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:

$$\int_A (\nabla \cdot \vec F) dV = \int_{\partial A} \vec F \cdot d \vec A$$

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