# Chapter 3 Vector Analysis

# Chapter Contents

1. Overview

2. 3.1 Basic Laws of Vector Algebra

3. 3.2 Orthogonal Coordinate Systems

4. 3.3 Transformations between Coordinate Systems

5. 3.4 Gradient of a Scalar Field

6. TB 5 Global Positioning System

7. 3.5 Divergence of a Vector Field

8. 3.6 Curl of a Vector Field

9. TB 6 X-Ray Computed Tomography

10. 3.7 Laplacian Operator

1. Chapter 3 Summary

2. Problems

# Objectives

Upon learning the material presented in this chapter, you should be able to:

1. Use vector algebra in Cartesian, cylindrical, and spherical coordinate systems.

2. Transform vectors between the three primary coordinate systems.

3. Calculate the gradient of a scalar function and the divergence and curl of a vector function in any of the three primary coordinate systems.

4. Apply the divergence theorem and Stokes’s theorem.

# Overview

In our examination of wave propagation on a transmission line in Chapter 2, the primary quantities we worked with were voltage, current, impedance, and power. Each of these is a scalar quantity, meaning that it can be completely specified by its magnitude if it is a positive real number or by its magnitude and phase angle if it is a negative or a complex number (a negative number has a positive magnitude and a phase angle of π (rad)). This chapter is concerned with vectors. A vector has a magnitude and a direction. The speed of an object is a scalar, whereas its velocity is a vector.

Starting in the next chapter and throughout the succeeding chapters in this book, the primary electromagnetic quantities we deal with are the electric and magnetic fields, E and H. These, and many other related quantities, are vectors. Vector analysis provides the mathematical tools necessary for expressing and manipulating vector quantities in an efficient and convenient manner. To specify a vector in three-dimensional space, it is necessary to specify its components along each of the three directions.

Vector algebra governs the laws of addition, subtraction, and “multiplication” of vectors. The rules of vector algebra and vector representation in each of the aforementioned orthogonal coordinate systems (including vector transformation between them) are two of the three major topics treated in this chapter. The third topic is vector calculus, which encompasses the laws of differentiation and integration of vectors, the use of special vector operators (gradient, divergence, and curl), and the application of certain theorems that are particularly useful in the study of electromagnetics, most notably the divergence and Stokes’s theorems.