Specifically, if we are given a vector field $F(x,y,z)=(F_1(x,y,z),F_2(x,y,z),F_3(x,y,z))^T$, how do we know if $F$ is conservative? I.e. does there exists an $f$ such that $\nabla f=F$? I am aware of the curl and the second derivative methods, as well as the fact that that all line integrals must be path independent.
However, are there any other results or conditions that would also give me existence of a solution? Preferably a method that does not rely on $F_1,$ $F_2$, and $F_3$ having derivatives.