In the classical complex analysis, the winding number of a plane curve with respect to a point is defined by
Let be open and bounded. Let and . Notation denotes the Jacobian of evaluated at . It is well-known that the Brouwer Degree Theory, usually denoted by , is constructed for continuous function, -class, via the following steps
For the -class: We assume then
For the -class: In case we want to remove the condition , we then define
where is any regular value of sufficiently closed to in the sense that
For the -class: In this case, we define
where is sufficiently closed to in the sense that
Now we prove the following fact
Theorem. Let be the unit ball and . Assume is a function and . Then
Proof. It is sufficient to prove in the case when
Let us assume
Then we need to show that
Take small enough such that the ‘s are disjoint, where . It is clear that
and the restriction of to is a homeomorphism for all .
Put then is a Jordan curve such that lies in its interior region. Besides, has the same orientation as if and the opposite orientation if .
Then in for some . We can divide into small rectangles such that on each latex . Since the image does not wind around , we have
and summing over all yields
Since the orientation of is determined by , winds exactly once around . Thus, we obtain
The proof now easily follows.