Page 19 - Engineering Electromagnetics, 8th Edition

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CHAPTER

Vector Analysis

ector analysis is a mathematical subject that is better taught by mathematicians
than by engineers. Most junior and senior engineering students have not had
V the time (or the inclination) to take a course in vector analysis, although it is
likely that vector concepts and operations were introduced in the calculus sequence.
These are covered in this chapter, and the time devoted to them now should depend
on past exposure.
The viewpoint here is that of the engineer or physicist and not that of the mathe-
matician. Proofs are indicated rather than rigorously expounded, and physical inter-
pretation is stressed. It is easier for engineers to take a more rigorous course in the
mathematics department after they have been presented with a few physical pictures
and applications.
Vector analysis is a mathematical shorthand. It has some new symbols and some
new rules, and it demands concentration and practice. The drill problems, first found
at the end of Section 1.4, should be considered part of the text and should all be
worked. They should not prove to be difficult if the material in the accompanying
section of the text has been thoroughly understood. It takes a little longer to “read”
the chapter this way, but the investment in time will produce a surprising interest. ■

1.1 SCALARS AND VECTORS
The term scalar refers to a quantity whose value may be represented by a single
(positive or negative) real number. The x, y, and z we use in basic algebra are scalars,
and the quantities they represent are scalars. If we speak of a body falling a distance
L in a time t,or the temperature T at any point in a bowl of soup whose coordinates
are x, y, and z, then L, t, T, x, y, and z are all scalars. Other scalar quantities are
mass, density, pressure (but not force), volume, volume resistivity, and voltage.
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A vector quantity has both a magnitude and a direction in space. We are con-
cerned with two- and three-dimensional spaces only, but vectors may be defined in

1 We adopt the convention that magnitude infers absolute value; the magnitude of any quantity is,
therefore, always positive.

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