1 MathematicalPreliminaries

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1 1 MathematicalPeliminaies The enomous usefulness of mathematics in the natual sciences is something bodeing on the mysteious. Eugene Wigne ( Intoduction This chapte pesents a collection of mathematical notation, definitions, identities, theoems, and tansfomations that play an impotant ole in the study of electomagnetism. A bief discussion accompanies some of the less familia topics and only a few poofs ae given in detail. Fo moe details and complete poofs, the eade should consult the books and papes listed in ouces, Refeences, and Additional Reading at the end of the chapte. Appendix C at the end of the book summaizes the popeties of Legende polynomials, spheical hamonics, and Bessel functions. 1.2 ectos A vecto is a geometical object chaacteized by a magnitude and diection. 1 Although not necessay, it is convenient to discuss an abitay vecto using its components defined with espect to a given coodinate system. An example is the ight-handed coodinate system with othogonal unit basis vectos (ê 1, ê 2, ê 3 shown in Figue 1.1, whee ê 1 ê 1 = 1 ê 2 ê 2 = 1 ê 3 ê 3 = 1 (1.1 ê 1 ê 2 = 0 ê 2 ê 3 = 0 ê 3 ê 1 = 0 (1.2 ê 1 ê 2 = ê 3 ê 2 ê 3 = ê 1 ê 3 ê 1 = ê 2. (1.3 We expess an abitay vecto in this basis using components k = ê k, A vecto can be decomposed in any coodinate system we please, so = 1 ê ê ê 3. (1.4 3 k ê k = k=1 3 k=1 kê k. (1.5 1 A moe pecise definition of a vecto is given in ection 1.8.

2 2 MATHEMATICAL PRELIMINARIE: DEFINITION, IDENTITIE, AND THEOREM ê 3 ê 1 Figue 1.1: An othonomal set of unit vectos ê 1, ê 2, ê 3. is an abitay vecto. ê 2 z z z y y x (a Cylindical x (b pheical Figue 1.2: Two cuvilinea coodinate systems Catesian Coodinates Ou notation fo Catesian components and unit vectos is = x ˆx + y ŷ + z ẑ. (1.6 In paticula, k always denotes the Catesian components of the position vecto, = x ˆx + yŷ + zẑ. (1.7 It is not obvious geometically (see Example 1.7 in ection 1.8, but the gadient opeato is a vecto with the Catesian epesentation The divegence, cul, and Laplacian opeations ae, espectively, = ( z y y z =ˆx x + ŷ y + ẑ z. (1.8 = x x + y y + z z ˆx + ( x z z x ŷ + (1.9 ( y x x ẑ (1.10 y 2 A = 2 A x A y A z 2. ( Cylindical Coodinates Figue 1.2(a defines cylindical coodinates (ρ,φ,z. Ou notation fo the components and unit vectos in this system is The tansfomation to Catesian coodinates is = ρ ˆρ + φ ˆφ + z ẑ. (1.12 x = ρ cosφ y = ρ sin φ z= z. (1.13

3 1.2 ectos 3 The volume element in cylindical coodinates is d 3 = ρdρdφdz. The unit vectos ( ˆρ, ˆφ, ẑ foma ight-handed othogonal tiad. ẑ is the same as in Catesian coodinates. Othewise, ˆρ = ˆx cosφ + ŷ sin φ ˆx = ˆρ cosφ ˆφ sin φ (1.14 ˆφ = ˆx sin φ + ŷ cosφ ŷ = ˆρ sin φ + ˆφ cosφ. (1.15 The gadient opeato in cylindical coodinates is The divegence, cul, and Laplacian opeations ae, espectively, = [ 1 z ρ φ ] φ ˆρ + z 2 A = 1 ρ pheical Coodinates = ˆρ ρ + ˆφ ρ φ + ẑ z. (1.16 = 1 (ρ ρ + 1 φ ρ ρ ρ φ + z z [ ρ z ] z ˆφ + 1 [ (ρφ ρ ρ ρ ( ρ A ρ ρ (1.17 ] ρ ẑ (1.18 φ + 1 ρ 2 2 A φ A z 2. (1.19 Figue 1.2(b defines spheical coodinates (, θ, φ. Ou notation fo the components and unit vectos in this system is The tansfomation to Catesian coodinates is = ˆ + θ ˆθ + φ ˆφ. (1.20 x = sin θ cos φ y = sin θ sin φ z= cosθ. (1.21 The volume element in spheical coodinates is d 3 = 2 sinθddθdφ. The unit vectos ae elated by ˆ = ˆx sin θ cosφ + ŷ sin θ sin φ + ẑ cosθ ˆx = ˆ sin θ cosφ + ˆθ cosθ cosφ ˆφ sin φ (1.22 ˆθ = ˆx cosθ cos φ + ŷ cosθ sin φ ẑ sin θ ŷ = ˆ sin θ sin φ + ˆθ cosθ sin φ + ˆφ cosφ (1.23 ˆφ = ˆx sin φ + ŷ cosφ ẑ = ˆ cosθ ˆθ sin θ. (1.24 The gadient opeato in spheical coodinates is =ˆ + ˆθ θ + ˆφ sin θ The divegence, cul, and Laplacian opeations ae, espectively, = 1 ( (sin θ θ + 1 φ 2 sin θ θ sin θ φ = 1 [ (sin θφ ] θ ˆ sin θ θ φ + 1 [ 1 sin θ φ ( ] φ ˆθ + 1 [ (θ 2 A = 1 2 ( 2 A sin θ ( sin θ A θ θ φ. ( ] ˆφ θ (1.26 ( A 2 sin 2 θ φ. (1.28 2

4 4 MATHEMATICAL PRELIMINARIE: DEFINITION, IDENTITIE, AND THEOREM he Einstein ummation Convention Einstein (1916 intoduced the following convention. An index which appeas exactly twice in a single tem of a mathematical expession is implicitly summed ove all possible values fo that index. The ange of this dummy index must be clea fom context and the index cannot be used elsewhee in the same expession fo anothe pupose. In this book, the ange fo a oman index like i is fom 1 to 3, indicating a sum ove the Catesian indices x, y, andz. Thus, in (1.6 and its dot poduct with anothe vecto F ae witten = 3 k ê k k ê k F = k=1 3 k F k k F k. (1.29 In a Catesian basis, the gadient of a scala ϕ and the divegence of a vecto D can be vaiously witten k=1 ϕ = ê k k ϕ = ê k k ϕ = ê k ϕ k (1.30 D = k D k = k D k = D k k. (1.31 If an N N matix C is the poduct of an N M matix A and an M N matix B, C ik = M A ij B jk = A ij B jk. (1.32 j= hekonecke and Levi-Civitàymbols The Konecke delta symbol δ ij and Levi-Cività pemutation symbol ǫ ijk have oman indices i, j,and k which take on the Catesian coodinate values x, y, andz. They ae defined by and δ ij = { 1 i = j, 0 i j, 1 ijk = xyz yzx zxy, ǫ ijk = 1 ijk = xzy yxz zyx, 0 othewise. ome useful Konecke delta and Levi-Cività symbol identities ae (1.33 (1.34 ê i ê j = δ ij δ kk = 3 (1.35 k j = δ jk k δ kj = j (1.36 [ F] i = ǫ ijk j F k [ A] i = ǫ ijk j A k (1.37 δ ij ǫ ijk = 0 ǫ ijk ǫ ijk = 6. (1.38 A paticulaly useful identity involves a single sum ove the epeated index i: ǫ ijk ǫ ist = δ js δ kt δ jt δ ks. (1.39 A genealization of (1.39 when thee ae no epeated indices to sum ove is the deteminant δ km δ im δ lm ǫ kil ǫ mpq = δ kp δ ip δ lp δ kq δ iq δ lq. (1.40

5 1.2 ectos 5 Finally, let C bea3 3 matix with matix elements C 11,C 12, etc. The deteminant of C can be witten using eithe an expansion by columns, o an expansion by ows, A closely elated identity we will use in ection is det C = ǫ ijk C i1 C j2 C k3, (1.41 det C = ǫ ijk C 1i C 2j C 3k. (1.42 ǫ lmn det C = ǫ ijk C li C mj C nk. ( ecto Identities in Catesian Components The Konecke and Levi-Cività symbols simplify the poof of vecto identities. An example is Using the left side of (1.37, the ith component of a (b cis a (b c = b(a c c(a b. (1.44 [a (b c] i = ǫ ijk a j (b c k = ǫ ijk a j ǫ klm b l c m. (1.45 The definition (1.34 tells us that ǫ ijk = ǫ kij. Theefoe, the identity (1.39 gives [a (b c] i = ǫ kij ǫ klm a j b l c m = (δ il δ jm δ im δ jl a j b l c m = a j b i c j a j b j c i. (1.46 The final esult, b i (a c c i (a b, is indeed the ith component of the ight side of (1.44. The same method of poof applies to gadient-, divegence-, and cul-type vecto identities because the components of the opeato tansfom like the components of a vecto [see above (1.8]. The next thee examples illustate this point. Example 1.1 Pove that ( g = 0. olution: Begin with ( g = i ǫ ijk j g k = 1 2 i j g k ǫ ijk i j g k ǫ ijk. Exchanging the dummy indices i and j in the last tem gives g = 1 2 i j g k ǫ ijk j i g k ǫ jik = 1 2 {ǫ ijk + ǫ jik } i j g k = 0. The final zeo comes fom ǫ ijk = ǫ jik, which is a consequence of (1.34. Example 1.2 Pove that (A B = A B (A B + (B A B A. olution: Focus on the ith Catesian component and use the left side of (1.37 to wite [ (A B] i = ǫ ijk j (A B k = ǫ ijk ǫ kst j (A s B t. The cyclic popeties of the Levi-Cività symbol and the identity (1.39 give [ (A B] i = ǫ kij ǫ kst j (A s B t = (δ is δ jt δ it δ js (A s j B t + B t j A s. Theefoe, [ (A B] i = A i j B j A j j B i + B j j A i B i j A j. This poves the identity because the choice of i is abitay.

6 6 MATHEMATICAL PRELIMINARIE: DEFINITION, IDENTITIE, AND THEOREM Example 1.3 Pove the double-cul identity ( A = ( A 2 A. olution: Conside the ith Catesian component. The identity on the left side of (1.37 and the invaiance of the Levi-Cività symbol with espect to cyclic pemutations of its indices give [ ( A] i = ǫ ijk j ( A k = ǫ ijk j ǫ kpq p A q = ǫ kij ǫ kpq j p A q. Now apply the identity (1.39 to get [ ( A] i = (δ ip δ jq δ iq δ jp j p A q = i j A j j j A i = i ( A 2 A i. The double-cul identity follows because 2 A = 2 (A x ˆx + A y ŷ + A z ẑ = ˆx 2 A x + ŷ 2 A y + ẑ 2 A z ecto Identities in Cuvilinea Components Cae is needed to intepet the vecto identities in Examples 1.2 and 1.3 when the vectos in question ae decomposed into spheical o cylindical components such as A = A ˆ + A θ ˆθ + A φ ˆφ. This can be seen fom Example 1.3 whee the final step is no longe valid because ˆ, ˆθ, and ˆφ ae not constant vectos. In othe wods, 2 A = (A ˆ + A θ ˆθ + A φ ˆφ ˆ 2 A + ˆθ 2 A θ + ˆφ 2 A φ. (1.47 One way to poceed is to wok out the components of (A ˆ, (A θ ˆθ, and (A φ ˆφ. Altenatively, we may simply define the meaning of the opeation 2 A when A is expessed using cuvilinea components. Fo example, [ 2 A] φ φ ( A [ ( A] φ, (1.48 and similaly fo ( 2 A and ( 2 A θ. Exactly the same issue aises when we examine the last step in Example 1.2, namely [ (A B] i = A i B (A B i + (B A i B i A. (1.49 By constuction, this equation makes sense when i stands fo x, y, oz. It does not make sense if i stands fo, say,, θ, oφ. On the othe hand, the full vecto vesion of the identity is coect as long as we etain the, θ,andφ vaiations of ˆ, ˆθ,and ˆφ. Fo example, (A B = [ A + A θ θ + A φ sin θ φ ] (B ˆ + B θ ˆθ + B φ ˆφ. (1.50 Application 1.1 Two Identities fo L The h = 1 vesion of the quantum mechanical angula momentum opeato, L = i,playsa useful ole in the analysis of classical spheical systems. In this Application, we pove two opeato identities which will appea late in the text: (A L = i 2 + ( i (1 + 1 (B L = (ˆ L + ˆ i L2.

7 1.3 Deivatives 7 Poof of Identity (A: We use (1.37, (1.39, and the cyclic popety of the Levi-Cività symbol to evaluate the kth component of L acting on a scala function φ: [ L] k φ = i[ ( ] k φ = iǫ mkl ǫ mst l s t φ = i[ l k l φ l l k φ]. (1.51 Because l l = 3and l k = δ lk, Howeve, Lφ = [ i 2 + i2 +i( ]φ. (1.52 k [ l l φ] = k φ + l l k φ, (1.53 which is the kth component of ( φ = φ + ( φ. ubstituting the latte into (1.53 gives Identity (A. Poof of Identity (B: We decompose the gadient opeato into its adial and angula pieces: =ˆ(ˆ ˆ (ˆ = ˆ i ˆ L. (1.54 Equation (1.54 and the Levi-Cività fomalism poduce the intemediate esult L = (ˆ L i (ˆ L L = (ˆ L i [ˆk LL k ˆL 2]. (1.55 Howeve, the angula momentum opeato obeys commutation elations which can be summaized as L L = il. Theefoe, ˆ (L L = iˆ L ˆ k LL k ˆ k L k L = iˆ L. (1.56 On the othe hand, k L k = 0 because L is pependicula to both and. Theefoe, ˆ k LL k = iˆ L, which we can substitute into (1.55. The esult is identity (B because, fo any scala function φ, [ ] 1 (φ = φ + φ. ( Deivatives Functions of and The position vecto is = ˆ with = x 2 + y 2 + z 2.Iff ( is a scala function and f ( = df/d, = ˆ = 0 (1.58 f = f ˆ 2 f = (2 f 2 (1.59 (f = (3 f 2 (f = 0. (1.60 imilaly, if g( is a vecto function and c is a constant vecto, g = g ˆ g = ˆ g (1.61 (g = g ( g = g (1.62

8 8 MATHEMATICAL PRELIMINARIE: DEFINITION, IDENTITIE, AND THEOREM ( g = g + ( g (g = 2g + g ( g (g = 0 (1.63 (c = c. ( Functions of Let R = = (x x ˆx + (y y ŷ + (z z ẑ. Then, Moeove, because f (R = f (R ˆR g(r = g (R ˆR g(r = ˆR g (R. (1.65 =ˆx x + ŷ y + ẑ z it it staightfowad to confim that heconvective Deivative and = ˆx x + ŷ y + ẑ z, (1.66 f (R = f (R. (1.67 Let φ(,t be a scala function of space and time. An obseve who epeatedly samples the value of φ at a fixed point in space,, ecods the time ate of change of φ as the patial deivative φ/t. Howeve, the same obseve who epeatedly samples φ along a tajectoy in space (t that moves with velocity υ(t = ṙ(t ecods the time ate of change of φ as the convective deivative, dφ = φ t + dx φ x + dy φ y + dz φ z = φ + (υ φ. (1.68 t Fo a vecto function g(,t, the coesponding convective deivative is Taylo s heoem dg = g t + (υ g. (1.69 Taylo s theoem in one dimension is f (x = f (a + (x a df dx + 1 x=a 2! (x d2 f a2 dx 2 +. (1.70 x=a An altenative fom follows fom (1.70 if x x + ǫ and a x: Equivalently, f (x + ǫ = [ f (x + ǫ = f (x + ǫ df dx + 1 d2 f 2! ǫ2 +. (1.71 dx2 1 + ǫ d dx + 1 2! This genealizes fo a function of thee vaiables to ( f (x + ǫ x,y+ ǫ y,z+ ǫ z = exp ǫ x x o f ( + ǫ = exp (ǫ f( = ( ǫ d 2 ( + ] f (x = exp ǫ d f (x. (1.72 dx dx exp ( ( ǫ y exp ǫ z f (x,y,z, (1.73 y z [ 1 + ǫ + 1 ] 2! (ǫ 2 + f (. (1.74

9 1.4 Integals Integals Jacobian Deteminant The deteminant of the Jacobian matix J elates volume elements when changing vaiables in an integal. Fo example, suppose x and y ae N-dimensional space vectos in two diffeent coodinate systems, e.g., Catesian and spheical. The volume elements d N x and d N y ae elated by x 1 x 1 x 1 y 1 y 2 y N d N x = J(x, y d N y = d N y. (1.75 x N x N x N y 1 y 2 y N hedivegenceheoem Let F( be a vecto function defined in a volume enclosed by a suface with an outwad nomal ˆn.Ifd = dˆn, the divegence theoem is d 3 F = d F. (1.76 pecial choices fo the vecto function F( poduce vaious integal identities based on (1.76. Fo example, if c is an abitay constant vecto, the eade can confim that the choices F( = cψ(and F( = A( c substituted into (1.76 espectively yield d 3 ψ = dψ ( Geen s Identities d 3 A = d A. (1.78 The choice F( = φ( ψ( in (1.76 leads to Geen s fist identity, d 3 [φ 2 ψ + φ ψ] = d φ ψ. (1.79 Witing (1.79 with the oles of φ and ψ exchanged and subtacting that equation fom (1.79 itself gives Geen s second identity, d 3 [φ 2 ψ ψ 2 φ] = d [φ ψ ψ φ]. (1.80 The choice F = P Qin (1.76 and the identity (A B = B A A Bpoduces a vecto analog of Geen s fist identity: d 3 [ P Q P Q] = d (P Q. (1.81

10 10 MATHEMATICAL PRELIMINARIE: DEFINITION, IDENTITIE, AND THEOREM Witing (1.81 with P and Q intechanged and subtacting that equation fom (1.81 gives a vecto analog of Geen s second identity: d 3 [Q P P Q] = d [P Q Q P]. ( tokes heoem tokes theoem applies to a vecto function F( definedonanopensuface bounded by a closed cuve C. Ifdl is a line element of C, d F = dl F. (1.83 The cuve C in (1.83 is tavesed in the diection given by the ight-hand ule when the thumb points in the diection of d. As with the divegence theoem, vaiations of (1.83 follow fom the choices F = cψ and F = A c: d ψ = dlψ (1.84 C C dl A = C d k A k hetime Deivative ofaflux Integal Leibniz Rule fo the time deivative of a one-dimensional integal is d x 2(t x 1(t dxb(x,t = b(x 2,t dx 2 b(x 1,t dx 1 d( A. ( x 2(t x 1(t dx b t. (1.86 This fomula genealizes to integals ove cicuits, sufaces, and volumes which move though space. Ou teatment of Faaday s law makes use of the time deivative of a suface integal whee the suface (t moves because its individual aea elements move with velocity υ(,t. In that case, d (t d B = (t Poof: We calculate the change in flux fom [ ] δ B d = [ d υ( B (υ B + B ]. (1.87 t δb d + B δ(ˆnd. (1.88 The fist tem on the ight comes fom time vaiations of B. The second tem comes fom time vaiations of the suface. Multiplication of evey tem in (1.88 by 1/δt gives d B B d = d + 1 B δ(ˆnd. (1.89 t δt We can focus on the second tem on the ight-hand side of (1.89 because the fist tem appeas aleady as the last tem in (1.87. Figue 1.3 shows an open suface(t with local nomal ˆn(t which moves and/o distots to the suface (t + δt with local nomal ˆn(t + δt in time δt.