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Homework Help: Quick Green-Gauss theorem Question

  1. Sep 13, 2006 #1
    "Quick" Green-Gauss theorem Question

    HELLO!! I am a grad student in Mech & Aero Engineering and have come across a bit of trouble with one of my problems. I'd appreciate your assistance.

    "Given a general closed surface S for which the position vector and normal are known at every point, derive a formula for the volume enclosed by S. Verify your relation for the special case of a sphere."

    I know to begin with the Green-Gauss theorem, which relates surface integrals to volume integrals (sorry I don't know how to display mathematics in here), but I'm not sure how to manipulate the bounded volume and isolate!

    thank you - Ciao
     
  2. jcsd
  3. Sep 13, 2006 #2

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    Gauss's theorem states:

    [tex]\int_A (\nabla \cdot \vec F) dV = \int_{\partial A} \vec F \cdot d \vec A[/tex]

    Where [itex]A[/itex] is a volume in space and [itex]\partial A[/itex] is its bounding surface. If you can find a vector function [itex] \vec F[/itex] with [itex]\nabla \cdot \vec F=1[/itex], then the LHS, and so also the RHS, is equal to the volume of A.
     
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