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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)