[Math] Show that this is a diffeomorphism

Question

I have a function $F:(0,2\pi) \times \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}^2$
with $(\phi,r)\mapsto(r(\phi-\sin(\phi)),r(1-cos(\phi)))$ and want to show that this is a smooth(meaning $C^{\infty}$ ) diffeomorphism. actually, i have already shown that this is ja bijection, but it seems to be difficult to show that the inverse function is also arbitrarily often differentiable.

Answer 1

There are three things to check.

1. The map $F$ is $C^\infty$ smooth.
2. Its Jacobian is nowhere zero.
3. It is a bijection.

Once you are done with 1 and 2, the inverse function theorem applies and shows that the map is locally a $C^\infty$ diffeomorphism. But Part 3 is still necessary because you are asked to show it's a global diffeomorphism.