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Outward flux through an ellipsoid

  1. Dec 5, 2008 #1

    kdr

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    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. jcsd
  3. Dec 5, 2008 #2

    HallsofIvy

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    Staff Emeritus
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    The bounds depend on the coordinate system you are using. Using spherical coordinates, the bound for the sphere of radius 100 are, of course, [itex]\rho from 0 to 100[/itex], [itex]\theta[/itex] from 0 to [itex]2\pi[/itex], and [itex]\phi[/itex] from 0 to [itex]\pi[/itex]. For the ellipse the bounds on [itex]\theta[/itex] and [itex]\phi[/itex] are the same but the bounds on [itex]\rho[/itex] will be functions of [itex]\theta[/itex] and [itex]\phi[/itex]. 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. [itex]x= 3 sin(\phi)cos(\theta)[/itex], [itex]y= 2 sin(\phi)sin(\theta)[/itex] and [itex]z= \sqrt{6}cos(\phi)[/itex] with [itex]\theta[/itex] from 0 to [itex]2\pi[/itex] and [itex]\phi[/itex] from 0 to [itex]\pi[/itex] is an appropriate parameterization for the surface of the ellipsoid.
     
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