Is there a quantity which will measure the number of twists in a vector field along a closed path?

Question

The vector field $\vec A=-y\hat{i}+x\hat{j}$ that turns by an amount $\pi$ when we complete half the circle $x^2+y^2=1$ and by $2\pi$ when we complete the full circle. It is also possible to imagine and draw a vector field that turns by an amount $2\pi$ when we complete half the circle $x^2+y^2=1$ and by $4\pi$ when we complete the full circle respectively. In the second case, the vector field has a twist along the complete circle. Is there a mathematical quantity that can tell us the first field has zero twists while the second field has one twist?

Answer 1

Yes! This quantity is called the "index of a singular point of the vector field", the singular points being point where the vector field is $0$. For non-singular points the index will always be $0$ if I'm not mistaken. You can find a quite extensive and geometric explanation for this in Needham's Visual Differential Geometry and Forms and it also features quite a bit in his Visual Complex Analysis if I'm not mistaken.