Just to lay the backdrop, suppose that I have an n-dimensional manifold $M^n$ and choose a particular coordinate chart on the manifold $(U,\phi)$, $\phi: U \rightarrow \mathbb{R^n}$.

I understand that $\phi$ is the local system of coordinates. My question concerns $\phi^{-1}$. In many books or lecture notes, $\phi^{-1}$ is called a local parameterization of $M$, and it is left at that. I am having trouble envisioning how the inverse map of a coordinate map is a parameterization.

Would someone please walk me through a simple example to clear up my confusion as to $\phi$ and $\phi^{-1}$? The thing that makes the confusion worse is that I somehow think that it should be obvious and that I already know this from coordinate transformations I have done in calculus and in physics.

Thanks.