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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.)

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