Abstract
This paper explores the application of the Theory of Functional Connections (TFC) to various mathematical tools that can be expressed using a minimal number of parameters. Through a series of illustrative examples, we demonstrate the versatility of TFC in addressing a diverse range of mathematical constructs. Specifically, TFC is applied to symmetric, anti-symmetric, and orthogonal matrices, to \(n\) -dimensional probability distributions, to curved surfaces, to quadric functions, and to multivectors in the context of geometric algebra. The study highlights the potential of TFC to extend its applicability to these mathematical tools, particularly in optimization problems where such mathematical tools plays a central role. Although the findings are primarily based on a limited set of examples rather than comprehensive mathematical proofs, the results support the conjecture that TFC is broadly applicable to any mathematical structure that admits a minimal parameterization. Additionally, the paper includes an appendix that presents an innovative use of TFC in conjunction with Heaviside’s step function. This approach demonstrates the ability to construct constrained functionals for addressing various inequality constraints, further expanding the scope of TFC applications. By showcasing these capabilities, the paper aims to underline the significant potential of TFC as a unifying framework for addressing diverse mathematical and optimization problems in both theoretical and practical contexts.
