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Chapter 2
COORDINATE SYSTEMS
AND TRANSFORMATION
Education makes a people easy to lead, but difficult to drive; easy to govern but
impossible to enslave.
—HENRY P. BROUGHAM
2.1 INTRODUCTION
In general, the physical quantities we shall be dealing with in EM are functions of space
and time. In order to describe the spatial variations of the quantities, we must be able to
define all points uniquely in space in a suitable manner. This requires using an appropriate
coordinate system.
A point or vector can be represented in any curvilinear coordinate system, which may
be orthogonal or nonorthogonal.
An
orthogonal system is
one
in which the coordinates arc mutually perpendicular.
Nonorthogonal systems are hard to work with and they are of little or no practical use.
Examples of orthogonal coordinate systems include the Cartesian (or rectangular), the cir
cular cylindrical, the spherical, the elliptic cylindrical, the parabolic cylindrical, the
conical, the prolate spheroidal, the oblate spheroidal, and the ellipsoidal.
1
A considerable
amount of work and time may be saved by choosing a coordinate system that best fits a
given problem. A hard problem in one coordi nate system may turn out to be easy in
another system.
In this text, we shall restrict ourselves to the three bestknown coordinate systems: the
Cartesian, the circular cylindrical, and the spherical. Although we have considered the
Cartesian system in Chapter 1, we shall consider it in detail in this chapter. We should bear
in mind that the concepts covered in Chapter 1 and demonstrated in Cartesian coordinates
are equally applicable to other systems of coordinates. For example, the procedure for
'For an introductory treatment of these coordinate systems, see M. R. Spigel,
Mathematical Hand
book of Formulas and Tables.
New York: McGrawHill, 1968, pp. 124130.
28
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View Full Document2.3 CIRCULAR CYLINDRICAL COORDINATES (R, F, Z)
29
finding dot or cross product of two vectors in a cylindrical system is the same as that used
in the Cartesian system in Chapter 1.
Sometimes, it is necessary to transform points and vectors from one coordinate system
to another. The techniques for doing this will be presented and illustrated with examples.
2.2 CARTESIAN COORDINATES (X, Y, Z)
As mentioned in Chapter 1, a point
P
can be represented as
(x, y, z)
as illustrated in
Figure 1.1. The ranges of the coordinate variables
x, y,
and
z
are
00
< X <
00
00<y<o> (2.1)
— 00 <
I
<
A vector A in Cartesian (otherwise known as rectangular) coordinates can be written as
(A
x
,A
y
,AJ
or
A A +
A
y
a
y
+ A
z
a
z
(2.2)
where
a
x
,
a
y
, and a
z
are unit vectors along the
x, y,
and zdirections as shown in
Figure 1.1.
2.3 CIRCULAR CYLINDRICAL COORDINATES
(p,
cj>,
z)
The circular cylindrical coordinate system is very convenient whenever we are dealing
with problems having cylindrical symmetry.
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 Fall '10
 DinavahiandIyer

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