Quick Green-Gauss theorem Question

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In summary, the Green-Gauss theorem is a useful tool for relating surface integrals to volume integrals. By finding a suitable vector function, the volume enclosed by a closed surface can be calculated using this theorem.
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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.
 

1. What is the Quick Green-Gauss Theorem?

The Quick Green-Gauss Theorem is a mathematical concept used in computational fluid dynamics to simplify the calculation of fluid flow properties. It is based on the Green-Gauss Theorem, which relates the surface integral of a function to the volume integral of its divergence.

2. How is the Quick Green-Gauss Theorem different from the regular Green-Gauss Theorem?

The Quick Green-Gauss Theorem uses a simplified version of the Green-Gauss Theorem by only considering the surface integrals of the function, rather than both surface and volume integrals. This makes it more efficient for computational purposes.

3. Why is the Quick Green-Gauss Theorem important in computational fluid dynamics?

The Quick Green-Gauss Theorem allows for a more efficient and accurate calculation of fluid flow properties, such as velocity and pressure, in computational fluid dynamics simulations. It reduces the computational cost and improves the accuracy of the results.

4. What are the limitations of the Quick Green-Gauss Theorem?

The Quick Green-Gauss Theorem is only applicable to certain types of problems, such as those involving incompressible fluids. It also assumes a regular and structured grid, which may not always be the case in real-world scenarios.

5. Are there any alternatives to the Quick Green-Gauss Theorem?

Yes, there are other methods for calculating fluid flow properties in computational fluid dynamics, such as the Least Squares or Galerkin methods. These methods may be more accurate for certain types of problems, but they also come with their own limitations and computational costs.

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