TY - JOUR T1 - On the Change of Variables Formula for Multiple Integrals AU - Liu , Shibo AU - Zhang , Yashan JO - Journal of Mathematical Study VL - 3 SP - 268 EP - 276 PY - 2017 DA - 2017/09 SN - 50 DO - http://doi.org/10.4208/jms.v50n3.17.04 UR - https://global-sci.org/intro/article_detail/jms/10620.html KW - Change of variables, surface integral, divergent theorem, Cauchy-Binet formula. AB -

We develop an elementary proof of the change of variables formula 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 $\mathbb{R}^m$, 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.