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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 $z = 16 - x^2 - y^2$

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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).

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Is that $\frac{D(x,y,z)}{D(r,θ,z)} = r$ notation referring to the Jacobian? –  Cactus BAMF Oct 19 '12 at 18:09
    
@CactusBAMF It's the determinant of the Jacobian matrix. Indeed $$dx\,dy\,dz = \left|\frac{D(x,y,z)}{D(r,\theta,z)}\right|\,dr\,d\theta\,dz$$ Sorry for the obscure notation. That's how I learned calculus, the Courant style. –  Pragabhava Oct 19 '12 at 18:24

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