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Let $X$ be a vector field on a manifold $M$ of dimension $m$ (compact, connected, oriented) with isolated zero $z$. Let $B$ be a ball around $z$ such that $X$ has no other zeroes in $B$. We define the index of $X$ at $z$ is the degree of the map $f$ from boundary of $B$ to $S^{m-1}$ such that $f$ maps $y$ to $\frac{X(y)}{|X(y)|}$. I need to prove that this definition is well-defined (may be for the easy case $M$ is a submanifold of $\mathbb{R}^k$).

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Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. To get the best possible answers, it is helpful to give the context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. The phrasing "I need to prove" rubs me the wrong way. Consider rephrasing to show that you care about understanding the math and not just credit on an assignment. – Noah Snyder Oct 3 '12 at 15:05
Thank you for your comment. Can you show me how I can type Latex in the question I submit to this website? I know it's impolite to write something informal, but I don't know how to correct them. – PAM Oct 3 '12 at 15:23
For some basic information about writing math at this site see e.g. here, here, here and here. – Noah Snyder Oct 3 '12 at 15:32

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