Calculate the inverse of a multi-variable function

Question

How would I calculate the inverse of

$F(x,y)=\left( \textrm{arctan} \left(\frac{ay}{x}\right),\frac{x^2+a^2y^2}{2a}\right)$

$a$ is a constant.

Answer 1

First set $F(x,y) = (X,Y)$

Then you have $$ay = x \textrm{tan}X$$ and $$x^2 + (a y )^2 = 2 a Y$$ Sub the first eqn into the second to get $$x^2 = 2a Y /(1 + (\textrm{tan} X)^2 )$$

This should get you started.