Suppose that we have a vector valued function $D(x)$ with derivative $H(x)$ and that both of these are smooth. Under what conditions does there exist a function $f(x)$ such that $\nabla f(x) = D(x)$? Is there a functional form for it? I was looking for the multivariate equivalent of the second fundamental theorem of calculus, but was coming up empty.
In my particular case, $H$ is symmetric and semi-definite.