I am reading Milnor's Topology from the differentiable viewpoint.
There is a lemma as below,in page 33-35:
Suppose that the vector field $v$ in $U$ corresponds to $v'=df \circ v \circ f^{-1} $ on $U'$, under a diffeomorphism $f: U \to U'$. Then the index of $v$ at an isolated zero is equal to the index of $v'$ at $f(z)$
He prove the orientation preserving case as follows:
we construct a one-parameter family of embeddings $$ f_{t}: U \rightarrow R^{m} $$ with $f_{0}=$ identity, $f_{1}=f$, and $f_{t}(0)=0$ for all $t .$ Let $v_{t}$ denote the vector field $d f_{t} \circ v \circ f_{t}^{-1}$ on $f_{t}(U)$, which corresponds to $v$ on $U$. These vector fields are all defined and nonzero on a sufficiently small sphere centered at 0 . Hence the index of $v=v_{0}$ at 0 must be equal to the index of $v^{\prime}=v_{1}$ at 0 . This proves Lemma 1 for orientation preserving diffeomorphisms.
I know the idea is $v_0$ homotopic to $v_1$ which means they have the same index.The first question is about writing down the homotopy between $v_0$ and $v_1$ explicitly which involves choosing some domain.
The question is the choice of small neighborhood around 0,I mean for each $f_t$ we may choose some small sphere around 0 inside $f_t(U)$ if we can find a uniform small sphere,then we can define $v_t$ on the same domain say all $v_t$ defined on some $S^{n-1}(\epsilon)$ with $\epsilon$ independent of time $t$,hence the desired map is homotopic,I have no idea how to construct such small sphere.
The second question is where do we use the assumption that $U$ is convex?