<s>[INST] How to find inverse of general curvilinear coordinates Lets say I have a curvilinear coordinate system 
$A=A(x,y,z) = \frac{x^2+y^2+z^2}{2z} $, $B=B(x,y,z)= \frac{x^2+y^2+z^2}{2\sqrt{x^2+y^2}}$, $C=C(x,y,z)=\tan^{-1}(y/x)$
How do I find the inverse of those i.e
$x=x(A,B,C)$, $y=y(A,B,C)$, $z=z(A,B,C)$
I know that in cylindrical or spherical coordinates I could do it based on the geometry, but I dont have the geometry, I just have the equations for A, B, and C in terms of x, y, and z
 [/INST] I will assume that $x,y,z>0$. You will need to check if it make sense to have them negative.
The first equation we are going to use is $$\tan C=\frac{y}{x}$$
Since everything else we have squares, I will write this as $$y^2=x^2\tan^2C\tag{1}$$
The second equation can be written from the ratio $A/B$ or in fact we can use the square of that:
$$\frac{A^2}{B^2}=\frac{x^2+y^2}{z^2}$$
This yields $$z^2=\frac{B^2}{A^2}(x^2+y^2)\tag{2}$$
The third equation is $$\frac{1}{A^2}+\frac{1}{B^2}=\frac{4(x^2+y^2+z^2)}{(x^2+y^2+z^2)^2}=\frac{4}{x^2+y^2+z^2}$$
I will rewrite this as $$x^2+y^2+z^2=\frac{4}{\frac{1}{A^2}+\frac{1}{B^2}}\tag{3}$$
Now it should be trivial to find $x^2$, $y^2$, and $z^2$ from $(1)$, $(2)$, and $(3)$.
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