What type of vector field has an zero integral of divergence over all 3D space?

Question

Assume $\mathbf{V}(\mathbf{x})$ is a vector field in 3D space. Its divergence is $\nabla\cdot \mathbf{V}(\mathbf{x})$. The integration of this divergence over all 3D space is $I=\displaystyle{\int\nabla\cdot \mathbf{V}(\mathbf{x})d\mathbf{x}}$. My question is: under what condition can $I$ always be zero? The more general the condition is, the better the answer would be. Thanks a lot.

Answer 1

One classic condition is that $V$ vanishes at infinity, in which case the integral of divergence is zero by Stokes's Theorem.