I recently realized that I have never really seen a rigorous definition of the electric polarization field in matter (and for that matter, magnetization). On the one hand, I know what its physical meaning is, but on the other, I don't believe that I'll really trust it until I come up with one. Depending on how I proceed, I run into certain issues. If I define it as a density of individual dipole moments, I do not have very much "calculatory" power, and I cannot even reproduce basic results like the fact that its divergence should be the opposite of the bound charge density and the time derivative should be the bound current density. I also tried defining it as the (decaying) quantity whose divergence is the opposite of the bound charge density, and whose curl is zero. Though this gives me more calculatory power, and also reproduces the dipole density approach, it runs into such problems such as the fact that it requires the current density to be curl-free in order to correctly reproduce the bound current density. Does anyone have any other suggestions?