On the Change of Variables Formula for Multiple Integrals (Journal of Mathematical Study, 2017)
We develop an elementary proof of the change of variables formula in multipleintegrals. Our proof is based on an induction argument. Assuming the formula for $(m-1)$-integrals, we define the integral over hypersurface in $\mathbb{R}^m$, establish the divergenttheorem 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 BrouwerFixed Point Theorem as a corollary.
Keywords: Change of variables; surface integral; divergent theorem; Cauchy-Binet formula.