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 - <p style="text-align: justify;">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.</p>