= Solution
A sufficient condition is that $G$ be an open <simply connected> domain, with $\mathbf F$ continuously differentiable. Under this condition a <curl-free vector field> has a global <potential of a conservative vector field> and hence <path independence> by the <fundamental theorem for line integrals>. One way to see the global step is to deform the closed loop formed by two paths into a point inside $G$: the integral is unchanged during the deformation because the <curl> is zero, as expressed by <Stokes theorem>. Therefore \b[simple connectivity is a sufficient domain hypothesis]. It is not a claim that every particular <curl>-free field needs such a domain; a given field can have a global potential on a domain with holes.
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