This is a very simple question from introductory differential geometry. Suppose I have an 2-dimensional manifold $M^2$ that is, for simplicity, a subset of $\mathbb{R}^2$. Now suppose $(U,\phi)$ is a local coordinate chart on $M$ such that $\phi:M \rightarrow \mathbb{R}^2$. This much I'm clear on.
My question concerns $\phi^{-1}$. A lot of books call this a local parameterization of $U$ and leave it at that. Could someone give me a very simple example of the explicit manifestation of $\phi$ and $\phi^{-1}$ for the case where $M$ is a subset of $ \mathbb{R}^2$? (Not the circle example which I've seen.)
What is confusing me is how $\phi$ and $\phi^{-1}$ are manifest, especially $\phi^{-1}$ such as where do the parameters come from.
I suppose a simple example could be the graph of the function $f(x) = x^3$. Let $X = \{(x, x^3) \; | \; x \in \mathbb{R}\}$. Clearly $X$ is a submanifold of $\mathbb{R}^2$. You can define a global coordinate chart $\varphi:\mathbb{R} \to X$ given by $\varphi(x) = (x, f(x))$ which is clearly smooth. Note $\pi:X \to \mathbb{R}$ given by $\pi(x,x^3) = x$ (i.e. projection onto the first coordinate), we see that $\pi\circ \varphi: \mathbb{R} \to \mathbb{R}$ satisfies $\pi(\varphi(x)) = x$ and $\varphi\circ \pi:X \to X$ satisfies $\varphi(\pi(x,x^3)) = (x,x^3)$, hence $\pi$ is the inverse to $\varphi$, hence $\pi = \varphi^{-1}$.
Does this answer your question how inverses arise in coordinate parameterizations?