Working with Vector Field Indices

[Plasma_Wolf]

I'm starting to do some work on Index theory, starting with the index of a vector field at point P (in this case it's going to be the origin).

I can understand what I've read intuitively but I can't work anything out on the mathematical grounds.
For the definition of an Index at the origin, I have a vector field V(x,y) in R^2.
A Disk D(r) (r=radius) with boundary C(r).
Then the restriction V|_{C(r)} of V to C(r) is a mapping from this circle to R^2 without the origin (shouldn't it be the other way around since V is a vector field and C(r) is a circle).

V|_C(r) has a winding number and with it, we can define the index of V at point P as Index_PV = W(V|_{C(r)},0).
Now, I am looking at the following vector fields to see if I understand it:
V(x,y) = (x^2-y^2,0) has index 0 at the origin (This vector field just consists of phase lanes colinear to the x-axis IE horizontal lines)
V(x,y)= (-y,x) has index 1 at the origin (This VF consists of circles around the origin, going counter-clockwise, the clockwise version has the same index because of the direction of your tangent vector while you walk around the origin along that circle you created).

V(x,y)=(y,x) will have index -1 (the direction of your tangent vector turns around at some point while you walk along the circle).
V(x,y)=(x^2-y^2,2xy) has index 2. The best way to describe this vector field is by thinking of a magnetic field (image below, but then the straight line would be the x-axis) - You'll meet the vector field twice if you go round the origin.



So intuitively, everything is right, but I simply can't get anything done on the mathematical front.

I'm having the definition of a winding number, it is as follows:

W(\gamma,0)=\frac{1}{2\pi}\int_{\gamma}\frac{xdy-ydx}{x^2+y^2}

for a curve gamma.

Now, the problem is, that if I take gamma to be a simple circle, I get a winding number of 1, but if I walk around the origin in a circle, in a vector field like one of the above, I don't necessarily get an index of +1 and that brings me to my question:
How do those two interact? I can't get anywhere on the mathematical front. It might be something small that I just forgot (I suppose so because I'm fairly certain that I've had everything required to do this...)
Any help would be much appreciated (and I hope I really did understand what I read because otherwise there are so many errors in it that no one would know where to start fixing it)

[mfb]

With the definition of the winding number, you have to integrate over the the values of your vector field along the curve C.
This is a bit tricky because you use "x" and "y" for different things here.
Let's use (a,b)=V(x,y). Then, the winding number is given by

W(\gamma,0)=\frac{1}{2\pi}\int_{\gamma}\frac{a\, db-b\, da}{a^2+b^2}

Index_PV = W(V|_{C(r)},0)

This looks a bit ill, as the right side depends on r and the left side does not. Maybe you want to use the limit r->0?

[Plasma_Wolf]

I'm sorry about the notation of C(r). I was planning to write it as the notation I got from the literature: C_r, but decided otherwise because of the double subscript in V|_C

[Ben-oni]

In the initial explanation, you create a loop C_r(p). Then, using V to induce a function, you have f:C_r(p) -> R²\{0}, which has a winding number associated with it. (Remember that vector fields never map to 0; the reason for this will become clear later.)

But now, you're trying to find the winding number, and you introduce γ as a path without defining it. The definition for the winding number is a good one: by the Residue theorem of complex analysis, we know that the path integral of a loop on 1/z is determined entirely by the number of times it winds around 0.

But then you made the mistake of assuming γ to be C_r(p), which it is not. Here is the definition: γ : [0,1] -> R²\{0} is given by γ = f ○ e, where e : [0,1] -> C_r is defined as e_{r, p}(t) = (r cos(2πt), r sin(2πt)) + p, and f is given above from the vector field V.

With this definition, you now have a relationship between the winding number and the vector field, which was lacking before. Hope that makes sense.