How to prove $\mathbf F$ is conservative, i.e. $\mathbf F = \mathbf\triangledown\Phi$ by using the fact that $\mathbf \triangledown\times\mathbf F=\mathbf0$ implying $\mathbf F=\triangledown\Phi$ and the domain is a simply connected region
Actually I can prove the converse, i.e. $\mathbf F=\triangledown\Phi$ implies $\triangledown\times\mathbf F=\mathbf0$. But I cannot understand how we prove the other direction. Particularly, how does the constraint simply-connected domain work here?
Could anyone elaborate how we judge $\mathbf F$ is conservative or not by this principle with some concrete examples and reasons? Thanks a lot!