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"Find a pair of fields having equal divergences in some region, having the same values on the boundary of that region, and yet having different curls"

Can anyone name a pair of fields that fit this requirement (It would be awesome if the region was the unit sphere)

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    Your gravatar is very fitting for this question :-)2011-09-13
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    Haha, Its a fluke though2011-09-13
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    Note that the difference $\mathbf{h}=\mathbf{F-G}$ will (1) be divergence-free, (2) vanish on $\partial D$, and (3) be a nonzero field (i.e. not identically the zero vector). It suffices to find such an $\mathbf{h}$, because then $\mathbf{G}$ can be chosen arbitrarily and $\mathbf{F}$ is determined. Helmholtz decomposition sounds like it might be relevant in some fashion for a general approach.2011-09-13
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    I'm confused by that, can you continue with how exactly, one would use that to find a pair of fields that match the conditions2011-09-13
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    @uguhu: Which point of mine are you referring to with the word "that"? I reduced the problem to a simpler one of finding a single vector field satisfying some properties and also mentioned Helmholtz decomposition because it just might be helpful; I don't actually have a solution in mind. Also, if you want to alert a user to a response you need to put "@anon" or equivalent to your comment so that the system pings them.2011-09-13

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