Brouwer’s Degree of Mapping

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.