Zachary Bednarke

DifferentialformsMaxwellLondon

Abstract: In this lecture I start with a superficial introduction to determinants, exterior derivatives, and their action on differential forms in order to convey arguments about the structure of Maxwell’s electromagnetic equation in a way that makes some heuristic sense but doesn’t have the aid of rigorous exposition. I start by reconstructing the first two equations, then I talk about splitting spacetime and the problems with last two equations. To construct those, I discuss the role of the metric on spacetime, the concepts of dual spaces and inner products and the energy density and lagrangian. Finally I expose the concepts of orientation and define the Hodge star, discuss ”div, grad, and curl”, and end with writing down the dual of Fµν and the second set of equations. Then, I take a detour to fix a few inconsistencies in the equations before discussing the London equations and superconductivity. Then I discuss their fewer restrictions and possible implications.

Or you could check out this arguably cooler one about topology, quantum mechanics and the role of potentials.Aharonov_Bohm_Effect