Electrodynamics: Polarization and Surface Charge Density

What is the polarization vector and how is it related to the surface charge density? Here is my very basic derivation along with a python calculation for the surface charge density on an insulating sphere. Code here: https://trinket.io/glowscript/8f720cc084ca Here is the playlist for the full course https://www.youtube.com/playlist?list=PLWFlMBumSLSYDQww1iwZXEyNJb6xnoxZL