An explicit expression for a diffeomorphism

Question

I know that the open unitary ball $B^n$ of $\mathbb R^n$ are diffeomorphic and I also know some explicit forms of a diffeomorphism, many can be found on this very website. Now I would like to find a more refined diffeomorphism with some additional properties. Let $R>r>0$. Can we find a diffeomorphism $B_R^n(0)\to \mathbb R^n$ which restricts to the identity on $B^n_r(0)$?

Answer 1

Take a smooth, monotone function $\psi\colon [0,R)\to\Bbb R$ with $$\psi(u) = \begin{cases} 0, & 0\le u\le r \\ 1, & \frac12(r+R)\le u < R \end{cases}\ .$$ The construction is standard when creating partitions of unity, for example.

Define your diffeomorphism $F\colon B_R(0)\to\Bbb R^n$ by $$F(x)= \left(1+\frac{\psi(\|x\|)}{\sqrt{R-\|x\|}}\right) x.$$ You can modify this idea in any number of ways according to your own aesthetics. :)