Volume 50, Issue 3
On the Change of Variables Formula for Multiple Integrals

J. Math. Study, 50 (2017), pp. 268-276.

Published online: 2017-09

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• Abstract

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.

• Keywords

Change of variables, surface integral, divergent theorem, Cauchy-Binet formula.

26B15, 26B20

liusb@xmu.edu.cn (Shibo Liu)

colourful2009@163.com (Yashan Zhang)

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@Article{JMS-50-268, author = {Shibo and Liu and liusb@xmu.edu.cn and 7077 and Department of Mathematics, Xiamen University, Xiamen 361005, P.R. China and Shibo Liu and Yashan and Zhang and colourful2009@163.com and 7078 and Department of Mathematics, University of Macau, Macau, P.R. China and Yashan Zhang}, title = {On the Change of Variables Formula for Multiple Integrals}, journal = {Journal of Mathematical Study}, year = {2017}, volume = {50}, number = {3}, pages = {268--276}, abstract = {

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.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v50n3.17.04}, url = {http://global-sci.org/intro/article_detail/jms/10620.html} }
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.

Shibo Liu & Yashan Zhang. (2020). On the Change of Variables Formula for Multiple Integrals. Journal of Mathematical Study. 50 (3). 268-276. doi:10.4208/jms.v50n3.17.04
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