On the change of variables formula for multiple integrals

TL;DR

This paper presents an elementary, inductive proof of the change of variables formula for multiple integrals, also deriving the Brouwer Fixed Point Theorem as a corollary, simplifying understanding of these fundamental concepts.

Contribution

It introduces a simple inductive proof method for the change of variables formula, connecting it with the Brouwer Fixed Point Theorem, enhancing conceptual clarity.

Findings

Elementary proof of change of variables formula

Derivation of Brouwer Fixed Point Theorem as a corollary

Inductive approach simplifies understanding of multiple integrals

Abstract

In this paper, we develop an elementary proof of the change of variables in multiple integrals. Our proof is based on an induction argument. Assuming the formula for (m-1)-integrals, we define the integral over hypersurface in Rm, establish the divergent theorem and then use the divergent theorem to prove the formula for m-integrals. In addition to its simplicity, an advantage of our approach is that it yields the Brouwer Fixed Point Theorem as a corollary.