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.
By Helmholtz's theorem -- the fundamental theorem of vector calculus -- any continuously differentiable vector field that vanishes at infinity sufficiently fast can be decomposed into irrotational and solenoidal parts of the form
$$\mathbf{D} = \nabla f + \nabla \times \mathbf{a}$$
We can find $f$ and $\mathbf{a}$ given the divergence and curl of the vector field since
$$\nabla \cdot \mathbf{D} = \nabla \cdot \nabla f + \nabla \cdot \nabla \times \mathbf{a} = \nabla^2f$$
and
$$\nabla \times \mathbf{D} = \nabla \times \nabla f + \nabla \times \nabla \times \mathbf{a} = \nabla \times \nabla \times \mathbf{a},$$
leading to linear partial differential equations that can be solved with suitable boundary conditions.
If $\mathbf{D}$ is curl-free (irrotational), then $\mathbf{D} = \nabla f$.