I've got the revolution surface $S$ generated by a regular, simple curve $\alpha:(a,b)\subset\mathbb{R}\to\mathbb{R}^3, \alpha(u)=(f(u),0,g(u))$. I have to prove that the map $$\vec{x}: (a, b) \times (c, d) \to S : (u, v) \mapsto (f(u) \cos( v), f(u) \sin( v), g(u))$$ being $(c,d)$ any real interval of length $2\pi$, is a local chart of $S$ and that any two different local charts of this kind are compatible, i.e., if $\vec{y}$ is another local chart then $\vec{x}\vec{y}^{-1}$ and $\vec{y}\vec{x}^{-1}$ belong to $\mathcal{C}^\infty$.
To prove that $\vec{x}$ is a local chart, I'm trying to show that it is homeomorphism onto its image. I've already shown that it is continuous and biyective, but I guess I need the inverse of $\vec{x}$ to prove what's left. Does anyone have any idea about how to compute it or al least show that it is continuous and $\mathcal{C}^\infty$?