Vector Calculus and Electromagnetism

Vector calculus is an extremely interesting and important branch of math with very relevant applications in physics. Specifically, vector calculus is the language in which (classical) electromagnetism is written. It is fascinating to me that Maxwell’s equations can so succinctly and elegantly express so many phenomena, from electric and magnetic interactions to light (electromagnetic waves). One popular formulation of Maxwell’s equations is the following:

So what does this all mean? Let’s break it down, arduously and tediously, from the nitty-gritty of the vector calculus requisite for understanding the elegance of these equations.

Note: I presume basic knowledge of calculus in this post. Obviously, this is not meant to be a substitute for a more rigorous foundation with more computational practice.