Question

I need help finding the flux through an ellipsoid (the surface) where F= and x^2+y^2+4z^2=16. How do I go about doing that? I have to evaluate it using the divergence theorem and through a method that involves a sphere through the origin. please help!

Answer 1

Use the divergence theorem
http://en.wikipedia.org/wiki/Divergence_theorem

Answer 2

Div(F)=2 x + 4 y + 6 z
The flux through the skin of the ellipsoid is equal to the integral of the Div(F) over the
meat inside that skin.
\[
\int _{-4}^4\int
_{-\sqrt{16-x^2}}^{\sqrt{16
-x^2}}\int _{-\frac{1}{2}
\sqrt{-x^2-y^2+16}}^{\frac{
1}{2} \sqrt{-x^2-y^2+16}}(2
x+4 y+6 z)dzdydx
\]
Since Div(F) is odd in x, in y and in z and the region is symmetric with respect to x, y and z, the above integral is equal to zero.