Brouwer’s Degree of Mapping

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


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
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.


Topological Mapping Product Theorem Integral Theorem Outer Domain Topological Sphere 
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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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