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On the change of variables formula for multiple integrals. (English) Zbl 1399.26023
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.)
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