Find the inverse of $f(x,y,z) = (\frac{x}{x^2+y^2+z^2}, \frac{y}{x^2+y^2+z^2}, \frac{z}{x^2+y^2+z^2})$

Question

I attempted to switch to cylindrical / spherical coordinates, but I keep getting stuck. Note that $(0,0,0)$ is not in the domain or codomain.

Answer 1

$f$ maps a vector with length $r=\sqrt{x^2+y^2+z^2}$ to the same-direction vector but with length $\frac1r$. Therefore $f^{-1}=f$.

Answer 2

Hint We can write this map in coordinate-free notation as $$f({\bf x}) = \frac{\bf x}{||{\bf x}||^2} .$$