I got stuck with one exercise from Chapter 3.5 in Guillemin and Pollack's book, which I used to study differential topology by myself:
Given a vector field $\overrightarrow{v}$ with isolated zeros in $\mathbb{R}^{k}$ and a compact $k$-dimensional submanifold $W$ of $\mathbb{R}^k$ with boundary. If $\overrightarrow{v}$ is never zero on the boundary $\partial W$, then we have that the sum of indices of $\overrightarrow{v}$ at its zeros inside $W$ equals the degree of the following map:
$$\frac{\overrightarrow{v}}{|\overrightarrow{v}|}: \partial W\rightarrow S^{k}$$
I have tried following the hint to delete (sufficiently small) balls around the zeros and follow the standard argument used to prove Poincare-Hopf's Theorem. It turns out that the manifold $W$ becomes a manifold with "two parts" of boundaries after we delete the balls: the original boundary $\partial W$ and the boundaries of the omitted the balls. I guess we just need to show that the degrees of the map $\frac{\overrightarrow{v}}{|\overrightarrow{v}|}$ on these two parts of boundaries sum up to zero, right? However, is there a way to consider the manifold $W$ as a whole to prove the claim?
Thanks in advance! Any hint/help would be greatly appreciated!