Vector Analysis in Three Dimensions

Transcription

1 Appendix 1 etor Analysis in Three Dimensions MULTIPLICATIE RELATIONHIP a (b ) = (a b) = b ( a) (A1.1) a (b ) = b(a ) (a b) (A1.2) a (b ) (b a) = b (a ) (A1.3) (a b) ( d) = (a )(b d) (a d)(b ) (A1.4) a [b ( d)] =(b d)(a ) (b )(a d) (A1.5) (a b) [(b ) ( a)] =[a (b )] 2. (A1.6) DIFFERENTIAL RELATIONHIP a and b are vetor point funtions; A and B are salar point funtions; all are provided with the neessary derivatives. grad(a + B) = grad A + grad B div(a + b) = div a + div b url(a + b) = url a + url b grad AB = A grad B + B grad A grad(a b) = a url b + b url a + (b grad)a + (a grad)b div(aa) = A div a + grad A a div(a b) = b url a a url b url(aa) = (grad A a) + A url a url(a b) = a div b b div a + (b grad)a (a grad)b url grad A = 0 (A1.7) (A1.8) (A1.9) (A1.10) (A1.11) (A1.12) (A1.13) (A1.14) (A1.15) (A1.16) Eletromagneti Fields, eond Edition, By Jean G. an Bladel Copyright 2007 the Institute of Eletrial and Eletronis Engineers, In. 1001

2 1002 Appendix 1 etor Analysis in Three Dimensions da dt = a t div url a = 0 url url a = grad div a 2 a div grad A = 2 A grad f (A) = f (A) grad A + a div v url(a v)(v = veloity) (A1.21) 2 (AB) = A 2 B + 2 grad A grad B + B 2 A 2 (Aa) = A 2 a + a 2 A + 2(grad A grad)a grad div (Aa) = (grad A) div a + A grad div a + grad A url a (A1.17) (A1.18) (A1.19) (A1.20) (A1.22) (A1.23) + (a grad) grad A + (grad A grad)a (A1.24) url url (Aa) = grad A url a a 2 A + (a grad) grad A + A url url a + grad A div a (grad A grad)a. (A1.25) INTEGRAL RELATIONHIP These integral relationships are valid for volumes bounded by regular surfaes, a preise definition of whih an be found in [158]. It is suffiient, for our purposes, to state that usually enountered surfaes with finite numbers of verties are regular. The basis for the various relations is the following theorem: φ d = φ( u i ) d x i (A1.26) where is the unit vetor along the outward-pointing normal. The theorem is valid when φ is a single-valued funtion in and on its boundary, and has a derivative φ/ x i that is ontinuous in the interiors of a finite number of regular regions of whih is the sum. Disontinuities in the derivatives are allowed at the boundaries between the regions. Coordinate x i is taken along an arbitrary axis with unit vetor u i. Appliation to three orthogonal diretions yields the following Gauss theorems: div v d = (v u n ) d (A1.27) url v d = ( v) d = r( url v) d (A1.28) grad fd= f d. (A1.29) The partial derivatives that appear in the formulas must have the ontinuity properties stated above for φ/ x i. They an eventually beome infinite at the boundary, but the integrals must then be understood to be improper integrals lim d. By hoosing speial vetors, suh as A grad B, for insertion in Gauss theorem, a whole series of Green s theorems an

3 Integral Relationships 1003 be obtained. The seond partial derivatives that appear in the formulas must now satisfy the onditions formerly required of φ/ x i. [A 2 B + (grad A grad B)] d = A B (A1.30) n d ) d ( (A 2 B B 2 A) d = A B n B A n (A1.31) (url a url b a url url b) d = (a url b) u n d (A1.32) (b url url a a url url b) d = [( a) url b (u n b) url a] d (A1.33) (div a div b + b grad div a) d = div a(b u n ) d (A1.34) (a grad div b b grad div a) d = [(a u n ) div b (b u n )div a] d (A1.35) [a 2 b + url a url b + div a div b) d = [( a) url b + (u n a) div b] d (A1.36) (a 2 b b 2 a) d = [( a) div b (b u n ) div a + ( a) url b (u n b) url a] d (A1.37) [a div b + b div a (a url b) (b url a)] d [a( b) + b( a) (a b)] d (A1.38) [b div a + (a grad)b] d = ( a)b d (A1.39) [A url url v + v 2 A + (div v) grad A] d = [A url v + ( v) grad A + ( v) grad A] d (A1.40) grad A url v d = A url v u n d = (v grad A) u n d. (A1.41) When is a regular two-sided surfae, and when the various partial derivatives that appear in the formulas are ontinuous in a region ontaining the surfae in its interior, the following tokes theorems hold: ( url v) d = v d (A1.42) ( grad f ) d = fd (A1.43)

4 1004 Appendix 1 etor Analysis in Three Dimensions ( grad) v d = d v (grad A grad B) u n d = A grad B d = B grad A d (A1.44) (A1.45) where is a ontour on surfae. This ontour must be desribed in the positive sense with respet to. DEFINITION OF THE MAIN OPERATOR The usual definition of url in terms of first derivatives is not valid at points at whih some of these derivatives do not exist, for example, at a surfae of disontinuity. A more general definition of the url is obtained by onsidering the expression url a = lim a d, (A1.46) 0 where is a volume surrounding P and bounded by. If this expression approahes a unique limit when approahes zero, the limit is termed the url of a at P and is denoted, in Weyl s notation, 1 by url a. The two definitions of the url are equivalent when a is ontinuously differentiable. Further, many of the properties of the usual url (tokes theorem, for example) an be extended to the url operator when the latter is ontinuous [201]. In a similar manner, a definition of the divergene is obtained by onsidering div a 1 = lim a d. (A1.47) 0 If this expression has a unique limit, the limit is termed divergene and is denoted by div a. Examples of vetors having a div but no div are given in [201]. Here, again, the two operators are idential when a is ontinuously differentiable. Finally, the gradient is defined by grad f = lim f d. (A1.48) 0 The starred definitions have the advantage of being independent of the oordinate system. The basi operators an also be given a weak definition, 2 in partiular the distributional forms (A8.72), (A8.75), (A8.78), (A8.79), whih an be applied to symboli funtions suh as δ(r). One an also introdue a url a operator, based on the notion of frational derivative of order a, where a an be real or omplex. 3 The linguisti notations div, grad, url used in this book are frequently replaed by the Gibbs version, based on the nabla operator = n=1 1, h n v n (A1.49) here expressed in the general orthogonal system v 1, v 2, v 3 disussed in Appendix 2. Using that operator, the basi operators are written as f, a and a. The impliation is that is a onstituent of these operators, and this through salar and vetor produts.

5 Helmholtz Theorem in Infinite pae 1005 A sholarly monograph disputes the desirability of this approah [173] and shows that a as a salar produt an lead to inorret results in the derivation of formulas valid in a general oordinate system. 4 It is therefore more appropriate to write ( )a and ( )a, where the objet between parentheses is an operator. Following that idea, Tai proposes the notations f, a and a, where is the expression (A1.49) and = = n=1 n=1 h n v n. h n v n (A1.50) (A1.51) This form, valid for orthogonal oordinate systems, emphasizes the independene of the three operators. The Laplaian is now written as f or a, grad div a as a, and url url a as a. The forms (A1.50), (A1.51) are atually derived from definitions of the kind shown in (A1.46) to (A1.48). Tai also proves the invariane of the operators by showing that, in any two urvilinear orthogonal systems, n=1 = h n v n u n h n=1 n where represents a null, a dot, or a ross. The proof an be extended to nonorthogonal urvilinear systems. Whatever the notation, a formula suh as (A2.102), whih gives grad div v in spherial oordinates, has been obtained by first expressing div v from (A2.92) and subsequently applying the gradient operation (A2.91) to the result. This safe approah has been used onsistently in the urrent text. v n HELMHOLTZ THEOREM IN INFINITE PACE Helmholtz theorem onsists in splitting a vetor field in the form f(r) = grad φ + url v. (A1.52) The term grad φ is the irrotational (lamellar or longitudinal) part, the term url v the solenoidal (or transverse) part. The first operator may be thought of as a diagonal matrix, the seond one as a skew matrix. The splitting appears on p. 38 of Helmholtz original paper, 5 in whih the author, although mostly onerned with hydrodynamis, mentions the relevane of his theory to eletromagneti problems. Only potential flows in inompressible fluids had been studied at the time, with veloity w equal to grad φ. This explains Helmholtz interest in vortex motion, whih he measures by the vortiity vetor 1 url w. Also relevant 2 is the heliity h = f url f d, (A1.53) whih is a topologial measure of how muh the field rotates about itself in a given volume. 6,7 The onept finds appliation in magneto-hydrodynamis, where f is the vetor potential. The heliity is gauge invariant in simply onneted domains and provides a measure of the linkage of the field lines [24].

6 1006 Appendix 1 etor Analysis in Three Dimensions Going bak to (A1.52), let f be a field that is bounded, approahes zero at large distanes, and has soures, div f and url f, whih are of finite support in spae. In (A1.52), the two parts keep the same value when a onstant is added to φ and a gradient to v. With the additional onstraint div v = 0, φ and v satisfy 2 φ = div f 2 v = url f. (A1.54) The solution beomes unique if we require φ and v (and the two terms of the splitting) to satisfy the lassial onditions of regularity at infinity defined in etions 3.1 and 6.1. Thus, φ(r) = 1 div 1 f 4π r r d v(r) = 1 url 1 f 4π r r d. (A1.55) The operators div and url must be understood in the sense of distributions, that is, aording to (A8.90) and (A8.91) if f is pieewise ontinuous. To show that the deomposition is unique, let us assume that two sets of funtions, (φ, v) and (φ, v ), are appropriate for (A1.52). The vetor α = grad(φ φ ) = url(v v ) is harmoni, with div α = 0 and url α = 0. It is shown in etion 3.6 that suh a vetor, if bounded, must have a onstant value in spae. It follows that α must vanish beause the two parts of the splitting are required to approah zero at infinity. This shows that grad φ = grad φ and url v = url v. The splitting into longitudinal and transverse omponents an be written formally as [133] [ f(r) = δ l (r r ) + δ t (r r )] f(r ) d, (A1.56) where Aording to (A4.48), δ l (r r ) + δ t (r r ) = Iδ(r r ). δ l (r r ) = 1 4π grad grad 1 r r. (A1.57) It is obvious that δ l and δ t are not onentrated on r = r. For example: grad grad 1 R = 1 R 3 (3u Ru R I) (R = 0). (A1.58) The appropriateness of the terms longitudinal and transverse is onfirmed by onsidering the spae transform f(k) of f(r), given in (A7.53). From (A7.56) and (A7.57), the transforms of div f and url f are, respetively, F [div f] =jk f(k) F [url f] =jk f(k).

7 Helmholtz Theorem in a Finite pae 1007 The div f soure is assoiated with the omponent of f in the diretion of the propagation vetor k, and the url f soure with the transverse omponents with respet to k [46]. HELMHOLTZ THEOREM IN A FINITE PACE A representation suh as (A1.52) an be generated from any solution of the equation f = grad div a url url a = 2 a. (A1.59) Potential theory immediately yields the possible solution (Fig. A1.1a) φ = div a = 1 4π v = url a = 1 4π div f(r ) r r d + 1 4π url f(r ) r r d 1 4π u n f(r ) r r d f(r ) r r d. (A1.60) Partiular splittings result from boundary onditions imposed on φ or a. In the eletri splitting, φ is required to vanish on. The solution is unique, and grad φ is easily interpreted as the eletrostati field generated by a volume harge ρ = ɛ 0 div f enlosed in a metallized. In a multiply bounded volume, however, (A1.52) should inlude harmoni (soure-free) terms. 8,9 In the doubly bounded region II of Figure A1.1b, one should write f = f 0 + grad φ + url v. (A1.61) etor f 0 is proportional to the eletri field e 0 resulting from a unit differene of potential applied between the metallized eletrodes 1 and 2. Although f 0 is solenoidal, it annot generally be represented as a url. In a doubly bounded volume, indeed, suh a representation requires the flux of the vetor through both 1 and 2 to vanish. 8,10 Beause f 0, grad φ, and url v are funtionally orthogonal [a property easily proven by means of the divergene theorem (A1.27)], f 0 an be determined from the relationship f 0 = f e 0 d e 0 2 d e 0 (A1.62) 2 C 1 (a) (b) () Figure A1.1 Finite volumes: (a) simply bounded and onneted region I; (b) doubly bounded region II; () doubly onneted region III.

8 1008 Appendix 1 etor Analysis in Three Dimensions where e 0 = grad φ 0. Potential φ 0 satisfies 2 φ 0 = 0 φ 0 = 1on 2 φ 0 = 0on 1. (A1.63) When f is solenoidal, it an only be written as a url when the term in f 0 is absent (i.e., when the flux of f vanishes through 1 and 2 individually). In the magneti splitting, the potential must satisfy the boundary ondition φ/ n = f on. The solution for φ is defined to within an additive onstant, whih has no influene on the value of grad φ. The url v term is tangent to, hene v may be hosen perpendiular to. In a ring-like region (Fig. A1.1) splitting (A1.52) takes the form 11 f = h 0 f h 0 d + grad φ + url v. (A1.64) Field h 0, introdued in etion 4.10, is soureless. It an be expressed as a url or a gradient, but the relevant salar potential is multivalued. In fluid dynamis, h 0 is the veloity of an inompressible fluid flowing irrotationally in the ring-like volume III. A Helmholtz theorem an also be written on a surfae. Let f be a tangential vetor funtion. The splitting is now 2 f = grad φ + grad θ. (A1.65) From (A3.40), 2 φ = div f 2 θ = div (f ). (A1.66) The solution of (A1.66) an be effeted, in priniple at least, by means of either a Green s funtion or an expansion in the eigenfuntions of 2. OTHER PLITTING The representation of fields by means of salar funtions failitates satisfation of boundary onditions. The mother funtions are typially less singular than the original fields, for example in the viinity of sharp metalli edges. For a general vetor funtion f, one may write f = grad φ + url(u) + url url(tu) where u is a onstant unit vetor. Another expansion is f = grad φ + url(r) + url url(tr) (A1.67) = grad φ + grad r + url(grad T r) (A1.68)

9 Notes 1009 where and T are Debye potentials. Representation (A1.68) is useful for a multipole analysis in spherial oordinates. olutions of Maxwell s equations in a soureless region are afforded by the expressions E = url url(tr) jk 0 url(r) R 0 H = url url(r) + jk 0 url(tr). (A1.69) By setting L = r grad, a vetor f an be written as f = Lφ + L v + v. (A1.70) To within an imaginary fator, the operator L is the orbital angular momentum enountered in quantum mehanis. 12 Finally, a solenoidal vetor f an be expressed in terms of Clebsh potentials and T. Thus, 13,14 f = url( grad T) = grad grad T. (A1.71) NOTE 1. H. Weyl, The method of orthogonal projetion in potential theory, Duke Math. J. 7, , J. an Bladel, A disussion of Helmholtz theorem on a surfae, AEÜ 47, , N. Engheta, Frational url operator in eletromagnetis, Mirowave Opt. Teh. Lett. 17, 86 91, P. Moon and D. E. pener. etors. D. an Nostrand Co., Prineton, N.J., H. Helmholtz, Uber Integrale der hydrodynamishen Gleihungen welhe den Wirbelbewegungen entsprehen. J. Reine Angew. Math. 55, 25 55, N. Anderson and A. M. Arthurs, The heliity of the eletromagneti field, Int. J. Eletronis 56, , A. Lakhtakia and B. hanker, Beltrami fields within ontinuous soure regions, volume integral equations, sattering algorithms and the extended Maxwell-Garnett model, Int. J. Applied Eletrom. Materials 4, 65 82, J. an Bladel, On Helmholtz theorem in multiplybounded and multiply-onneted regions, J. Franklin Inst. 269, , J. an Bladel, A disussion of Helmholtz theorem, Eletromagn. 13, , A. F. tevenson, Note on the existene and determination of a vetor potential, Quart. Appl. Math. 12, , A. Bossavit, Magnetostati problems in multiply onneted regions: some properties of the url operator. IEE Pro. 135-A, , J.. Lemont and H. E. Moses, An angular momentum Helmholtz theorem, Comm. Pure Appl. Math. 14, 69 76, P. R. Kotiuga, Clebsh potentials and the visualisation of three-dimensional vetor fields, IEEE Trans. MAG 27, , H. Rund, Clebsh potentials in the theory of eletromagneti fields admitting eletri and magneti harge distributions, J. Math. Phys. 18, 84 95, 1977.