Definition ($C^{r}$-diffeomorphism)

Let $U \subset \mathbb{R}^{n}$ and $V \subset \mathbb{R}^{n}$ be Open subsets. A $C^{\infty}$ map $F : U → V$ is called a diffeomorphism if it is Bijective and has a $C^{\infty}$ inverse $F^{-1} : V → U$.