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I was asked to calculate the flux of the field $$\mathbf A = (1/R^2)\hat r$$ where $R$ is the radius, through the surface of the ellipsoid $$\left(\frac{x^2}{a^2}\right) + \left(\frac{y^2}{b^2}\right) + z^2 = 1$$

Without doing the formal calculations (I have never encountered a problem like this), I neverless suspect that it's equivalent to the flux of the same field through the surface of a sphere whose radius is the mean radius of the ellipsoid (i.e. $4 \pi$)...am I wrong and/or how to solve?

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The field you are asked about is the prominent gravitational / Coulomb field which main characteristics is the fact that its divergence is $0$ everywhere except for the origin point $(0,0,0)$: $$ \nabla\cdot{\mathbf A}=4\pi\delta({\mathbf r}), $$ which is easy to check.

Therefore by divergence theorem the flux over the surface is $4\pi$ since the origin point is inside the surface.

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