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The answer to this question: Divergence-free vector field on a 2-sphere. shows that every divergence-free vector field on the 2-sphere can be generated from a scalar function. (A commenter noted that, given the standard metric, the procedure to get the vector field is to take the gradient of the scalar field and rotate it 90 degrees.)

Can the same thing be said about divergence-free vector fields on the plane? What about on the 2-torus?

Matt Dickau
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    Mariano's answer in the linked post seems to apply without change replacing $\Bbb S^2$ by $\Bbb R^2$, because $\Bbb R^2$ is also symplectic and simply-connected, and his proof did not use compactness of $\Bbb S^2$. – Ivo Terek Apr 02 '20 at 23:08
  • I think this question is related also to this problem: https://math.stackexchange.com/q/3759470/532409 – Quillo Jul 17 '20 at 13:01
  • @IvoTerek , I am not an expert but the sphere is simply connected, as $R^2$, but the torus $T^2$ is not, maybe this changes something. – Quillo Jul 17 '20 at 13:26

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