Page 39 - Engineering Electromagnetics, 8th Edition

CHAPTER 1 Vector Analysis 21

Table 1.2 Dot products of unit vectors in spherical
and rectangular coordinate systems
a r a θ a φ
a x · sin θ cos φ cos θ cos φ − sin φ
a y · sin θ sin φ cos θ sin φ cos φ
a z · cos θ − sin θ 0

vector in the direction of the rectangular vector, the dot products with a z are found
to be

a z · a r = cos θ
a z · a θ =−sin θ
a z · a φ = 0
The dot products involving a x and a y require first the projection of the spherical
unit vector on the xy plane and then the projection onto the desired axis. For example,
a r · a x is obtained by projecting a r onto the xy plane, giving sin θ, and then projecting
sin θ on the x axis, which yields sin θ cos φ. The other dot products are found in a
like manner, and all are shown in Table 1.2.

EXAMPLE 1.4
We illustrate this procedure by transforming the vector field G = (xz/y)a x into
spherical components and variables.
Solution. We find the three spherical components by dotting G with the appropriate
unit vectors, and we change variables during the procedure:
xz xz
G r = G · a r = a x · a r = sin θ cos φ
y y
2
cos φ
= r sin θ cos θ
sin φ
xz xz
G θ = G · a θ = a x · a θ = cos θ cos φ
y y
2
2 cos φ
= r cos θ
sin φ
xz xz
Gφ = G · a φ = a x · a φ = (−sin φ)
y y
=−r cos θ cos φ
Collecting these results, we have
G = r cos θ cos φ (sin θ cot φ a r + cos θ cot φ a θ − a φ )
Appendix A describes the general curvilinear coordinate system of which the
rectangular, circular cylindrical, and spherical coordinate systems are special cases.
The first section of this appendix could well be scanned now.