Assume we are given for some sufficiently smooth function $g : \mathbb{R}^6 \to \mathbb{R}$ the following convergent integral.
$$ \int \limits_{V \subseteq \mathbb{R}^3} g(x,x') dx' $$
Is there known some criteria that can be used to find out whether the integral can be written as a divergence in a sense that if $x = (x_1, x_2, x_3)$ then there exists a function $f: \mathbb{R^3} \to \mathbb{R}^3$ such that:
$$\int \limits_{V \subseteq \mathbb{R}^3} g(x,x') dx' = \sum \limits_{i=1}^3 \frac{\partial f_i}{\partial x_i} (x).$$
This problem occurs in hydrodynamics where we are integrating the given integral. It would be nice if it could be written as divergence of some function because then the integral over volume would be reduced to an integral over surface.
If the function $G: \mathbb{R}^3 \to \mathbb{R}$ given by
$$G(x) = \int \limits_{V \subseteq \mathbb{R}^3} g(x,x') \,dx'$$
is sufficiently well behaved, then there exists a solution $\phi$ of the Poisson equation $\nabla^2 \phi = G$, e.g., in terms of a Green's function.
Thus,
$$\int \limits_{V \subseteq \mathbb{R}^3} g(x,x') \,dx' = \nabla \cdot f,$$
where $f = \nabla \phi$.