I would like to understand the proof by Fulton, Algebraic topology (in the chapter 8) that for any non-degenerate vector fields $V,W$ on a closed orientable surface $X$ we have $\sum ind_{p \in X}(V) = \sum_{p \in X} ind_p(W)$.
The argument goes as follow : triangulate $X$ so that a single triangle $T$ contains at most a singular point of $V$ or $W$. The claim is that on $T$, the number $\sum ind_{p \in T}(V) - \sum_{p \in T} ind_p(W)$ is the angular variation of the vector field $V-W$ along the boundary $\partial T$ and one conclude by taking the sum over all triangle $T$.
I don't understand why this true : indeed if we take $W' = \epsilon W$ for $\epsilon$ small enough, obviously $W$ and $W'$ have the same index but this should modify the angular variation, since for $\epsilon$ small enough $V-W'$ is homotopic to $V$. This is clearly not possible, e.g when $W$ has a single critical point and $V$ is nonzero on $T$.
It's probable that I misunderstood the argument, so any clarification would be appreciated. I'll also be glad with another argument using the same kind of ideas.