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An integral theorem and its applications to coincidence theorems. (English) Zbl 0703.26011
It is given an elementary proof of the following Gauss theorem: Let f,g: \(U\to {\mathbb{R}}^ n\) be maps of class \(C^ 2\) from an open set \(U\subset {\mathbb{R}}^ n\). Assume that \(K\subset U\) is a compact set such that \(f| \partial K=g| \partial K.\) Then \[ \int_{K}\det f'(x) dx=\int_{K}\det g'(x) dx. \] As an application of the theorem one can obtain an analytical proof of the Brouwer fixed point theorem.
Reviewer: W.Kulpa
MSC:
26B20 Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.)
54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
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