We introduce the polarization density P, which has the following relation to E and D : D = ε ε 0 E + P and the following relation to the bound charge: ρ ρ b o u n d = − − ∇ ∇ ⋅ ⋅ P Now, consider the three equations: ρ ρ b o u n d = ∇ ∇ ⋅ ⋅ ( − − P ) ρ ρ f r e e = ∇ ∇ ⋅ ⋅ D ρ ρ = ∇ ∇ ⋅ ⋅ ( ε ε 0 E ) The key insight is that the sum of the first two equations is the third equation. This completes the proof: The first equation is true by definition, and therefore the second equation is true if and only if the third equation is true. So the second and third equations are equivalent, which is what we wanted to prove.