-
--
- - ------
APPENDIX A
VECTOR ANALYSIS
We shall normally orient rectangular (x,y,z), cylindrical (p,e/>,z), and spherical (r,O,e/ coordinates as shown in Fig. A-I. Coordinate transformations are then given by
x = p cos e/> = r sin 0 cos e/> y = p sin e/> = r sin 0 sin e/> Z = r cos 0 p = y'-'x2-+-y--:2 = r sin 0 = e/>
=
tan-1 JL
(A-I)
r = y' x 2
o=
+ y2 + Z2 = tan-1 VX2 + y2 =
Z
y' p2 + z~
tan-1 e
Z
Transformations of the coordinate components of a vector among the three coordinate systems are given by
A: = Ap cos e/> - A</> sin e/> = Ar sin 0 cos e/> A s cos 0 cos e/> - A</> sin e/> All = Ap sin e/> + A</> cos e/> = AT sin 0 sin e/> A 8 cos 0 sin e/> A</> cos e/> A. = Ar cos 0 - As sin 0 Ap = Ax cos e/> All sin e/> = Ar sin 0 A8 cos 0 (A-2) A</> = - A: sin e/> + All cos If> Ar = Ax sin 0 cos If> All sin 0 sin If> A. cos 0 = Ap sin 0 + A. cos 0 A8 = A", cos 0 cos If> + All cos 0 sin If> - A. sin 0 = Ap cos 0 - A. sin (}
+ +
The coordinate-unit vectors in the three systems are denoted by (u:,ulI,u.), (up,u</>,u.), and (Ur ,U8,U</. Differential elements of volume are
d., = dx dy dz
= p dp dlf> dz = r2 sin 0 dr dO de/>
447
(A-3)
448
TIME-HARMONIC ELECTROMAGNETIC FIELDS
FIG. A-l. Normal coordinate orientation.
differential elements of vector area are
ds = u'" dy dz + U y dx dz + u. dx dy = UpP dcp dz + dp dz + Uzp dp d4> = u rr 2 sin 0 dO d4> + Uer sin 0 dr d4> + u",r dr dO
u'"
(A-4)
and differential elements of vector length are dl
= Uz dx + U7I dy + Uz dz = Up dp + u",p d4> + U z dz
= U dr r
+ Uer dO + u",r sin 0 d4>
(A-5)
The elementary algebraic operations are the same in all right-handed orthogonal coordinate systems. Letting (UI,U2,Ua) denote the unit vectors and (AI,A2,Aa) the corresponding vector components, we have addition defined by
A
+B
= uI(A I
+ B + u2(A 2 + B + u a(A a + Ba)
I) 2)
(A-6)
scalar multiplication defined by
A.B
AlB I
+ A 2B 2 + A aB a
UI Al
BI
(A-7)
and vector multiplication defined by U2
A2 B2
AX B
Ua Aa Ba
(A-8)
The above formula is a determinant, to be expanded in the usual manner. The differential operators that we have occasion to use are the gradient (Vw) , divergence (v. A), curl (V X A), and Laplacian (V' 2w). In rectangular coordinates we can think of del (V) as the vector operator
(A-g)
VECTOR ANALYSIS
449
and the various operations are
U",
Uy
Uz
vxA= a
a a ax ay az A", Ay A. 2 2 2 w 'V2 = a w + a w + a w ax2 ay2 az 2
(A-lO)
In cylindrical coordinates we have
In spherical coordinates we have
aw 1 aw 1 aw --o-a ar + U e- ao + U</> rsm r cp 1 0 2) 1 a. V . A = "2 a- (r Ar + r SIn ao (Ae sm 0) - . -0 r r 1 a . v x A = u. r sin 0 [ ao (A</> sm 0) - aAe] acp
Vw
Ur -
loA + --0 oA-</> r SIn
'I'
1 aAr a + Ue -r [ - .1 0 - - - -ar (r A</ ] sm acp + u</> -1 [ -a (rAe)
(A-12)
r ar
oAr] - -
00
'V2w =
r2 ar
! ~ (r2 aw) + _1_ ~ (sin 0 aw) +
ar r2 sin 0 ao ao
1 02W r2 sin 2 0 Ocp2
A number of useful vector identities, which are independent of the choice of coordinate system, are as follows . For addition and multiplica-
450
TIME-HARMONIC ELEC'l' ROMAGNETIC FIELDS
tion we have
A2 = A A
A A* A+B=B+A AB = BA A X B = -B X A (A + B) C = A C + B . C (A + B) X C = A X C + B X C ABxC=BCxA=CAxB A X (B X C) = (A C)B - (A B)C
IAI2=
(A-13)
For differentiation we have
V(v
V X
v . (A
+ B) (A + B )
+ w)
= V A + V . B = V X A+V X
= Vv
+ Vw
v(vw) = v Vw w Vv V (wA) = wV' A A VW V X (wA) = wV X A - A X Vw V . (A X B) = B V X A - A . V X B V2A = V(V . A) - V X V X A V X (v Vw) = V v X Vw
V X Vw
+ +
(A-14)
vvxA=O
For integration we have
fff V A dr 1fi A ds ff V X A ds = A . dl fff A dr -1fi A fff dr = 1fi ds ff n ds = w dl
=
V X
X ds
(A-15)
Vw
X Vw
Finally, we have the Helmholtz identity
47rA = - V
r- 2 at infinity.
Iff Ir _ r'l
V' A dr'
+V
Iff v' _ r'l Ir
X A dr'
(A-16)
valid if A is well-behaved in all space and vanishes at least as rapidly as
