Using the divergence theorem, evaluate where and is the...

The divergence theorem states that where is a closed surface, is the volume inside it and is a good enough vector field defined inside and on it. The symbol d the divergence operator.

In our case is an ellipsoid, a smooth closed surface, the vector field is defined everywhere and is infinitely differentiable. The divergence of is

The divergence theorem states that where is a closed surface, is the volume inside it and is a good enough vector field defined inside and on it. The symbol d the divergence operator.

In our case is an ellipsoid, a smooth closed surface, the vector field is defined everywhere and is infinitely differentiable. The divergence of is

where are the components of

In our case    thus

and the integral becomes a very simple one:

The equation of our ellipsoid is equivalent to    therefore the semiaxes are and and the final answer is  

I used WolframAlpha to sketch the ellipsoid.