I have a question concerning Milnor's book: Topology from the differentiable viewpoint.
First (p.32), he defines the index of $v$ at an isolated zero $z$, where $U$ is an open subset of $\mathbb{R}^m$ and $v: U \to \mathbb{R}^m$ is a smooth map:
Then (p.34), he defines the index of $w$ at an isolated zero $z$, where $M$ is a smooth manifold and $w: M \to TM$ is a smooth vector field on $M$ (using local charts):
What I do not understand is the following: technically, his map $dg^{-1} \circ w \circ g$ goes from $U$ to $TU$ (where the tangent bundle $TU$ of $U$ is not a subset of $\mathbb{R}^n$), so I do not understand this definition.
My first idea was the following: $TU$ is trivial, so actually, the set $\mathfrak{X}(U)$ of vector fields on $U$ is in bijection with the set $\mathcal{C}^1(U,\mathbb{R}^m)$ (let $\Phi$ be such a bijection). Thus, coming back to Milnor's definition, I would say that given a vector field $w: M \to TM$ on $M$, the index of $w$ at an isolated zero $z$ should be the index of $\Phi[dg^{-1} \circ w \circ g]$. So:
either this is what Milnor means, but in that case, I do not see why this definition does not depends on $\Phi$,
or this is not what he means, and in that case, I do not understand at all his definition.
Thanks for you help !