**8. Conclusions**

This paper extends to *n*-dimensional spaces the Univariate Theory of Connections (ToC), introduced in Ref. [1]. First, it provides a mathematical tool to express *all* possible surfaces subject to constraint functions and arbitrary-order derivatives in a boundary rectangular domain, and then it extends the results to the multivariate case by providing the Multivariate Theory of Connections, which allows one to obtain *n*-dimensional manifolds subject to any-order derivative boundary constraints.

In particular, if the constraints are provided along one axis only, then this paper shows that the univariate ToC, as defined in Ref. [1], can be adopted to describe *all* possible surfaces satisfying the constraints. If the boundary constraints are defined in a rectangular domain, then the constrained expression is found in the form *f*(*x*) = *<sup>A</sup>*(*x*) + *<sup>B</sup>*(*x*), where *<sup>A</sup>*(*x*) can be *any* function satisfying the constraints and *<sup>B</sup>*(*x*) describes *all* functions that are vanishing at the constraints. This is obtained by introducing a free function, *g*(*x*), into the function *<sup>B</sup>*(*x*) in such a way that *<sup>B</sup>*(*x*) is zero at the constraints no matter what the *g*(*x*) is. This way, by spanning all possible *g*(*x*) surfaces (even discontinuous, null, or piece-wise defined) the resulting *<sup>B</sup>*(*x*) generates *all* surfaces that are zero at the constraints and, consequently, *f*(*x*) = *<sup>A</sup>*(*x*) + *<sup>B</sup>*(*x*), describes all surfaces satisfying the constraints defined in the rectangular boundary domain. The function *<sup>A</sup>*(*x*) has been selected as a Coons surface [11] and, in particular, a Coons surface is obtained if *g*(*x*) = 0 is selected. All possible combinations of Dirichlet *and* Neumann constraints are also provided in Appendix A.

The last section provides the Multivariate Theory of Connections extension, which is a mathematical tool to transform *n*-dimensional constraint optimization problems subject to constraints on the boundary value and any-order derivative into unconstrained optimization problems. The number of applications of the Multivariate Theory of Connections are many, especially in the area of partial and stochastic differential equations: the main subjects of our current research.

**Author Contributions:** C.L. derived the table in Appendix A and the mathematical proof validating the tensor notation. All the remaining parts are provided by D.M.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors acknowledge Ergun Akleman for pointing out the Coons surface.

**Conflicts of Interest:** The authors declare no conflict of interest.
