Vector Calculus v2

8/2/2019 Vector Calculus v2

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APPENDIX A

VECTOR ANALYSIS

We shall normally orient rectangular (x,y,z), cylindrical (p,e/>,z), andspherical (r,O,e/ coordinates as shown in Fig. A-I. Coordinate transformations are then given byx = p cos e/> = r sin 0 cos e/>y = p sin e/> = r sin 0 sin e/>Z = r cos 0p = y'-'x2-+-y--:2 = r sin 0e/> = tan-1JLxr = y 'x2 + y2 + Z2 = y 'p2+o = tan-1 VX2 + y2 = tan-1e

Z Z

(A-I)

Transformations of the coordinate components of a vector among thethree coordinate systems are given byA: = Ap cos e/> - A sin e/>= Ar sin 0 cos e/> + A scos 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 0Ap = 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 0A8 = 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 (U r ,U8,U dz = r2 sin 0 dr dO de/>447

(A-3)


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448z

TIME-HARMONIC ELECTROMAGNETIC FIELDS

y

FIG. A-l. Normal coordinate orientation.

differential elements of vector area areds = u '" dy dz + U y dx dz + u. dx dy= UpP dcp dz + u" dp dz + Uzp dp d4> (A-4)= u rr2 sin 0 dO d4> + Uer sin 0 dr d4> + u",r dr dO

and differential elements of vector length aredl = Uz dx + UI dy + Uz dz= Up dp + u ",p d4> + Uz dz

= U dr + Uer dO + u",r sin 0 d4> (A-5)The elementary algebraic operations are the same in all right-handedorthogonal coordinate systems. Letting (UI,U 2Ua) denote the unitvectors and (AI,A2,Aa) the corresponding vector components, we haveaddition defined by

A + B = uI(A I + B I ) + u2(A 2+ B 2) + ua(Aa+ Ba) (A-6)scalar multiplication defined by

A . B = AlB I + A 2B2+ A aB aand vector multiplication defined by

UI U2 UaA X B = Al A2 AaBI B2 Ba

(A-7)

(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). Inrectangular coordinates we can think of del (V) as the vector operator(A-g)


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VECTOR ANALYSIS 449and the various operations are

U", Uy UzvxA= a a aax ay az (A-lO)A", Ay A.

'V2w = a2w + a2w + a2wax2 ay2 az 2In cylindrical coordinates we have

In spherical coordinates we haveaw 1 aw 1 awVw = U r -a + Ue- ao + U - - o - ar r rsm cp

1 0 2) 1 a. loA V . A = "2 a- (r Ar + - . -0 ao (Ae sm 0) + - - 0 oA-r r r SIn r SIn 'I '1 [ a . aAe]v x A = u. r sin 0 ao (A sm 0) - acp

1 [ 1 aAr a ]Ue - -. -- - - (r A


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450tion we have

TIME-HARMONIC ELEC'l'ROMAGNETIC FIELDS

A2 = A AIAI2= A A*A + B = B + AA B = B AA X B = -B X A

(A + B) C = A C + B . C(A + B) X C = A X C + B X CA B x C = B C x A = C A x BA X (B X C) = (A C)B - (A B)C

For differentiation we haveV(v + w) = Vv + Vwv . (A + B) = V A + V . B

V X (A + B ) = V X A + V X Bv(vw) = v Vw + w Vv

V (wA) = wV' A + A VWV X (wA) = wV X A - A X VwV . (A X B) = B V X A - A . V X B

V2A = V(V . A) - V X V X AV X (v Vw) = Vv X Vw

V X Vw = 0vvxA=O

For integration we havef V A dr = 1fi A dsf V X A ds = A . dlf f V X A dr = -1fi A X dsf f Vw dr = 1fi w dsf n X Vw ds = w dlFinally, we have the Helmholtz identity

Iff V ' A Iff v' X A47rA = - V Ir _ r'ldr' + V X Ir _ r' l dr'

(A-13)

(A-14)

(A-15)

(A-16)valid if A is well-behaved in all space and vanishes at least as rapidly asr - 2 at infinity.


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