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.

Not sure what the relationship is with those coordinates, but your map is $\mathbf y= \mathbf x/|\mathbf x|^2$, so $|\mathbf y|=1/|\mathbf x|$ and $\mathbf x = \mathbf y|\mathbf x|^2 = \mathbf y/|\mathbf y|^2$ which is the same map — Calvin Khor, Mar 13 '21 at 07:54

score 2 · Answer 1 · answered Mar 13 '21 at 07:55

2

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

answered Mar 13 '21 at 07:55

Travis Willse

score 2 · Accepted Answer · answered Mar 13 '21 at 07:57

2

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

answered Mar 13 '21 at 07:57

Parcly Taxel

2 Answers2