1
$\begingroup$

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

1 Answers 1

1

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).

  • 0
    @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.2012-10-19