Help with flux double integral I have a question regarding flux. 
The vector field $F(x,y,z) = 4xi + 4yj + 4zk$, and the surface $S$ is the sphere $x^2 +  y^2 + 4z^2 = 3$. Would it be best to attack this with through polar? How can I find the normal vector and $dS$ in the flux equation?
 A: First of all $x^2 + y^2 + 4z^2 = 3$ isn't a sphere, but an ellipsoid. Yeah, it seems that you can do the problem using the polar coordinates, as the vector field is rather simple-looking. But you shouldn't use "pure" polar coordinates, in fact you should use elliptic coordinates, just because of the reason I mentioned above. If you're not familiar with the elliptic coordinates they are just the usual polar coordinates with some coordinates multiplied with various constants.
An easier way to solve this problem is by using the Divergence (Gauss') Theorem if you're familiar with it and you're allowed to use it. So we have:
$$\iint \vec{F} \cdot \vec{n} \ dS = \iiint \text{div}(\vec{F}) \ dV$$
where the first integral is over the boundary of the sphere, while the second over it's volume. As $\text{div}(\vec{F}) = 12$ this should make stuff simplier, basically reducing the problem to just finding the volume of the ellipsoid. You will again need to use elliptic coordinates to find it, but the calculations will be much easier.
