Paraboloidal Coordinates How would one find the transforms for paraboloidal coordinate systems. ie) I want to find $x,y$, and $z$ in terms of other variables so that I can use the Jacobian to find the differential volume. The paraboloid in question is $z = 16 - x^2 - y^2$

If you use cylindrical coordinates \begin{align} x &= r \cos \theta \\\ y &= r \sin \theta \\\ z &= z \end{align} then $$ z = 16 - r^2 $$ and $$ \frac{D(x,y,z)}{D(r,\theta,z)} = r. $$ which leads to a pretty easy volume calculation (if top and bottom of paraboloid are simple enough).