Antenna Theory and Applications

Chapter 3

Vector Analysis

Scalars are defined by an amplitude only, for example temperature, charge. Vectors, for example forces, are defined not only by an amplitude but also by a direction. This means that simple arithmetic operations that can be applied for scalars, like addition, subtraction, multiplication and differentiation, have slightly more complicated counterparts for vectors. In this chapter we will briefly explain these operations and we will discuss the gradient-, divergence- and curl-operators.

A scalar is represented in print as a normal, though often italic, letter (e.g., charge q). A vector is represented in print with a bold face letter (e.g., electric field E). Since bold face letters are difficult to reproduce in writing, the convention of writing a small arrow over the letter is adapted. This arrow may be simplified to a half arrow or a line. Also a line underneath the letter may be used to represent a vector. Thus, the vector a may be represented as

3.1

The amplitude of this vector, |a|, is a scalar, |a| = a.

For a vector a in a Cartesian three-dimensional space, the relation between amplitude and direction (ϑ, φ), see Figure 3.1, is given by

3.2

Figure 3.1 Three-dimensional vector a having amplitude a and direction (ϑ,φ).

Here, , and are unit vectors ...