A vector field $F:\mathbb R^n\to \mathbb R^n$ is conservative if for some "potential function" $f:\mathbb R^n\to \mathbb R$, we have $F=\nabla f$.
I am intuitively wondering "how many" vector fields are conservative. Obviously this can be interpreted in multiple ways, which is why I have multiple questions:
if we define a "uniform" measure on the space of such vector fields $\mathcal F$ for say $n=2$, is the set of conservative vector fields then measured larger than $0$?
can we put certain unrestrictive assumptions, or "natural" assumptions, on $F$ to ensure that they are conservative?
how often do we encounter nonconservative vector fields in practice, e.g. in physics?