I'm an undergraduate mathematics educator and I teach a lot of multivariable calculus. I posed this question on MSE over four years ago and I haven't gotten any definitive answers (despite 12 upvotes and a bounty posted). It could be there's no answer, but someone suggested I ask on this forum.
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?
TL;DR: gradient is to conservative as curl is to ___?
Best Answer
I think it’s just called a solenoidal field (incompressible field), because by definition, if we have $\mathbf{\nabla}\times \mathbf{A}= \mathbf{V}$, $$\mathbf{\nabla}\cdot(\mathbf{\nabla}\times\mathbf{A})= \mathbf{\nabla}\cdot \mathbf{V }=0$$ because the divergence of the curl is 0. Because of this, any field that can be derived from a vector potential is automatically incompressible. Since every incompressible field can be expressed as the curl of some potential, they are precisely equivalent. Therefore, we already have a name for it, and it doesn’t need a new one.