Showing multivariable maps are smooth

Question

How does one in practice actually show a map is smooth?

Say I am considering: $$F:B^n\to \Bbb R^n,\qquad F(x) = \frac{x}{\sqrt{1-\|x\|^2}},$$ where $B^n$ is the open unit ball in $\Bbb R^n$ and $\|x\|$ is the euclidean norm on $\Bbb R^n$.

Smoothness means that all partial derivatives of all orders exist. I can compute the Jacobian matrix, which seems quite ugly, but even still, I can't just deduce that the partial derivatives of all orders exist (and they don't vanish in few steps).

Usually the maps I encounter are obviously smooth, but only because they are polynomial in each entry and what-not, but outside of these cases, I don't know how to approach this. Are there some results that I haven't seen?

How do you actually show such a map is smooth?

Answer 1

Your function is smooth because it can be obtained from smooth functions ($x$, square root, and $\lVert x\rVert^2$) using composition and arithmetic opeations.

Answer 2

I would go even further and say that smoothness requires all partial derivatives to exist and be continuous.

One way to prove smoothness is to find a closed form for each n-th partial derivative. This can be difficult in practice.

For easy functions like yours, you can use the standard result that a sum, multiplication, square root on $(0, +\infty)$, and composition of smooth functions is smooth. You can prove this using the sum/product rules for derivatives and the chain rule.