Vector d is a linear vector function of vector d when the following relationships hold:

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1 Appendix 4 Dyadic Analysis DEFINITION ecto d is a linea vecto function of vecto d when the following elationships hold: d x = a xxd x + a xy d y + a xz d z d y = a yxd x + a yy d y + a yz d z d z = a zxd x + a zy d y + a zz d z. A4.1 These elationships can be epesented in moe compact fom by means of the matix notation d = a d. The matix opeato itself can be expessed in tems of dyads as A4.2 a = a xx u x u x + a xy u x u y + a xz u x u z + a yx u y u x + a yy u y u y + a yz u y u z + a zx u z u x + a zy u z u y + a zz u z u z A4.3 povided, by convention, ab c stands fo ab c. The symbol ab is called a dyad, and a sum of dyads such as a is a dyadic. Also by convention, c ab stands fo c ab, so that the dot poduct of a dyad and a vecto is now defined fo ab acting as both a pefacto and a postfacto. The witing of a in nonion fom, as shown above, is athe cumbesome, and one often pefes to use the fom a = a xx u x + a yx u y + a zx u z u x + a xy u x + a yy u y + a zy u z u y + a xz u x + a yz u y + a zz u z u z = a x u x + a y u y + a z u z A4.4 whee the a ae the column vectos of the matix of a. Altenatively, a = u x a xx u x + a xy u y + a xz u z + u y a yx u x + a yy u y + a yz u z + u z a zx u x + a zy u y + a zz u z = u x a x + u y a y + u z a z, A4.5 Pofesso Lindell has been kind enough to check this appendix, make coections, and suggest additional fomulas. Electomagnetic Fields, econd Edition, By Jean G. an Bladel Copyight 2007 the Institute of Electical and Electonics Enginees, Inc. 1035

2 1036 Appendix 4 Dyadic Analysis whee the a ae the ow vectos of the matix of a. It is obvious that a d is, in geneal, diffeent fom d a. In othe wods, the ode in which a and d appea should be caefully espected. a d is equal to d a only when the dyadic is symmetic i.e., when a ik = a ki. The tanspose of a is a dyadic a t such that a d is equal to d a t. One may easily check that the tanspose is obtained by an intechange of ows and columns. Moe pecisely, a t = a x u x + a y u y + a z u z = u x a x + u ya y + u za z. A4.6 The tace of the dyadic is the sum of its diagonal tems. Thus, t a = a xx + a yy + a zz. A4.7 The tace is a scala i.e., it is invaiant with espect to othogonal tansfomations of the base vectos. The tace of ab is a b. Among dyadics endowed with special popeties we note 1. The unitay dyadic, which epesents a pue otation. The deteminant of its elements is equal to The identity dyadic Clealy, I = u x u x + u y u y + u z u z. A4.8 I d = d I = d. A The symmetic dyadic, chaacteized by a ik = a ki, fo which a t = a. The dyadic ab is symmetic when a b = 0. Futhe, a d = d a. A The antisymmetic dyadic, chaacteized by a ik = a ki. Fo such a dyadic a t = a, and a d = d a. A4.11 The diagonal elements ae zeo, and thee ae only thee distinct components. The dyadic can always be witten in tems of I and a suitable vecto b as a = b z u x u y + b y u x u z + b z u y u x b x u y u z b y u z u x + b x u z u y, = I b, A4.12 whee the skew poduct is the dyad bc d = bc d. A4.13 The antisymmetic a can also be expessed as a = cb bc. A4.14

3 Definitions The eflection dyadic f u = I 2uu, A4.15 whee u is a eal unit vecto.applied to the position vecto, it pefoms a eflection with espect to a plane pependicula to u. 6. The otation dyadic u = uu + sin θu I + cos θi uu. A4.16 Applied to a vecto, it pefoms a otation by an angle θ in the ight-hand diection aound the diection of u. The elements of a dyadic may be complex a case in point is the fee-space dyadic discussed in Chapte 7. It then becomes useful to intoduce concepts such as the Hemitian dyadic a ik = aki,otheanti-hemitian dyadic a ik = aki. Useful poducts of dyads ae defined as follows: ab cd = ab cd the diect poduct, a dyad. A4.17 ab : cd = a cb d the double poduct, a scala. A4.18 ab cd = a cb d the double coss-poduct, a dyad. A4.19 ab cd = a cb d a vecto. A4.20 ab cd = a cb d a vecto. A4.21 Geneal Multiplicative elationships b a c = b a c = b a c A4.22 b c a = b c a = c b a A4.23 a b c = a b c = a c b but not a b c A4.24 b a c = b a b A4.25 b a c = b a c A4.26 b a c = b a c = b a c A4.27 b c a = cb a ab c A4.28 bc cb d = c b d A4.29 c a b = c a b = c a b A4.30 a b c = a b c = a b c A4.31 c a b = c a b = c a b A4.32 a b c = a b c = a b c A4.33 a c b = a c b A4.34

4 1038 Appendix 4 Dyadic Analysis b a c = c a t b a b c = a b c. A4.35 A4.36 The identity dyadic satisfies the following elationships: I b c = b I c = b c A4.37 I b a = b a = b I a A4.38 I b c = cb bc. A4.39 DIFFEENTIAL ELATIONHIP Diffeentiation with espect to a Paamete d df f a = dt dt a + f d a dt d dt a b = d a db b + a dt dt d d a a b = b + a db dt dt dt d dt a b = d a dt b + a db dt. A4.40 A4.41 A4.42 A4.43 Basic Diffeential Opeatos The action of a linea opeato L on a dyadic is defined by the fomula La = La x u x + La y u y + La z u z. A4.44 In paticula, div a = a = div a x u x + div a y u y + div a z u z = a x x + a y y + a z cul a = a = cul a x u x + cul a y u y + cul a z u z az = u x y a y ax + u y a z ay + u z x x a x y 2 a = 2 a x a y a = gad div a cul cul a. 2 A4.47 A4.45 A4.46

5 Integal elationships 1039 Also a gad a = a = u x x + u a y y + u a z = gad a x u x + gad a y u y + gad a z u z A4.48 a gad = a =au x x + au y y + au z. A4.49 Deived elationships gadb c = gad b c gad c b A4.50 gad f b = gad f b + f gad b f is any scala function A4.51 a b gada = b x x + b a y y + b a z d gad a = da divbc = div bc + b gad c div cul a = 0 divf a = gad f a + f div a diva b = div a b + ta t gad b divb a = cul b a b cul a divbc cb = cul c b A4.52 A4.53 A4.54 A4.55 A4.56 A4.57 A4.58 A4.59 div f I = gad f divi a = cul a culbc = cul bc b gad c cul gad a = 0 cul f a = gad f a + f cul a cul f I = gad f I cula b = cul a b gad b a cul cul f I = culgad f I = gad gad f I 2 f. A4.60 A4.61 A4.62 A4.63 A4.64 A4.65 A4.66 A4.67 INTEGAL ELATIONHIP The integal elationships of vecto analysis have thei equivalent in dyadic analysis. The most impotant examples ae N M dc gad a = an am A4.68

6 1040 Appendix 4 Dyadic Analysis dc a = u n gad a d, c A4.69 whee the contou is descibed in the positive sense with espect to u n. dc a = u n cul ad c gad a d = u n a d div ad = u n ad cul ad = u n ad A4.70 A4.71 A4.72 A4.73 b gad div a gad div b a ] d = un b div a div bu n a ] d A4.74 cul cul b a b cul cul a ] d = un b cul a + u n cul b a ] d = un b cul a + u n cul b a ] d ] b 2 a 2 b a d = un b div a div bu n a A u n b cul a + u n cul b a ] d A4.76 a 2 f f 2 a d = u n gad f a f gad a d. A4.77 ELATIONHIP IN CYLINDICAL COODINATE Dyadic a can be witten as a = a u + a ϕ u ϕ + a z u z = u a + u ϕ a ϕ + u z a z. The basic diffeential opeatos ae then: gad a = gad a a ϕu ϕ u + a = u + u 1 a ϕ + u a z div a = div a a ϕϕ u + = 1 a + a + 1 a ϕ + a z gad a ϕ + a u ϕ div a ϕ + a ϕ u ϕ + gad a z u z u ϕ + div a z u z A4.78 A4.79

7 cul a = cul a + a ϕ u ϕ = u 1 In paticula: a z a ϕ elationships in pheical Coodinates 1041 u + cul a ϕ a u ϕ u ϕ + cul a z u z a + u ϕ a z aϕ + u z + a ϕ 1 a. A4.80 gad u = u ϕu ϕ gad u ϕ = u ϕu gad u z = 0 gadu = u u + u ϕ u ϕ = I u z u z. A4.81 A4.82 A4.83 A4.84 Note that the dyadic opeatos expessed in tems of the ow vectos a ae identical with thei vecto countepats povided bas ae put above scala pojections to tansfom them into ow vectos, and povided the unit vectos ae used as pefactos. This simple ule, which is also valid in spheical coodinates, allows one to wite composite opeatos such as gad div simply by efeing to the vecto fomula. Fo example: 2 a = u 2 a a 2 2 a ϕ 2 + u ϕ 2 a ϕ a ϕ a 2 + u z 2 a z. A4.85 ELATIONHIP IN PHEICAL COODINATE Dyadic a can be witten as a = a u + a θ u θ + a ϕ u ϕ = u a + u θ a θ + u ϕ a ϕ. The basic diffeential opeatos ae gad a = gad a a ϕu ϕ a θ u θ a + gad a ϕ + + a θ tan θ a = u + u 1 a θ θ + u 1 a ϕ sin θ div a = div a a θθ + a ϕϕ u + + div a ϕ + a ϕ + a ϕθ tan θ = a + 2a + 1 a θ θ + u + gad a θ + a u θ u ϕ ] u ϕ div a θ + a θ a ϕϕ tan θ u ϕ a θ tan θ + 1 sin θ a ϕ a ϕu ϕ u θ tan θ u θ A4.86 A4.87

8 1042 Appendix 4 Dyadic Analysis cul a = In paticula: cul a + a θ u θ + cul a ϕ a u ϕ a θ u ϕ tan θ 1 a ϕ = u θ + a ϕ tan θ 1 sin θ 1 a + u θ sin θ a ϕ a ϕ + a ϕ u ϕ u + cul a θ a u θ u ϕ a θ aθ + u ϕ + a θ 1 + a ϕ u θ u θ tan θ a. A4.88 θ gad u = u θ u θ gad u θ = u θ u gad u ϕ = u ϕu gadu = I. + u ϕu ϕ + u ϕu ϕ tan θ u ϕu θ tan θ A4.89 A4.90 A4.91 A4.92 NOTE In addition to 12, 165, 173] of the geneal bibliogaphy: I.. Lindell, Elements of Dyadic Algeba and Its Application in Electomagnetics. epot 126, adio Laboatoy, Helsinki Univesity of Technology, I.. Lindell, Complex ectos and Dyadics fo Electomagnetics. epot 36, Electomagnetics Laboatoy, Helsinki Univesity of Technology, C. T. Tai, ome essential fomulas in dyadic analysis and thei applications. adio ci. 22, , 1987.