What vector field property means “is the curl of another vector field?”

Solution 1:

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'.

Solution 2:

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.