Is it possible to nontrivially represent the cross product of a vector field [itex]\vec{f}(x,y,z)[/itex] with its conjugate as the gradient of some scalar field [itex]\phi(x,y,z)[/itex]?(adsbygoogle = window.adsbygoogle || []).push({});

In other words, can the PDE

[itex]\vec{\nabla}\phi(x,y,z) = \vec{f}(x,y,z)\times\vec{f}^\ast(x,y,z)[/itex]

be nontrivially (no constant field [itex]\vec{f}[/itex]) solved?

If not, why? If so, can you give an example of such a scalar field? This problem has popped up in my research and I'm afraid my PDE skills are lacking.

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# Cross product as a gradient?

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