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 .

Now set

.

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.

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Sweet! I will might use this in my thesis 🙂

But it seems that one needs Jordan curve Theorem in this proof. Also I haven’t found any good explanations for that “Jacobian sign and orientation connection”.

Any way, congratulations for your Ph.D!

I too self-study GR but on elementary level: I just finished Woodhouse’s book General Relativity.

Comment by JL — February 26, 2013 @ 12:49

I forgot to mention, there is a minor typo on the last line: right hand side should state .

Comment by JL — February 26, 2013 @ 18:17

Thanks JL, I wish you all the best.

Comment by Ngô Quốc Anh — February 26, 2013 @ 21:35