A study of Numerical Electromagnetics must rely on a firm base of knowledge in the foundations of electromagnetics as stated in Maxwell's equations. Accordingly, we undertake in this chapter a review of Maxwell's equations and associated boundary conditions.

All classical electromagnetic phenomena are governed by a compact and elegant set of fundamental rules known as Maxwell's equations. This set of four coupled partial differential equations was put forth as the complete classical theory of electromagnetics in a series of brilliant papers written by James Clerk Maxwell between 1856 and 1865, culminating in his classic paper [2]. In this work, Maxwell provided a mathematical framework for Faraday's primarily experimental results, clearly elucidated the different behavior of conductors and insulators under the influence of fields, imagined and introduced the concept of displacement current [3, Sec. 7.4], and inferred the electromagnetic nature of light. A most fundamental prediction of this theoretical framework is the existence of electromagnetic waves, a conclusion to which Maxwell arrived in the absence of experimental evidence that such waves can exist and propagate through empty space. His bold hypotheses were to be confirmed 23 years later (in 1887) in the experiments of Heinrich Hertz [4].

When most of classical physics was fundamentally revised as a result of Einstein's introduction [6] of the special theory of relativity, Maxwell's equations remained intact. To this day, they stand as the most general mathematical statements of fundamental natural laws which govern all of classical electrodynamics.