Let us suppose we have $m$-dimensional smooth topological manifold $M$. Let $(U,\varphi)$ and $(V,\psi)$ be two charts on the manifold and $U \cap V \neq \emptyset$. For a point $p \in U \cap V$, we then have two coordinate systems: \begin{equation} \varphi: U \to \mathbb{R}^m \;\;\; \text{and} \;\;\; \psi: V \to \mathbb{R}^m \end{equation} Thus, we can change coordinate by using the transition map $\varphi \circ \psi^{-1}$. This is considered to be a passive coordinate transformation.
On the other hand, we can use a diffeomorphism, $\phi : M \to M$, to change the manifold itself, which will result in a new coordinate system. This is considered to be an active coordinate transformation.
Now, in my physics courses I've always learnt that active and passive transformations are exactly the same, and it is just a matter of convention which one we choose. However, the above seems to suggest that it is always possible to change coordinate via an active transformation, whereas it is only possible to make a passive coordinate transformation if at least two open sets overlap each other. Is this true? Or is is it also possible to somehow make an passive coordinate transformation if the open sets do not overlap.
I guess, a different phrasing of this question would be:
For any chart $(U,\varphi)$, are we allowed to change the homeomorphic map $\varphi$ to another map in order to get a different coordinate system? I.e. is the map $\varphi$ corresponding to $U$ unique, or can we associate multiple maps with $U$?
I've tried to reason that we are in fact able to always choose a different map, but I'm not sure and none of the books I own mention this explicitly.
