# Outward flux through an ellipsoid

1. Dec 5, 2008

### kdr

1. The problem statement, all variables and given/known data
Let r(x,y,z)=<x,y,z>. Compute the outward flux of F=r/|r|^3 through the ellipsoid 4x^2+9y^2+6z^2=36.

2. Relevant equations

3. The attempt at a solution
I know that I can't use the divergence theorem on the region inside S because F isn't continuous at 0. But I can do it on the region between S and say, a sphere of radius 100 right? An how would I set that up? I'm having some trouble figuring out the bounds between the ellipsoid and sphere.

2. Dec 5, 2008

### HallsofIvy

The bounds depend on the coordinate system you are using. Using spherical coordinates, the bound for the sphere of radius 100 are, of course, $\rho from 0 to 100$, $\theta$ from 0 to $2\pi$, and $\phi$ from 0 to $\pi$. For the ellipse the bounds on $\theta$ and $\phi$ are the same but the bounds on $\rho$ will be functions of $\theta$ and $\phi$. And, of course, you will still need to find the flux through the sphere directly.

Perhaps it would be simpler just to find the flux through the ellipsoid by integrating over the surface. $x= 3 sin(\phi)cos(\theta)$, $y= 2 sin(\phi)sin(\theta)$ and $z= \sqrt{6}cos(\phi)$ with $\theta$ from 0 to $2\pi$ and $\phi$ from 0 to $\pi$ is an appropriate parameterization for the surface of the ellipsoid.

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