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Use the Divergence theorem to calculate the flux of the vector field: $$ \vec{F}=\frac{1}{(x^2+y^2+z^2)^{\frac{3}{2}}}\big<x,y,z\big> $$ over the upper part of the ellipsoid $S:x^2+y^2+\frac{z^2}{4}=1$, with normal pointing outwards.


I can't close the ellipsoid with the plane $z=0$ to apply the divergence theorem because I have a singularity at the origin. So what I tried to do was: $$ \iint_{S1}\vec{F}\cdot\vec{dS} - \iint_{S2}\vec{F}\cdot\vec{dS} = \iiint_V \text{div}(\vec{F})dV $$ where $S1$ is the boundary of the ellipsoid, $S2$ is the boundary of the unit sphere, and $V$ is the volume between them, like the following image:

enter image description here

I've subtracted $S2$ because the normal vector of it is actually going inward the center of the sphere.

And I'm done here. I don't know how to proceed since I don't know how to parametrize the volume $V$.

Can someone please help me? I really need to understand this!

Thanks!

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In cylindrical coordinates: $$ 0\leq\theta\leq2\pi,\quad 0\leq\rho\leq1,\quad \sqrt{1-x^2-y^2}\leq z\leq2\sqrt{1-x^2-y^2} $$

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