Liu, Shibo; Zhang, Yashan On the change of variables formula for multiple integrals. (English) Zbl 1399.26023 J. Math. Study 50, No. 3, 268-276 (2017). Summary: 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. MSC: 26B15 Integration of real functions of several variables: length, area, volume 26B20 Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.) Keywords:change of variables; surface integral; divergent theorem; Cauchy-Binet formula PDF BibTeX XML Cite \textit{S. Liu} and \textit{Y. Zhang}, J. Math. Study 50, No. 3, 268--276 (2017; Zbl 1399.26023) Full Text: DOI arXiv