Quick Green-Gauss theorem Question

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"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
 

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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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