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What's the "role" of $\psi \circ F \circ \phi^{-1}$ in diff. geometry?

It's used e.g. in the proof of "Diffeomorphism invariance of dimension":

Suppose $M$ is nonempty smooth $m$-manifold, $N$ is a nonempty smooth $n$-manifold and $f:M \rightarrow N$ is diffeomorphism. Choose any point $p \in M$ and let $(U, \phi)$ and $(V, \psi)$ be smooth coord. charts containing $p$ and $F(p)$. Then (the restriction of) $\hat{F}=\psi \circ F \circ \phi^{-1}$ is a diffeomorphism from open subset of $\mathbb{R}^m$ to an open subset of $\mathbb{R}^n$. Then from C.4 it follows that $n=m$.

But in the perspective of this question. I've been wondering, why does one "draw" the map

$$\hat{F}=\psi \circ F \circ \phi^{-1}$$

It seems like a common tool in diff. geom. I've seen it in other proofs as well. But I'm not sure why is it used.


Some reasoning:

$\phi^{-1}$ gives the topological representation of $U$ ( or $U$'s euclidean representation, since $\phi$ works to give the euclidean representation).

Then $F$ "acts" on that space and produces a topological space $N$ or subset of $N$.

Then $\psi$ maps this to euclidean coordinates (but it's a different chart, it acts on the image of $F$, whereas $\phi$ didn't).


But I find it hard to put it into words as to "why is this done".

Like is it in order to "infer through" all the sets between all the given transformations?

Or perhaps it's chosen in order to talk about the functions using "Euclidean terminology"? Since the map $\hat{F}$ is now $\mathbb{R}^m \rightarrow \mathbb{R}^n$. Thus one can apply concepts from Euclidean spaces to it, rather than be teased by topological concepts.

So for example, inferring the dimensionality is much easier after one has started from euclidean space and ended up into an euclidean space. Rather than trying to infer dimensionality in the topological spaces? Where they might not even have definitions.

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General set does not have coordinate systems. By introducing charts, these specific kind of sets [i.e. manifolds] could equip the familiar coordinate systems. Then almost all geometric properties could be deduced using tools from Multivariate Calculus and linear algebra etc. that are applicable to the familiar space $\mathbb R^m$.

Your reasoning added in the last several paragraphs is acceptable and convincing enough

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