Consider the equations that relate cylindrical and Cartesian coordinates in $\displaystyle \mathbb{R}^3$;

$\displaystyle x=rcos\theta$

$\displaystyle y=rsin\theta$

$\displaystyle z=z$

a. Near which points of $\displaystyle \mathbb{R}^3$ can we solve for r, $\displaystyle \theta$, and z in terms of the Cartesian coordinates?

b. Explain the geometry behind your answer in (a).

I started off by saying

$\displaystyle F: x-rcos\theta=0$

$\displaystyle G: y-rsin\theta=0$

but then what do you do for the third equation since $\displaystyle z-z=0$? If I use H:0 then that makes $\displaystyle \Delta J=0$ everywhere, meaning that it is not in general possible to solve for r, $\displaystyle \theta$, and z in terms of x, y, and z. That seems so odd that I assume it's wrong, and I have to think that my mistake is in H.

Any thoughts?