Could someone give me some help to get started with this question? Don't even have the slightest idea.. =(
Suppose a vector field v on $\mathbb{R}^n$ has exactly two isolated zeros $p, q$, and $p, q$ are connected by a flow-line of the vector field. Furthermore, assume one can modify the vector field in a compact neighborhood of the flow line and merge $p, q$ to give new vector field $w$ with a single isolated zero $r$. Show that the index of $r$ must be the sum of the indices of $p$ and $q$.
Thank you.