Appendix D Coordinate systems Rectangular coordinate system Coordinate variables u = x, ?∞ < x < ∞ (D.1) v = y, ?∞ < y < ∞ (D.2) w = z, ?∞ < z < ∞ (D.3) Vector algebra A = ?xA x + ?yA y + ?zA z (D.4) A · B = A x B x + A y B y + A z B z (D.5) A × B = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle ?x ?y ?z A x A y A z B x B y B z vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle (D.6) Dyadic representation ˉa = ?xa xx ?x + ?xa xy ?y + ?xa xz ?z + + ?ya yx ?x + ?ya yy ?y + ?ya yz ?z + + ?za zx ?x + ?za zy ?y + ?za zz ?z (D.7) ˉa = ?xa prime x + ?ya prime y + ?za prime z = a x ?x + a y ?y + a z ?z (D.8) a prime x = a xx ?x + a xy ?y + a xz ?z (D.9) a prime y = a yx ?x + a yy ?y + a yz ?z (D.10) a prime z = a zx ?x + a zy ?y + a zz ?z (D.11) a x = a xx ?x + a yx ?y + a zx ?z (D.12) a y = a xy ?x + a yy ?y + a zy ?z (D.13) a z = a xz ?x + a yz ?y + a zz ?z (D.14) Di?erential operations dl = ?x dx + ?y dy+ ?z dz (D.15) dV = dx dydz (D.16) dS x = dydz (D.17) dS y = dx dz (D.18) dS z = dx dy (D.19) ? f = ?x ?f ?x + ?y ?f ?y + ?z ?f ?z (D.20) ?·F = ?F x ?x + ?F y ?y + ?F z ?z (D.21) ?×F = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle ?x ?y ?z ? ?x ? ?y ? ?z F x F y F z vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle (D.22) ? 2 f = ? 2 f ?x 2 + ? 2 f ?y 2 + ? 2 f ?z 2 (D.23) ? 2 F = ?x? 2 F x + ?y? 2 F y + ?z? 2 F z (D.24) Separation of the Helmholtz equation ? 2 ψ(x, y, z) ?x 2 + ? 2 ψ(x, y, z) ?y 2 + ? 2 ψ(x, y, z) ?z 2 + k 2 ψ(x, y, z) = 0 (D.25) ψ(x, y, z) = X(x)Y(y)Z(z) (D.26) k 2 x + k 2 y + k 2 z = k 2 (D.27) d 2 X(x) dx 2 + k 2 x X(x) = 0 (D.28) d 2 Y(y) dy 2 + k 2 y Y(y) = 0 (D.29) d 2 Z(z) dz 2 + k 2 z Z(z) = 0 (D.30) X(x) = braceleftBigg A x F 1 (k x x)+ B x F 2 (k x x), k x negationslash= 0, a x x + b x , k x = 0. (D.31) Y(y) = braceleftBigg A y F 1 (k y y)+ B y F 2 (k y y), k y negationslash= 0, a y y + b y , k y = 0. (D.32) Z(z) = braceleftBigg A z F 1 (k z z)+ B z F 2 (k z z), k z negationslash= 0, a z z + b z , k z = 0. (D.33) F 1 (ξ), F 2 (ξ) = ? ? ? ? ? ? ? ? ? e jξ e ?jξ sin(ξ) cos(ξ) (D.34) Cylindrical coordinate system Coordinate variables u = ρ, 0 ≤ ρ<∞ (D.35) v = φ, ?π ≤ φ ≤ π (D.36) w = z, ?∞ < z < ∞ (D.37) x = ρ cosφ (D.38) y = ρ sinφ (D.39) z = z (D.40) ρ = radicalbig x 2 + y 2 (D.41) φ = tan ?1 y x (D.42) z = z (D.43) Vector algebra ?ρ = ?x cosφ + ?y sinφ (D.44) ? φ =??x sinφ + ?y cosφ (D.45) ?z = ?z (D.46) A = ?ρA ρ + ? φA φ + ?zA z (D.47) A · B = A ρ B ρ + A φ B φ + A z B z (D.48) A × B = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle ?ρ ? φ ?z A ρ A φ A z B ρ B φ B z vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle (D.49) Dyadic representation ˉa = ?ρa ρρ ?ρ+ ?ρa ρφ ? φ+ ?ρa ρz ?z + + ? φa φρ ?ρ+ ? φa φφ ? φ+ ? φa φz ?z + + ?za zρ ?ρ+ ?za zφ ? φ+ ?za zz ?z (D.50) ˉa = ?ρa prime ρ + ? φa prime φ + ?za prime z = a ρ ?ρ+ a φ ? φ+ a z ?z (D.51) a prime ρ = a ρρ ?ρ+ a ρφ ? φ+ a ρz ?z (D.52) a prime φ = a φρ ?ρ+ a φφ ? φ+ a φz ?z (D.53) a prime z = a zρ ?ρ+ a zφ ? φ+ a zz ?z (D.54) a ρ = a ρρ ?ρ+ a φρ ? φ+ a zρ ?z (D.55) a φ = a ρφ ?ρ+ a φφ ? φ+ a zφ ?z (D.56) a z = a ρz ?ρ+ a φz ? φ+ a zz ?z (D.57) Di?erential operations dl = ?ρdρ + ? φρ dφ + ?z dz (D.58) dV = ρ dρ dφ dz (D.59) dS ρ = ρ dφ dz, (D.60) dS φ = dρ dz, (D.61) dS z = ρ dρ dφ (D.62) ? f = ?ρ ?f ?ρ + ? φ 1 ρ ?f ?φ + ?z ?f ?z (D.63) ?·F = 1 ρ ? ?ρ parenleftbig ρF ρ parenrightbig + 1 ρ ?F φ ?φ + ?F z ?z (D.64) ?×F = 1 ρ vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle ?ρ ρ ? φ ?z ? ?ρ ? ?φ ? ?z F ρ ρF φ F z vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle (D.65) ? 2 f = 1 ρ ? ?ρ parenleftbigg ρ ?f ?ρ parenrightbigg + 1 ρ 2 ? 2 f ?φ 2 + ? 2 f ?z 2 (D.66) ? 2 F = ?ρ parenleftbigg ? 2 F ρ ? 2 ρ 2 ?F φ ?φ ? F ρ ρ 2 parenrightbigg + ? φ parenleftbigg ? 2 F φ + 2 ρ 2 ?F ρ ?φ ? F φ ρ 2 parenrightbigg + ?z? 2 F z (D.67) Separation of the Helmholtz equation 1 ρ ? ?ρ parenleftbigg ρ ?ψ(ρ,φ, z) ?ρ parenrightbigg + 1 ρ 2 ? 2 ψ(ρ,φ,z) ?φ 2 + ? 2 ψ(ρ,φ,z) ?z 2 + k 2 ψ(ρ,φ,z) = 0 (D.68) ψ(ρ,φ,z) = P(ρ)Phi1(φ)Z(z) (D.69) k 2 c = k 2 ? k 2 z (D.70) d 2 P(ρ) dρ 2 + 1 ρ dP(ρ) dρ + parenleftBigg k 2 c ? k 2 φ ρ 2 parenrightBigg P(ρ) = 0 (D.71) ? 2 Phi1(φ) ?φ 2 + k 2 φ Phi1(φ) = 0 (D.72) d 2 Z(z) dz 2 + k 2 z Z(z) = 0 (D.73) Z(z) = braceleftBigg A z F 1 (k z z)+ B z F 2 (k z z), k z negationslash= 0, a z z + b z , k z = 0. (D.74) Phi1(φ) = braceleftBigg A φ F 1 (k φ φ)+ B φ F 2 (k φ φ), k φ negationslash= 0, a φ φ + b φ , k φ = 0. (D.75) P(ρ) = ? ? ? ? ? a ρ lnρ + b ρ , k c = k φ = 0, a ρ ρ ?k φ + b ρ ρ k φ , k c = 0 and k φ negationslash= 0, A ρ G 1 (k c ρ)+ B ρ G 2 (k c ρ), otherwise. (D.76) F 1 (ξ), F 2 (ξ) = ? ? ? ? ? ? ? ? ? e jξ e ?jξ sin(ξ) cos(ξ) (D.77) G 1 (ξ), G 2 (ξ) = ? ? ? ? ? ? ? ? ? J k φ (ξ) N k φ (ξ) H (1) k φ (ξ) H (2) k φ (ξ) (D.78) Spherical coordinate system Coordinate variables u = r, 0 ≤ r < ∞ (D.79) v = θ, 0 ≤ θ ≤ π (D.80) w = φ, ?π ≤ φ ≤ π (D.81) x = r sinθ cosφ (D.82) y = r sinθ sinφ (D.83) z = r cosθ (D.84) r = radicalbig x 2 + y 2 + z 2 (D.85) θ = tan ?1 radicalbig x 2 + y 2 z (D.86) φ = tan ?1 y x (D.87) Vector algebra ?r = ?x sinθ cosφ + ?y sinθ sinφ + ?z cosθ (D.88) ? θ = ?x cosθ cosφ + ?y cosθ sinφ ? ?z sinθ (D.89) ? φ =??x sinφ + ?y cosφ (D.90) A = ?rA r + ? θA θ + ? φA φ (D.91) A · B = A r B r + A θ B θ + A φ B φ (D.92) A × B = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle ?r ? θ ? φ A r A θ A φ B r B θ B φ vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle (D.93) Dyadic representation ˉa = ?ra rr ?r + ?ra rθ ? θ+ ?ra rφ ? φ+ + ? θa θr ?r + ? θa θθ ? θ+ ? θa θφ ? φ+ + ? φa φr ?r + ? φa φθ ? θ+ ? φa φφ ? φ (D.94) ˉa = ?ra prime r + ? θa prime θ + ? φa prime φ = a r ?r + a θ ? θ+ a φ ? φ (D.95) a prime r = a rr ?r + a rθ ? θ+ a rφ ? φ (D.96) a prime θ = a θr ?r + a θθ ? θ+ a θφ ? φ (D.97) a prime φ = a φr ?r + a φθ ? θ+ a φφ ? φ (D.98) a r = a rr ?r + a θr ? θ+ a φr ? φ (D.99) a θ = a rθ ?r + a θθ ? θ+ a φθ ? φ (D.100) a φ = a rφ ?r + a θφ ? θ+ a φφ ? φ (D.101) Di?erential operations dl = ?r dr + ? θrdθ + ? φr sinθ dφ (D.102) dV = r 2 sinθ dr dθ dφ (D.103) dS r = r 2 sinθ dθ dφ (D.104) dS θ = r sinθ dr dφ (D.105) dS φ = rdrdθ (D.106) ? f = ?r ?f ?r + ? θ 1 r ?f ?θ + ? φ 1 r sinθ ?f ?φ (D.107) ?·F = 1 r 2 ? ?r parenleftbig r 2 F r parenrightbig + 1 r sinθ ? ?θ (sinθF θ )+ 1 r sinθ ?F φ ?φ (D.108) ?×F = 1 r 2 sinθ vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle ?r r ? θ r sinθ ? φ ? ?r ? ?θ ? ?φ F r rF θ r sinθF φ vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle (D.109) ? 2 f = 1 r 2 ? ?r parenleftbigg r 2 ?f ?r parenrightbigg + 1 r 2 sinθ ? ?θ parenleftbigg sinθ ?f ?θ parenrightbigg + 1 r 2 sin 2 θ ? 2 f ?φ 2 (D.110) ? 2 F = ?r bracketleftbigg ? 2 F r ? 2 r 2 parenleftbigg F r + cosθ sinθ F θ + 1 sinθ ?F φ ?φ + ?F θ ?θ parenrightbiggbracketrightbigg + + ? θ bracketleftbigg ? 2 F θ ? 1 r 2 parenleftbigg 1 sin 2 θ F θ ? 2 ?F r ?θ + 2 cosθ sin 2 θ ?F φ ?φ parenrightbiggbracketrightbigg + + ? φ bracketleftbigg ? 2 F φ ? 1 r 2 parenleftbigg 1 sin 2 θ F φ ? 2 1 sinθ ?F r ?φ ? 2 cosθ sin 2 θ ?F θ ?φ parenrightbiggbracketrightbigg (D.111) Separation of the Helmholtz equation 1 r 2 ? ?r parenleftbigg r 2 ?ψ(r,θ,φ) ?r parenrightbigg + 1 r 2 sinθ ? ?θ parenleftbigg sinθ ?ψ(r,θ,φ) ?θ parenrightbigg + + 1 r 2 sin 2 θ ? 2 ψ(r,θ,φ) ?φ 2 + k 2 ψ(r,θ,φ)= 0 (D.112) ψ(r,θ,φ)= R(r)Theta1(θ)Phi1(φ) (D.113) η = cosθ (D.114) 1 R(r) d dr parenleftbigg r 2 dR(r) dr parenrightbigg + k 2 r 2 = n(n + 1) (D.115) (1 ?η 2 ) d 2 Theta1(η) dη 2 ? 2η dTheta1(η) dη + bracketleftbigg n(n + 1)? μ 2 1 ?η 2 bracketrightbigg Theta1(η) = 0, ?1 ≤ η ≤ 1 (D.116) d 2 Phi1(φ) dφ 2 +μ 2 Phi1(φ) = 0 (D.117) Phi1(φ) = braceleftBigg A φ sin(μφ)+ B φ cos(μφ), μ negationslash= 0, a φ φ + b φ ,μ= 0. (D.118) Theta1(θ) = A θ P μ n (cosθ)+ B θ Q μ n (cosθ) (D.119) R(r) = braceleftBigg R(r) = A r r n + B r r ?(n+1) , k = 0, A r F 1 (kr)+ B r F 2 (kr), otherwise. (D.120) F 1 (ξ), F 2 (ξ) = ? ? ? ? ? ? ? ? ? j n (ξ) n n (ξ) h (1) n (ξ) h (2) n (ξ) (D.121)