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Brouwer’s Degree of Mapping

  • Friedrich Sauvigny
Part of the Universitext book series (UTX)

Abstract

Let the function f:[a,b]→ℝ be continuous with the property f(a)<0<f(b). Due to the intermediate value theorem, there exists a number ξ∈(a,b) satisfying f(ξ)=0. When we assume that the function f is differentiable and each zero ξ of f is nondegenerate - this means \(f'(\xi)\not=0\) holds true - we name by
$$i(f,\xi):=\mbox{sgn}\,f'(\xi)$$
the index of f at the point ξ. We easily deduce the following index-sum formula
$$\sum_{\xi\in(a,b):\ f(\xi)=0}i(f,\xi)=1,$$
where this sum possesses only finitely many terms. In this chapter we intend to deduce corresponding results for functions in n variables. We start with the case n=2, which is usually treated in a lecture on complex analysis.

Keywords

Topological Mapping Product Theorem Integral Theorem Outer Domain Topological Sphere 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag London 2012

Authors and Affiliations

  • Friedrich Sauvigny
    • 1
  1. 1.Mathematical Institute, LS AnalysisBrandenburgian Technical UniversityCottbusGermany

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