TABLE OF CONTENTS Vector calculus deals with vectors and with their derivatives and integrals. Vector calculus plays a central role in the study of electromagnetic theory. The equations for the electric and magnetic fields (Maxwell's equations) link components of these fields in different directions; such equations can be expressed most concisely in vector notation. In fact, much of vector calculus was invented for the specific purpose of simplifying the equations of electromagnetic theory. To appreciate the advantages of vector notation, one need only compare the cumbersome formulas in Maxwell's old treatise Electricity and Magnetism with the much neater formulas in modern textbooks.
This first chapter is mostly mathematics. It introduces a concise superscript notation for vectors and tensors, and it establishes some valuable mathematical results, which will be much used in later chapters. Only in the last section of this chapter will we deal with an application of our mathematical formulas to a question in physics, that is, the conservation of electric charge. [*]
Roughly, a vector is a quantity that has magnitude and direction and that behaves like the position vector under addition and multiplication, and under rotations. Thus, the position vector is to be regarded as the prototype for all vectors. In this section, we begin with the precise definition of the position vector and afterward spell out in detail how any other vector "behaves like" the position vector.
We will describe points of space by means of their Cartesian coordinates x, y, and z
