Gauss's law states that the total electric flux P through any closed surface is equal to the total charge enclosed by that surface.

Thus

$\psi = Q_{enc}$...(1)

that is,

$\Psi=\oint d \Psi=\oint_{S} \mathbf{D} \cdot d \mathbf{S}$...(2)

= Total charge enclosed $Q=\int \rho_{v} d v$

or

$\boxed{Q=\oint_{S} \mathbf{D} \cdot d \mathbf{S}=\int_{v} \rho_{v} d v}$ ..(3)

By applying divergence theorem to the middle term in eqs.(3)

$\oint_{S} \mathbf{D} \cdot d \mathbf{S}=\int_{\mathbf{v}} \nabla \cdot \mathbf{D} d v$..(4)

Comparing the two volume integrals in eqs. (3) and (4) results in

$\boxed{\rho_{v}=\nabla \cdot \mathbf{D}}$ ..(5)

Equation (5) states that the volume charge density is the same as the divergence of the electric flux density.