I know that a vector field $\mathbf{F}$ is called irrotational if $\nabla \times \mathbf{F} = \mathbf{0}$ and conservative if there exists a function $g$ such that $\nabla g = \mathbf{F}$. Under suitable smoothness conditions on the component functions (so that Clairaut's theorem holds), conservative vector fields are irrotational, and under suitable topological conditions on the domain of $\mathbf{F}$, irrotational vector fields are conservative.
Moving up one degree, $\mathbf{F}$ is called incompressible if $\nabla \cdot \mathbf{F} = 0$. If there exists a vector field $\mathbf{G}$ such that $\mathbf{F} = \nabla \times \mathbf{G}$, then (again, under suitable smoothness conditions), $\mathbf{F}$ is incompressible. And again, under suitable topological conditions (the second cohomology group of the domain must be trivial), if $\mathbf{F}$ is incompressible, there exists a vector field $\mathbf{G}$ such that $\nabla \times\mathbf{G} = \mathbf{F}$.
It seems to me there ought to be a word to describe vector fields as shorthand for “is the curl of something” or “has a vector potential.” But a google search didn't turn anything up, and my colleagues couldn't think of a word either. Maybe I'm revealing the gap in my physics background. Does anybody know of such a word?
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$\begingroup$Differential geometry answer: Coboundary (see comment of Circonflexe).
Elementary calculus answer: Try 'curl vector field' (see comments of JonathanRayner and myself).
Actually, in some textbooks, e.g. Thomas Calculus, the $F = \nabla f$ is more an equivalent definition of conservative vector field with $F = \nabla f$ as the definition of a gradient vector field. Then we can say conservative if and only if gradient (under such and such). The definition of conservative in Thomas Calculus is in terms of path independence of line integrals.
Calling $F$ with $F = \nabla f$ as a gradient vector field instead of a conservative vector field could analogously lead to $G$ with $G = curl H = \nabla \times H$ as something like 'curl vector field'.
$\endgroup$ $\begingroup$The concept of a pseudovector is quite similar, being associated with the curl of a vector. An apparent generalization is to the concept of a pseudovector field, for which I found a reference here. There is also the related notion of a vorticity arising as the curl of a vector field but this is not generic.
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