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Compute the integral $\int_S{\vec{r} \cdot \hat{n}dS}$, where $S$ is the surface of the ellipsoid $x^2/a^2+y^2/b^2+z^2/c^2=1$.

I have calculated and thoroughly checked each step and the result I am getting is $4πabc$ which is a very nice looking answer but I cannot verify it in any way (answer not provided). Can someone please verify this?

The steps I used was as follows: 1. Calculate the normal vector to the surface. 2. Replace $(x/a)$ and $(y/b)$ with $u$ and $v$ respectively after protecting it onto the $xy$ plane. Then I have transferred the integral into plane polar coordinates. Then multiplication by $2$ gives me $4πabc$.

[ $\vec{r}= x\hat{i} +y\hat{j}+ z\hat{k}$]

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  • $\begingroup$ Apply divergence theorem and notice $\nabla\cdot \vec{r} = 3$, the surface integral is $3$ times the volume of the ellipsoid. So your answer is correct. $\endgroup$ – achille hui May 9 at 19:49
  • $\begingroup$ @achillehui got it! $\endgroup$ – Subhasis Biswas May 9 at 19:52

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