ x-{x}^{2}/2\\ {x}^{2}/2\end{array}\right\}$ | ||

✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | $\left\{\begin{array}{c}1\\ 1-3{x}^{2}+2{x}^{3}\\ x-2{x}^{2}+{x}^{3}\\ 3{x}^{2}-2{x}^{3}\\ -{x}^{2}+{x}^{3}\end{array}\right\}$ |

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**Figure 2.**Surface obtained using function $g(x,y)={x}^{2}\phantom{\rule{0.166667em}{0ex}}y-sin\left(5x\right)cos\left(4\phantom{\rule{0.166667em}{0ex}}\mathrm{mod}(y,1)\right)$.

**Figure 4.**Surface obtained using function $g(x,y)=3{x}^{2}y-2sin\left(15x\right)cos\left(2y\right)$.

**Figure 6.**ToC surface subject to multiple constraints on two axes: using $g(x,y)=0$ (

**left**); and using $g(x,y)=\mathrm{mod}(x,0.5)cos\left(19y\right)-x\phantom{\rule{0.166667em}{0ex}}\mathrm{mod}(3y,0.4)$ (

**right**).

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**MDPI and ACS Style**

Mortari, D.; Leake, C.
The Multivariate Theory of Connections. *Mathematics* **2019**, *7*, 296.
https://doi.org/10.3390/math7030296

**AMA Style**

Mortari D, Leake C.
The Multivariate Theory of Connections. *Mathematics*. 2019; 7(3):296.
https://doi.org/10.3390/math7030296

**Chicago/Turabian Style**

Mortari, Daniele, and Carl Leake.
2019. "The Multivariate Theory of Connections" *Mathematics* 7, no. 3: 296.
https://doi.org/10.3390/math7030296