The following seemingly-simple problem came up when working on a problem in the fluid theory of plasmas.



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> Given a vector field $\mathbf{A}$, find a symmetric tensor $\mathbf{P}$ such that $\boldsymbol{\nabla}\times\mathbf{A} = \boldsymbol{\nabla}\cdot\mathbf{P}$.
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This isn't very hard if you don't require $\mathbf{P}$ to be symmetric (e.g. $P\_{ij} = \epsilon\_{ijk}A\_k$ works), but the symmetry requirement is proving harder than I thought. I tried this over on Math Stack Exchange and got no responses, so I'm hoping you guys might be able to help.



Relation between curl and tensor divergence
More generally, you can consider a given vector field v, not necessarily a curl. To solve $\nabla\cdot P=v$, set $P=\nabla u+(\nabla u)^T$. This results in the equation $\Delta u+\nabla(\nabla\cdot u)=v$, which is an elliptic system for u. 


