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Calculate the flux of $F=(3xy^2,3x^2y,z^3),$ $ S$ the sphere of radius 1.

I have done $\iint F\cdot n dS= \frac{12\pi}5$

I am not sure about of that result because when I try to solve this by using the Divergence Theorem I get another value:

$$\iint_S\vec F\cdot \vec ndS=\iiint_V\nabla\cdot \vec FdV=3\iiint_V(x^2+y^2+z^2)dV$$ Then, since S is the sphere of radius 1 $x^2+y^2+z^2=1$ $$=3\iiint_VdV=3(\frac{4\pi}{3})=4\pi$$

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$$\iint_S\vec F\cdot \vec ndS=\iiint_V\nabla\cdot \vec FdV=3\iiint_V(x^2+y^2+z^2)dV$$ $$=3\iiint_Vr^2dV=3\int_0^14\pi r^4dr=\frac{12}{5}\pi$$

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  • $\begingroup$ but since S is the sphere of radio 1 x2+y2+z2=1, isn't it? $\endgroup$ May 20 '18 at 2:14
  • $\begingroup$ @AlexTurner It's equal to 1 on the surface of the sphere, but in this case you have to integrate $x^2+y^2+z^2$ over the whole volume of the sphere, so its not always 1. $\endgroup$
    – Macrophage
    May 20 '18 at 2:16
  • $\begingroup$ @AlexTurner Hope you have understood the argument :) $\endgroup$
    – Macrophage
    May 21 '18 at 10:05
  • $\begingroup$ Yes, I understood. Thanks $\endgroup$ May 22 '18 at 15:32
  • $\begingroup$ @AlexTurner You are welcome! $\endgroup$
    – Macrophage
    May 23 '18 at 10:32

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