In reviewing the familiar Poincare-Hopf theorem I come across the following question:
Suppose $x$ an isolated 0 of $V$. Pick up a disk around $x$ in its neighborhood. Calculate the degree of the map $u:\partial D \rightarrow S^{m-1},u(z)=\frac{V(z)}{|V(z)|}$
where $V(z)$ is the the map $M\rightarrow TM$ which represents a vector field.
I am confused because while in case 1 and 2 it is clear the degree $S^{1}\rightarrow S^{1}$ must be 1 since they are orientation preserving maps combined with rotation, I do not know how to calculate case 3. The author claimed the index is -1. But how? A hint would be mostly welcome.