Obtaining Poisson's equation is exceedingly simple, for from the point form of Gauss's law, $$ \nabla \cdot \mathbf{D}=\rho_{v} $$ ...(1)

the definition of D

$$ \mathbf{D}=\epsilon \mathbf{E}$$ ...(2)

and the gradient relationship, $$ \mathbf{E}=-\nabla V $$ ...(3)

by substitution we have

$$ \nabla \cdot \mathbf{D}=\nabla \cdot(\epsilon \mathbf{E})=-\nabla \cdot(\epsilon \nabla V)=\rho_{v} …