How can you prove that the winding number around two zeros of a vector field is the sum of the two indices?

Question

If v is a continuous vector field with two isolated zeros, then the winding number around one zero is its index. The winding number on a circle with both zeros in its interior is the sum of the two indices. How can you prove that?

Answer 1

For the intuitive picture, draw a simple closed curve counterclockwise around both zeros. For simplicity, say it's a circle with one zero in the upper half, the other in the lower half. Now draw a diameter of the circle separating the two. Put together, you have two half circles, each surrounding one zero (by a half circle here, I mean a semicircle with the diameter added to make a closed curve). Now track the change in the angle $\theta$ of the vector field with some fixed direction as you go around the two half circles. The total change around each will be $2\pi$ times the change in $\theta$, right? Now look at the change along the diameter, which is part of each half circle – but traversed in opposite directions. Adding the changes together, the changes along the diameter cancel, and you're left with the change taken around the full circle.

A variant of this picture takes a small circle surrounding each zero. Now add a curve exterior to each circle, joining one point on one circle to a point on the other. Put everything together into a curve traversing one circle, crossing along the given curve to the other, going around that circle once, and going back. Again, the two trips along the new curve cancel out.