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Is there a function $F\in C^1(\mathbb R^3,\mathbb R^3)$ with div($F$)=$0$ and $\int_{\partial B_1(0)}\langle F(x),x\rangle dS_{\partial B_1(0)}=1$?
This looks a little bit like the theorem of Gauß but I dont know how to prove or disprove it.

andy
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  • did you mean to put an $=0$ at the end of your first line? I assume yes; as a hint, what is the unit outward normal to $\partial B_1(0)$? – peek-a-boo Aug 14 '22 at 15:01
  • @peek-a-boo it is a "=1". I corrected it. – andy Aug 14 '22 at 15:17
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    As the surface is $\partial B_1(0)$, the normal to the surface at any point $x$ is $x$ itself, which means the integral is just the flux of $F$. On the other hand indeed the divergence theorem tells us this flux has to be zero, so I don't think it is possible. – Theorem Aug 14 '22 at 15:28
  • A divergence free vector field can be written as the curl of something. Perhaps using this fact will lead you to using Stokes's theorem. – K.defaoite Aug 14 '22 at 16:17

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