I am trying to determine the outward flux integral of the vector field $$F(x,y,z) = \frac {(x-y)i + (x+y)j + zk}{(x^2+y^2+4z^2)^{3/2}}$$ across the ellipsoid $x^2 + y^2 + 4z^2=4$.
Could I have a hint for this? I can't use the divergence theorem because of the singularity, so I am trying to evaluate it directly. I have replaced the denominator with 8 but I'm not sure what to do next.
Parametrizing is unnecessary after a simplification. The integrand of the surface integral simplifies to
$$F(x,y,z) = \frac{(x-y,x+y,z)}{8}$$
by plugging in $x^2+y^2+4z^2=4$. This is a vector field divergence theorem applies to:
$$\nabla\cdot F = \frac{3}{8}$$
which means the integral evaluates to $\frac{3}{8}\operatorname{vol}(x^2+y^2+4z^2\leq 4) = 2\pi$