CHAPTER
1
VECTOR
ANALYSIS
Vector analysis is a mathematical subject which is much better taught by math
ematicians than by engineers. Most junior and senior engineering students, how
ever, have not had the time (or perhaps the inclination) to take a course in vector
analysis, although it is likely that many elementary vector concepts and opera
tions were introduced in the calculus sequence. These fundamental concepts and
operations are covered in this chapter, and the time devoted to them now should
depend on past exposure.
The viewpoint here is also that of the engineer or physicist and not that of
the mathematician in that proofs are indicated rather than rigorously expounded
and the physical interpretation is stressed. It is easier for engineers to take a more
rigorous and complete course in the mathematics department after they have
been presented with a few physical pictures and applications.
It is possible to study electricity and magnetism without the use of vector
analysis, and some engineering students may have done so in a previous electrical
engineering or basic physics course. Carrying this elementary work a bit further,
however, soon leads to linefilling equations often composed of terms which all
look about the same. A quick glance at one of these long equations discloses little
of the physical nature of the equation and may even lead to slighting an old
friend.
Vector analysis is a mathematical shorthand. It has some new symbols,
some new rules, and a pitfall here and there like most new fields, and it demands
concentration, attention, and practice. The drill problems, first met at the end of
Sec. 1.4, should be considered an integral part of the text and should all be
1
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worked. They should not prove to be difficult if the material in the accompany
ing section of the text has been thoroughly understood. It take 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 used 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, and volume
resistivity. Voltage is also a scalar quantity, although the complex representation
of a sinusoidal voltage, an artificial procedure, produces a
complex scalar
, or
phasor
, which requires two real numbers for its representation, such as amplitude
and phase angle, or real part and imaginary part.
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 Spring '10
 Wilton
 Cartesian Coordinate System, Electromagnet, Polar coordinate system, Engineering Electromagnetics, Coordinate systems

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