This chapter is on the Brouwer degree, a very useful concept with numerous and important applications. The degree can be used to prove some difficult theorems in topology such as the Brouwer fixed point theorem, the Jordan separation theorem, and the invariance of domain theorem. A couple of these big theorems have been presented earlier, but when you have degree theory, they get much easier. Degree theory is also used in bifurcation theory and many other areas in which it is an essential tool. The degree will be developed for ℝ^{n} first. When this is understood, it is not too difficult to extend to versions of the degree which hold in Banach space. There is more on degree theory in the book by Deimling [?] and much of the presentation here follows this reference. Another more recent book which is really good is [?]. This is a whole book on degree theory.
The original reference for the approach given here, based on analysis, is [?] and dates from 1959. The degree was developed earlier by Brouwer and others using different methods.
To give you an idea what the degree is about, consider a real valued C^{1} function defined on an interval, I, and let y ∈ f
In the above picture, d
The amazing thing about this is the number you obtain in this simple manner is a specialization of something which is defined for continuous functions and which has nothing to do with differentiability. An outline of the presentation is as follows. First define the degree for smooth functions at regular values and then extend to arbitrary values and finally to continuous functions. The reason this is possible is an integral expression for the degree which is insensitive to homotopy. It is very similar to the winding number presented in the part of the book introducing complex analysis. The difference between the two is that with the degree, the integral which ties it all together is taken over the open set while the winding number is taken over the boundary.
In this chapter Ω will refer to a bounded open set.
Definition 21.0.1 For Ω a bounded open set, denote by C

If the functions take values in ℝ^{n} write C^{m}

Of course if m = ∞, the notation means that there are infinitely many derivatives. Also, C
Note that, by applying the Tietze extension theorem to the components of the function, one can always extend a function continuous on Ω to all of ℝ^{n} so there is no loss of generality in simply regarding functions continuous on Ω as restrictions of functions continuous on ℝ^{n}. Next is the idea of a regular value.
Definition 21.0.2 For W an open set in ℝ^{n} and g ∈ C^{1}
Also, ∂Ω will often be referred to. It is those points with the property that every open set (or open ball) containing the point contains points not in Ω and points in Ω. Then the following simple lemma will be used frequently.
Lemma 21.0.3 Define ∂U to be those points x with the property that for every r > 0, B

Let C be a closed subset of ℝ^{p} and let K denote the set of components of ℝ^{p} ∖C. Then if K is one of these components, it is open and

Proof: Let x ∈U ∖ U. If B
Why is K open for K a component of ℝ^{p} ∖C? This is obvious because in ℝ^{p} an open ball is connected. Thus if k ∈ K,letting B
Now for K a component of ℝ^{p} ∖ C, why is ∂K ⊆ C? Let x ∈ ∂K. If x