Existence of smooth diffeomorphism $f$ of open ball onto itself with $f(0) = p$.

Question

I am trying to show that for every point $p$ of the open $n$-disk $B^n$, there exists a smooth diffeomorphism $B^n \to B^n$ sending $0$ to $p$. Certainly, it seems intuitively obvious for points $p$ that are very close to $0$, but I am not used to explicit constructions of continuous/differentiable maps.

Thanks.

Answer 1

HINT: Look for a diffeomorphism that is the identity outside a neighborhood of a path joining $0$ and $p$. You might think of getting this diffeomorphism by finding the flow of a vector field (which should therefore be $0$ outside said neighborhood).