Editing Polarization density (section)

===Differential form===

By the divergence theorem, Gauss's law for the field '''P''' can be stated in ''differential form'' as:
<math display="block">-\rho_b = \nabla \cdot \mathbf P,</math>
where {{math|∇ · '''P'''}} is the divergence of the field '''P''' through a given surface containing the bound charge density <math>\rho_b</math>.

{{math proof|proof=
By the divergence theorem we have that
<math display="block">-Q_b = \iiint_V \nabla \cdot \mathbf P\ \mathrm{d} V,</math>
for the volume ''V'' containing the bound charge <math>Q_b</math>. And since <math>Q_b</math> is the integral of the bound charge density <math>\rho_b</math> taken over the entire volume ''V'' enclosed by ''S'', the above equation yields
<math display="block">-\iiint_V \rho_b \ \mathrm{d} V = \iiint_V \nabla \cdot \mathbf{P}\ \mathrm{d} V ,</math>
which is true if and only if
<math>-\rho_b = \nabla \cdot \mathbf{P}</math>
}}