Compute the integral $\int_S{\vec{r} \cdot \hat{n}dS}$

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 $$4πabc$$.

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

• 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. – achille hui May 9 at 19:49
• @achillehui got it! – Subhasis Biswas May 9 at 19:52