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I tried to prove this problem using the Helmholtz decomposition theorem, but it seems the two are entirely contradictory--thus leaving me with empty hands. Does anyone know how to proceed?

problem

Thanks

Moehringer

Willie Wong
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moehringer
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    it is locally, but you are asked about cohomology. Anyway, see http://math.stackexchange.com/questions/81405/anti-curl-operator – Will Jagy Dec 12 '13 at 02:14
  • @moehringer : I don't know anything about the Helmholtz decomposition theorem, but how far did you get? Did you do (a) and (b)? How far did you get on (c) and (d)? – Stefan Smith Dec 12 '13 at 02:17
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    I see, http://en.wikipedia.org/wiki/Helmholtz_decomposition which says nothing more here than the Poincare Lemma, that is there is a solution in a small disk around any point, or ball in three dimensions. http://en.wikipedia.org/wiki/De_Rham_cohomology – Will Jagy Dec 12 '13 at 03:06
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    Saying that $F$ is a vector field "in the plane" is a bit misleading. It is, as noted in (c), a vector field on the punctured plane. On a simply connected domain, a curl-free vector field is a gradient. – ronno Dec 18 '13 at 09:55
  • The curl of your vector field F is the Dirac delta measure on the origin, i.e. it's everywhere zero except at the origin, and it integrates to 1. – Jules Jan 11 '18 at 13:43

2 Answers2

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Read an answer here: Helmholtz decomposition contradictions PDF https://www.academia.edu/34072970/Helmholtz_Decomposition_Remains_Opened_for_Future_Researchers

Alexandr
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Read an answer here: Helmholtz decomposition contradictions PDF http://analysis3.com/Helmholtz-decomposition-contradictions-pdf-e117376.html

Alexandr
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