I know that $\operatorname{ind}_0(\vec{v})$ simply counts the number of times $\vec{v}$ rotates completely while we walk counterclockwise around the circle.

However, if I follow the definition on Guillemin and Pollack's Differential Topology Page 133, I couldn't see why the degree of the directional map is the number of times $\vec{v}$ rotates:

The index of $\vec{v}$ at $0$, $\operatorname{ind}_0(\vec{v})$, is defined to be the degree of the directional map $S_\epsilon \to S^{k-1}$: with radius $\epsilon$ so small that $\vec{v}$ has no zeros inside $S_\epsilon$ except at the origin.

($\vec{v}$ is a smooth map $\vec{v}: X \to \mathbb{R}^n$ such that $\forall x, \vec{v}(x) \in T_x(X)$. Assume that we are in $\mathbb{R}^k$ and that $\vec{v}$ has an isolated zero at the origin.)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.