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 .