Introduction

The reader is supposed to be familiar with undergraduate calculus, analytic geometry and differential equations. In this introductory chapter, some additional mathematical concepts and techniques are outlined, that are essential for the understanding and manipulation of Maxwell's equations, on which all applied electromagnetism is based.

First-order differential operators are introduced in rectangular Cartesian coordinates and some simple examples are given of their calculations. The reader should memorize their definitions (the most important formulas are put in a box). In many applications, the expression of these operators are needed in other coordinate systems, such as circular cylindrical or spherical polar coordinates; however, however, there is no need to commit these other expressions to memory.

Second-order vector differential operators are discussed next. Special attention is given to the laplacian operator, which will play a key role in Poisson's equation and in the wave equation.

Vector integral calculus is here limited to two fundamental theorems, the divergence (or Gauss') theorem and Stokes' theorem. More general theorem, such as the various forms of Green's theorem, are left for more advanced courses.

Curvilinear orthogonal coordinates are introduced, and two coordinate systems are examined in detail: the circular cylindrical coordinates and the spherical polar coordinates. The next section is devoted to general orthogonal coordinates and its title is followed by an asterisk (*), meaning that the section is not necessary to the understanding of the remainder of this textbook and may be skipped if necessary. However, the reader is urged to read all asterisked sections, so as to be able to refer to the material contained in them when necessary. Phasors have been introduced in any first course on circuits, and the reader should already be conversant with them. Their many properties are summarized in Section A.5, which also contains an optional subsection on comparison between phasors of a time-dependent quantity and the Fourier transform on that same quantity.

The physical quantities considered in this textbook may be scalar, or vector, or tensor functions of position and time. Vectors are underlined (e.g, electric field $\underline{\mathcal{E}}$ and tensors are superlined (e.g., electric permittivity $\overline{\varepsilon}$ of an anisotropic dielectric such as a crystal). The position in space is indicated by three scalar coordinates in any chosen coordinate system (e.g, $x$, $y$, and $z$ in rectangular Cartesian coordinates) or, more succinctly, by the vector distance $\underline{r}$ of the point being considered from the origin of coordinates, where

$$\underline{r}=x \hat{x} + y\hat{y} + z \hat{z}$$ (2.1)

and $\hat{x}$, $\hat{y}$ and $\hat{z}$ are unit vectors (i.e., vectors of unit magnitude) directed parallel to the positive $x$, $y$ and $z$ directions, respectively (see Fig A.1). The time is indicated by t. Thus, $\underline{\mathcal{ E}}(\underline{r},t)$ means the electric field vector evaluated at position $\underline{r}$ in space and time $t$.

Figure A.1: Position in space using rectangular coordinates.