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APPENDIXVII
Vector Analysis
VII.1 VECTOR TRANSFORMATIONS
In this appendix we present the vector transformations from rectangular to cylindrical
(and vice versa), from cylindrical to spherical (and vice versa), and from rectangular
to spherical (and vice versa). The three coordinate systems are shown in Figure VII.1.
VII.1.1 Rectangular to Cylindrical (and Vice Versa)
The coordinate transformation from rectangular (x, y, z) to cylindrical (,,z) is
given, referring to Figure VII.1(b)
x = cos
y = sin
z = z
(VII-1)
In the rectangular coordinate system, we express a vector A as
A = axAx + ayAy + azAz (VII-2)
where ax , ay , az are the unit vectors and Ax , Ay , Az are the components of the vector
A in the rectangular coordinate system. We wish to write A as
A = aA + aA + azAz (VII-3)
where a , a , az are the unit vectors and A , A , Az are the vector components in the
cylindrical coordinate system. The z-axis is common to both of them.
Referring to Figure VII.2, we can write
ax = a cos a sin
ay = a sin + a cos
az = az
(VII-4)
Antenna Theory: Analysis Design, Third Edition, by Constantine A. BalanisISBN 0-471-66782-X Copyright 2005 John Wiley & Sons, Inc.
1079
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1080 APPENDIX VII
Figure VII.1 Rectangular, cylindrical, and spherical coordinate systems.
Using (VII-4) reduces (VII-2) to
A = (a cos a sin)Ax + (a sin + a cos)Ay + azAz
A = a(Ax cos + Ay sin) + a(Ax sin + Ay cos) + azAz (VII-5)
which when compared with (VII-3) leads to
A = Ax cos + Ay sin
A = Ax sin + Ay cos
Az = Az
(VII-6)
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APPENDIX VII 1081
Figure VII.2 Geometrical representation of transformations between unit vectors of rectangu-
lar and cylindrical coordinate systems.
In matrix form, (VII-6) can be written as
AA
Az
=
cos sin 0 sin cos 0
0 0 1
AxAy
Az
(VII-6a)
where
[A]rc = cos sin 0
sin cos 00 0 1
(VII-6b)
is the transformation matrix for rectangular-to-cylindrical components.
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1082 APPENDIX VII
Since [A]rc is an orthonormal matrix (its inverse is equal to its transpose), we can
write the transformation matrix for cylindrical-to-rectangular components as
[A]cr = [A]1rc = [A]
trc =
cos sin 0
sin cos 0
0 0 1
(VII-7)
or AxAy
Az
=
cos sin 0sin cos 0
0 0 1
AA
Az
(VII-7a)
orAx = A cos A sin
Ay = A sin + A cos
Az = Az
(VII-7b)
VII.1.2 Cylindrical to Spherical (and Vice Versa)
Referring to Figure VII.1(c), we can write that the cylindrical and spherical coordinates
are related by
= r sin
z = r cos
(VII-8)
In a geometrical approach similar to the one employed in the previous section, we can
show that the cylindrical-to-spherical transformation of vector components is given by
Ar = A sin + Az cos
A = A cos Az sin
A = A
(VII-9)
or in matrix form by
Ar
AA
=
sin 0 cos
cos 0 sin 0 1 0
A
AAz
(VII-9a)
Thus the cylindrical-to-spherical transformation matrix can be written as
[A]cs =
sin 0 cos cos 0 sin
0 1 0
(VII-9b)
The [A]cs matrix is also orthonormal so that its inverse is given by
[A]sc = [A]1cs = [A]
tcs =
sin cos 00 0 1
cos sin 0
(VII-10)
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APPENDIX VII 1083
and the spherical-to-cylindrical transformation is accomplished by
AA
Az
=
sin cos 00 0 1
cos sin 0
ArA
A
(VII-10a)
orA = Ar sin + A cos
A = AAz = Ar cos A sin
(VII-10b)
This time the component A and coordinate are the same in both systems.
VII.1.3 Rectangular to Spherical (and Vice Versa)
Many times it may be required that a transformation be performed directly fromrectangular-to-spherical components. By referring to Figure VII.1, we can write that
the rectangular and spherical coordinates are related by
x = r sin cos
y = r sin sin
z = r cos
(VII-11)
and the rectangular and spherical components by
Ar = Ax sin cos + Ay sin sin + Az cos
A = Ax cos cos + Ay cos sin Az sin
A = Ax sin + Ay cos
(VII-12)
which can also be obtained by substituting (VII-6) into (VII-9). In matrix form, (VII-12)
can be written as
ArA
A
=
sin cos sin sin cos
cos cos cos sin sin
sin cos 0
AxAy
Az
(VII-12a)
with the rectangular-to-spherical transformation matrix being
[A]rs =
sin cos sin sin cos cos cos cos sin sin
sin cos 0
(VII-12b)
The transformation matrix of (VII-12b) is also orthonormal so that its inverse can
be written as
[A]sr = [A]1rs = [A]
trs =
sin cos cos cos sinsin sin cos sin cos
cos sin 0
(VII-13)
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1084 APPENDIX VII
and the spherical-to-rectangular components related by
AxAy
Az
=
sin cos cos cos sinsin sin cos sin cos
cos sin 0
ArAA
(VII-13a)
orAx = Ar sin cos + A cos cos A sin
Ay = Ar sin sin + A cos sin + A cos
Az = Ar cos A sin
(VII-13b)
VII.2 VECTOR DIFFERENTIAL OPERATORS
The differential operators of gradient of a scalar (), divergence of a vector ( A),
curl of a vector ( A), Laplacian of a scalar (2), and Laplacian of a vector
(
2
A) frequently encountered in electromagnetic field analysis will be listed in therectangular, cylindrical, and spherical coordinate systems.
VII.2.1 Rectangular Coordinates
= ax
x+ ay
y+ az
z(VII-14)
A =Ax
x+
Ay
y+
Az
z(VII-15)
A = ax
Az
y
Ay
z
+ ay
Ax
z
Az
x
+ az
Ay
x
Ax
y
(VII-16)
= 2 =2
x2+
2
y2+
2
z2(VII-17)
2A = ax2Ax + ay
2Ay + az2Az (VII-18)
VII.2.2 Cylindrical Coordinates
= a
+ a 1
+ az z
(VII-19)
A =1
(A) +
1
A
+
Az
z(VII-20)
A = a
1
Az
A
z
+ a
A
z
Az
+ az
1
(A)
1
A
(VII-21)
2 =1
+
1
2
2
2+
2
z2(VII-22)
2A = ( A) A (VII-23)
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APPENDIX VII 1085
or in an expanded form
2A = a
2A
2+
1
A
A
2+
1
2
2A
2
2
2
A
+
2A
z2
+ a2A
2 +
1
A
A
2 +
1
2
2A
2 +
2
2
A
+
2A
z2
+ az
2Az
2+
1
Az
+
1
2
2Az
2+
2Az
z2
(VII-23a)
In the cylindrical coordinate system 2A = a2A + a
2A + az2Az because the
orientation of the unit vectors a and a varies with the and coordinates.
VII.2.3 Spherical Coordinates
= ar
r+ a
1
r
+ a
1
r sin
(VII-24)
A =1
r2
r(r2Ar) +
1
r sin
(A sin ) +
1
r sin
A
(VII-25)
A =ar
r sin
(A sin )
A
+
a
r
1
sin
Ar
r(rA)
+a
r
r
(rA) Ar
(VII-26)
2 =1
r2
r
r2
r
+
1
r2 sin
sin
+
1
r2 sin2
2
2(VII-27)
2A = ( A) A (VII-28)
or in an expanded form
2A = ar 2Ar
r2
+2
r
Ar
r
2
r2Ar +
1
r2
2Ar
2
+cot
r2
Ar
+1
r2
sin2
2Ar
2
2
r2
A
2 cot
r2A
2
r2 sin
A
+ a
2A
r2+
2
r
A
r
A
r2 sin2 +
1
r2
2A
2+
cot
r2
A
+1
r2 sin2
2A
2+
2
r2
Ar
2cot
r2 sin
A
+ a2A
r2 +2
r
A
r 1
r2 sin2 A +
1
r2
2A
2 +cot
r2
A
+1
r2 sin2
2A
2+
2
r2 sin
Ar
+
2 cot
r2 sin
A
(VII-28a)
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1086 APPENDIX VII
Again note that 2A = ar2Ar + a
2A + a2A since the orientation of the unit
vectors ar , a, and a varies with the r, , and coordinates.
VII.3 VECTOR IDENTITIES
VII.3.1 Addition and Multiplication
A A = |A|2 (VII-29)
A A = |A|2 (VII-30)
A + B = B + A (VII-31)
A B = B A (VII-32)
A B = B A (VII-33)
(A + B) C = A C + B C (VII-34)
(A + B) C = A C + B C (VII-35)
A B C = B C A = C A B (VII-36)
A (B C) = (A C)B (A B)C (VII-37)
(A B) (C D) = A B (C D)
= A (B DC B CD)
= (A C)(B D) (A D)(B C) (VII-38)
(A B) (C D) = (A B D)C (A B C)D (VII-39)
VII.3.2 Differentiation
( A) = 0 (VII-40)
= 0 (VII-41)
( + ) = + (VII-42)
() = + (VII-43)
(A + B) = A + B (VII-44)
(A + B) = A + B (VII-45)
(A) = A + A (VII-46)
(A) = A + A (VII-47)
(A B) = (A )B + (B )A + A ( B) + B ( A) (VII-48)
(A B) = B A A B (VII-49)
(A B) = A B B A + (B )A (A )B (VII-50)
A = ( A) 2A (VII-51)
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APPENDIX VII 1087
VII.3.3 Integration
C
A dl =
S
( A) ds Stokes theorem (VII-52)
#S
A ds =
V
( A)dv Divergence theorem (VII-53)
#S
(n A)ds =
V
( A)dv (VII-54)
#S
ds =
V
dv (VII-55)
C
dl =S
n ds (VII-56)
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