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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
##### Keywords:
Gauss-Stokes theorem; Brouwer fixed point theorem
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