Page 35 - Engineering Electromagnetics, 8th Edition

CHAPTER 1 Vector Analysis 17

vectors is generally required. That is, we may be given a rectangular vector

A = A x a x + A y a y + A z a z
where each component is given as a function of x, y, and z, and we need a vector in
cylindrical coordinates

A = A ρ a ρ + A φ a φ + A z a z
where each component is given as a function of ρ, φ, and z.
To find any desired component of a vector, we recall from the discussion of the
dot product that a component in a desired direction may be obtained by taking the
dot product of the vector and a unit vector in the desired direction. Hence,
and
A ρ = A · a ρ A φ = A · a φ
Expanding these dot products, we have
(12)
A ρ = (A x a x + A y a y + A z a z ) · a ρ = A x a x · a ρ + A y a y · a ρ
(13)
A φ = (A x a x + A y a y + A z a z ) · a φ = A x a x · a φ + A y a y · a φ
and

A z = (A x a x + A y a y + A z a z ) · a z = A z a z · a z = A z (14)
since a z · a ρ and a z · a φ are zero.
In order to complete the transformation of the components, it is necessary to
know the dot products a x · a ρ , a y · a ρ , a x · a φ , and a y · a φ . Applying the definition
of the dot product, we see that since we are concerned with unit vectors, the result
is merely the cosine of the angle between the two unit vectors in question. Refer-
ring to Figure 1.7 and thinking mightily, we identify the angle between a x and a ρ
as φ, and thus a x · a ρ = cos φ,but the angle between a y and a ρ is 90 − φ, and
◦
a y · a ρ = cos (90 − φ) = sin φ. The remaining dot products of the unit vectors
◦
are found in a similar manner, and the results are tabulated as functions of φ in
Table 1.1.
Transforming vectors from rectangular to cylindrical coordinates or vice versa
is therefore accomplished by using (10) or (11) to change variables, and by using the
dot products of the unit vectors given in Table 1.1 to change components. The two
steps may be taken in either order.

Table 1.1 Dot products of unit vectors in cylindrical
and rectangular coordinate systems

a ρ a φ a z
a x · cos φ − sin φ 0
a y · sin φ cos φ 0
a z · 0 0 1