Partial Differential Equations 1 pp 175-214 | Cite as

# Brouwer’s Degree of Mapping

Chapter

## Abstract

Let the function the where this sum possesses only finitely many terms. In this chapter we intend to deduce corresponding results for functions in

*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$$i(f,\xi):=\mbox{sgn}\,f'(\xi)$$

*index of**f**at the point**ξ*. We easily deduce the following*index-sum formula*$$\sum_{\xi\in(a,b):\ f(\xi)=0}i(f,\xi)=1,$$

*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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© Springer-Verlag London 2012