Abstract

Higher - dimensional calculus and integral transformation play crucial roles in advancing our understanding of complex systems in mathematics and theoretical physics . Integral transformations are instrumental in simplifying complex differential equations, enabling the resolution of multi - dimensional problems that arise in various scientific fields . This paper aims to delve into a specific higher - dimensional integral transformation defined by the axioms $F[q, s, l, \alpha]$ and $G[q, s, l, \beta, c]$ . We start by outlining the axioms which define the functions $F$ and $G$ . Specifically, Axiom 1 defines $F$ as a function of four variables : $q$, $s$, $l$, and $\alpha$, whereas Axiom 2 defines $G$ as a function that additionally includes variables $\beta$ and $c$ . Axiom 3 relates $h$ and $l$ via a sine function . The core of our investigation is the integral transformation expressed as a five - dimensional integral involving $G$ and proving its equivalence to $F$, provided a specific condition on $c$ holds . We approach this problem by first deriving the expression for $c$ through detailed differentiation of $F$ and equating it to $G$ . The derivation involves advanced calculus techniques and symbolic mathematics to solve the resulting equations . We then verify the derived expression for $c$ by substituting it back into the relationship between $F$ and $G$, ensuring that the equality holds under integral transformation . Finally, to corroborate our findings, we employ visualizations through multidimensional contour plots to illustrate the relationship between the derived expressions . This provides an intuitive confirmation of the mathematical consistency and validity of the transformation . This paper contributes to the field by providing a nuanced and detailed examination of higher - dimensional integral transformations and their underlying mathematical structures . The results have potential implications for theoretical physics, particularly in areas involving complex systems and multi - dimensional analyses .