Annotations updated 2/28/2014: Introduction to electromagnetic theory, Maxwell’s Equations in S.I. units, and the classical vector calculus of the linear differential operator del. We construct the differential operators grad, div, curl aka. gradient, divergence and curl from del. We discuss non-relativistic time and space as a topological manifold having a topological time variable and a Euclidean flat coordinate space. Maxwell’s equations formulated on this topological product manifold are perfectly good and valid. Later in this series we introduce a geometric time variable on the four dimensional semi-Riemann manifold of special relativistic Spacetime.
We use vector calculus to show that Maxwell’s equations imply conservation of electric charge and derive a local differential equation for conservation of charge. We develop a physical picture of charge conservation using Gauss’s divergence theorem.
We also discuss gauge invariance of Maxwell’s theory and other important properties of the theory including the linearity of Maxwell’s equations.
Homework problem: What if we replace the “physical space” Euclidean coordinate manifold E^3 with the three sphere S^3 as “physical space?” Can we write down Maxwell’s equations on the Riemann manifold S^3 x Topological time? Extra credit for a theory of Maxwell’s equations in Space-Time having a Minkowski metric and the topology of S^3 x T^1.