Gradient of Divergence is Conservative

Theorem
Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions.

Let $\mathbf V$ be a vector field on $\R^3$:

Then the gradient of the divergence of $\mathbf V$ is a conservative vector field.

Proof
The divergence of $\mathbf V$ is by definition a scalar field.

Then from Vector Field is Expressible as Gradient of Scalar Field iff Conservative it follows that $\grad \operatorname {div} \mathbf V$ is a conservative vector field.