# The index of zero of vector field is well defined

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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