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Let $u=((x+y)z,y+xz,z^2+xz)$ be a vector field and $\Sigma$ be the surface $\frac{x^2}{9}+\frac{y^2}{4}+\frac{z^2}{36}=1$. Calculate $\iint_{\Sigma}(u\cdot N)dS$.

So $\text{div }u=1+3z+x$ and by using gauss theorem $\iiint_K (1+x+3z)dV=\iiint_K1dV$, by symmetry. The volume of $K$ is $\frac{4}{3}\pi*3*2*6=48\pi$ but the answer should be $8\pi$ or $-8\pi$. What is wrong here?

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    $\begingroup$ What "symmetry" are you referring to here? $\endgroup$
    – PrincessEev
    Commented Apr 19 at 6:51
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    $\begingroup$ Symmetry in the $+-z$, $+-x$ direction. I asked this question 2 days ago math.stackexchange.com/questions/4900361/… $\endgroup$
    – per persson
    Commented Apr 19 at 6:56
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    $\begingroup$ I don't see anything wrong with your work as written. The supposed "answer" is wrong. $\endgroup$
    – Ninad Munshi
    Commented Apr 19 at 7:01
  • $\begingroup$ The answer is given by my teacher. He calculated the volume and derived the answer from it without explaining. $\endgroup$
    – per persson
    Commented Apr 19 at 7:08

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