Appendix D Coordinate systems Rectangular coordinate system Coordinate variables l=x,-0<x< Vector algebra =XAr+yAy+2A A·B=AxBx+AyBy+A2B A×B=A,A,A Br B, B. Dyadic representation Rarvy + Taxi+ gay&+yayy y+yay 2+ a= Xay yay+ ia,= axx+avy+a2 i a arx+ axyy +axzZ a, arx+ ayyy +azz +ayx+azz =axi+ayyy+axy Z ax2 i+avy+azz @2001 by CRC Press LLC
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 = xˆ Ax + yˆ Ay + zˆ Az (D.4) A · B = Ax Bx + Ay By + Az Bz (D.5) A × B = xˆ yˆ zˆ Ax Ay Az Bx By Bz (D.6) Dyadic representation a¯ = xˆaxxxˆ + xˆaxyyˆ + xˆaxzzˆ + + yˆayxxˆ + yˆayyyˆ + yˆayzzˆ + + zˆazxxˆ + zˆazyyˆ + zˆazzzˆ (D.7) a¯ = xaˆ x + yaˆ y + zaˆ z = axxˆ + ayyˆ + azzˆ (D.8) a x = axxxˆ + axyyˆ + axzzˆ (D.9) a y = ayxxˆ + ayyyˆ + ayzzˆ (D.10) a z = azxxˆ + azyyˆ + azzzˆ (D.11) ax = axxxˆ + ayxyˆ + azx zˆ (D.12) ay = axyxˆ + ayyyˆ + azy zˆ (D.13) az = axzxˆ + ayzyˆ + azzzˆ (D.14)
dl=&dx+y dy+id (D.15) dSx dy dz ds. dx d Vf afr afaF- (D21) V×F (D.22) af af a-f f F=XV Fr+yV Fy+iV-Fx Separation of the Helmholtz equation v(x,y.3+32y(x,y.3)+32y(x,y,2 a y2 +k2y(x,y,z)=0(D5) (x,y,z)=X(x)r()Z(z) (D.26) d2x(x)+k2X(x)=0 key (D.29) d-z(z) d2+k2Z(z)=0 (D.30) X(x)= Ar Fi(rx)+ Br F2(krx),kx#0, arx+br 0. Y()=AyFi(k, y)+B, F2(kyy), ky+0. (D.32) a,y+b @2001 by CRC Press LLC
Differential operations dl = xˆ dx + yˆ dy + zˆ dz (D.15) dV = dx dy dz (D.16) d Sx = dy dz (D.17) d Sy = dx dz (D.18) d Sz = dx dy (D.19) ∇ f = xˆ ∂ f ∂x + yˆ ∂ f ∂y + zˆ ∂ f ∂z (D.20) ∇ · F = ∂Fx ∂x + ∂Fy ∂y + ∂Fz ∂z (D.21) ∇ × F = xˆ yˆ zˆ ∂ ∂x ∂ ∂y ∂ ∂z Fx Fy Fz (D.22) ∇2 f = ∂2 f ∂x 2 + ∂2 f ∂y2 + ∂2 f ∂z2 (D.23) ∇2 F = xˆ∇2Fx + yˆ∇2Fy + zˆ∇2Fz (D.24) Separation of the Helmholtz equation ∂2ψ(x, y,z) ∂x 2 + ∂2ψ(x, y,z) ∂y2 + ∂2ψ(x, y,z) ∂z2 + k2 ψ(x, y,z) = 0 (D.25) ψ(x, y,z) = X(x)Y (y)Z(z) (D.26) k2 x + k2 y + k2 z = k2 (D.27) d2X(x) dx 2 + k2 x X(x) = 0 (D.28) d2Y (y) dy2 + k2 yY (y) = 0 (D.29) d2Z(z) dz2 + k2 z Z(z) = 0 (D.30) X(x) = Ax F1(kx x) + Bx F2(kx x), kx = 0, ax x + bx , kx = 0. (D.31) Y (y) = Ay F1(ky y) + By F2(ky y), ky = 0, ay y + by , ky = 0. (D.32)
Az Fi(2z)+B F2(2z),kz#0, F1(),F2()= (D.34) COS(E) Cylindrical coordinate system Coordinate variables U=,-0<7<a x=pcosφ y=psinφ (D.39) 41 y Vector algebra p=scosφ+ysinφ 44 q=-sinφ+ycosφ A=pAa+φAφ+ZA (D.47) A.B=ApBp+AφB+A2Bz pφ B。BB2 @2001 by CRC Press LLC
Z(z) = Az F1(kzz) + Bz F2(kzz), kz = 0, azz + bz, kz = 0. (D.33) F1(ξ), F2(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 cos φ (D.38) y = ρ sin φ (D.39) z = z (D.40) ρ = x 2 + y2 (D.41) φ = tan−1 y x (D.42) z = z (D.43) Vector algebra ρˆ = xˆ cos φ + yˆ sin φ (D.44) φˆ = −xˆ sin φ + yˆ cos φ (D.45) zˆ = zˆ (D.46) A = ρˆ Aρ + φˆ Aφ + zˆ Az (D.47) A · B = Aρ Bρ + Aφ Bφ + Az Bz (D.48) A × B = ρˆ φˆ zˆ Aρ Aφ Az Bρ Bφ Bz (D.49)
Dyadic representation papp+pap中+paaz2 +apP+φaφ+oasz2+ +2ap+22中+2az2 (D.50) a=pa+a+ia=app+a,+a2 (D.51) a=ap+a中+a92 a2=ap+a中+a2 (D.57) Differential operations pdp dv= pdpdo dz (D.59) dSp=pdφd dSo= dp dz, (D.61) Vf=pa +6+ (D.63) 1a(PFp)+p a 1aF。,aF2 V×F 能 (D65) a-f f 2 d VF=PV2Fp +中(VFaE9 +2v2F2(D.67) @2001 by CRC Press LLC
Dyadic representation a¯ = ρˆ aρρρˆ + ρˆ aρφφˆ + ρˆ aρzzˆ + + φˆ aφρρˆ + φˆ aφφφˆ + φˆ aφzzˆ + + zˆazρρˆ + zˆazφφˆ + zˆazzzˆ (D.50) a¯ = ρˆ a ρ + φˆ a φ + zaˆ z = aρρˆ + aφφˆ + azzˆ (D.51) a ρ = aρρρˆ + aρφφˆ + aρzzˆ (D.52) a φ = aφρρˆ + aφφφˆ + aφzzˆ (D.53) a z = azρρˆ + azφφˆ + azzzˆ (D.54) aρ = aρρρˆ + aφρφˆ + azρzˆ (D.55) aφ = aρφρˆ + aφφφˆ + azφzˆ (D.56) az = aρzρˆ + aφzφˆ + azzzˆ (D.57) Differential operations dl = ρˆ dρ + φˆ ρ dφ + zˆ dz (D.58) dV = ρ dρ dφ dz (D.59) d Sρ = ρ dφ dz, (D.60) d Sφ = dρ dz, (D.61) d Sz = ρ dρ dφ (D.62) ∇ f = ρˆ ∂ f ∂ρ + φˆ 1 ρ ∂ f ∂φ + zˆ ∂ f ∂z (D.63) ∇ · F = 1 ρ ∂ ∂ρ ρFρ + 1 ρ ∂Fφ ∂φ + ∂Fz ∂z (D.64) ∇ × F = 1 ρ ρˆ ρφˆ zˆ ∂ ∂ρ ∂ ∂φ ∂ ∂z Fρ ρFφ Fz (D.65) ∇2 f = 1 ρ ∂ ∂ρ ρ ∂ f ∂ρ + 1 ρ2 ∂2 f ∂φ2 + ∂2 f ∂z2 (D.66) ∇2 F = ρˆ ∇2Fρ − 2 ρ2 ∂Fφ ∂φ − Fρ ρ2 + φˆ ∇2Fφ + 2 ρ2 ∂Fρ ∂φ − Fφ ρ2 + zˆ∇2Fz (D.67)
Separation of the Helmholtz equation 1a2y(p,中, +k2y(p,p,z)=0(D.68) y(p,φ,3)=P(p)(φ)Z(z) (D.69) d-P(p) 1 dP(p) P(p)=0 dp a2Φ 2-+k中(p)=0 (D72) d-z(z) kaz dz2 z()=14:F1(k2x)+B2F2(k2),k≠0 k=0. Φ(φ) AφF1(k)+BF2(中),k≠0, φ+b ap Inp+be P(p) k=0 and ko≠0 Ap GI(ep)+ BeG2(kcp), otherwise F1(),F2()= 人() G1(),G2()= (D.78) Spherical coordinate system l=r,0≤r<∝ U=6,0≤6≤丌 (D.80) φ,-π≤φ≤丌 @2001 by CRC Press LLC
Separation of the Helmholtz equation 1 ρ ∂ ∂ρ ρ ∂ψ(ρ, φ,z) ∂ρ + 1 ρ2 ∂2ψ(ρ, φ,z) ∂φ2 + ∂2ψ(ρ, φ,z) ∂z2 + k2 ψ(ρ, φ,z) = 0 (D.68) ψ(ρ, φ,z) = P(ρ)(φ)Z(z) (D.69) k2 c = k2 − k2 z (D.70) d2P(ρ) dρ2 + 1 ρ d P(ρ) dρ + k2 c − k2 φ ρ2 P(ρ) = 0 (D.71) ∂2(φ) ∂φ2 + k2 φ(φ) = 0 (D.72) d2Z(z) dz2 + k2 z Z(z) = 0 (D.73) Z(z) = Az F1(kzz) + Bz F2(kzz), kz = 0, azz + bz, kz = 0. (D.74) (φ) = Aφ F1(kφφ) + Bφ F2(kφφ), kφ = 0, aφφ + bφ, kφ = 0. (D.75) P(ρ) = aρ ln ρ + bρ, kc = kφ = 0, aρρ−kφ + bρρkφ , kc = 0 and kφ = 0, AρG1(kcρ) + BρG2(kcρ), otherwise. (D.76) F1(ξ), F2(ξ) = e jξ e− jξ sin(ξ) cos(ξ) (D.77) G1(ξ), G2(ξ) = Jkφ (ξ) Nkφ (ξ) H(1) kφ (ξ) H(2) kφ (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=rsin6cosφ (D82) r=√x2+y2+z2 6=t √x2+y2 (D86) y (D.87) Vector algebra f= sin a cosφ+ D sin 6 sinφ+2cosb b= xcos 0 cosφ+ v cos e sinφ-isin6 (D89) φ=-8sinφ+ycosφ (D.90) TA+0Ag+Ag (D.91) A·B=ABr+AB+ e Be Bo a=farr+rare+raro Haer r+ eaee6+ Baes+ +φasb+ass r+ae0 a'=arf+are8+aroa (D.96) ae=ae,f+aeea+aes arf+arb+asr中 ae=anf+aee6+ase中 (D.101) @2001 by CRC Press LLC
x = r sin θ cos φ (D.82) y = r sin θ sin φ (D.83) z = r cos θ (D.84) r = x 2 + y2 + z2 (D.85) θ = tan−1 x 2 + y2 z (D.86) φ = tan−1 y x (D.87) Vector algebra rˆ = xˆ sin θ cos φ + yˆ sin θ sin φ + zˆ cos θ (D.88) θˆ = xˆ cos θ cos φ + yˆ cos θ sin φ − zˆ sin θ (D.89) φˆ = −xˆ sin φ + yˆ cos φ (D.90) A = rˆ Ar + θˆ Aθ + φˆ Aφ (D.91) A · B = Ar Br + Aθ Bθ + Aφ Bφ (D.92) A × B = rˆ θˆ φˆ Ar Aθ Aφ Br Bθ Bφ (D.93) Dyadic representation a¯ = rˆarr rˆ + rˆarθθˆ + rˆarφφˆ + + θˆ aθr rˆ + θˆ aθθθˆ + θˆ aθφφˆ + + φˆ aφr rˆ + φˆ aφθθˆ + φˆ aφφφˆ (D.94) a¯ = raˆ r + θˆ a θ + φˆ a φ = ar rˆ + aθθˆ + aφφˆ (D.95) a r = arr rˆ + arθθˆ + arφφˆ (D.96) a θ = aθr rˆ + aθθθˆ + aθφφˆ (D.97) a φ = aφr rˆ + aφθθˆ + aφφφˆ (D.98) ar = arr rˆ + aθrθˆ + aφrφˆ (D.99) aθ = arθ rˆ + aθθθˆ + aφθφˆ (D.100) aφ = arφrˆ + aθφθˆ + aφφφˆ (D.101)
Differential operations dl= fdr+ Or de or dv=r2 sing dr de dp (D.103) ds=rasin g de do (D.104) dSe=rsin6drdφ dSo=drdo (D.106) dr Vf rar(Fr)Ursine ag(sin8F0)+aFg (D.108) rsing d r2 sin e 品a Fr r Fe 1 af =-2(+0+如面+m)+ cos e +a7Fe-r2sin26Fe-2a0+ dFr cos a Fe Separation of the Helmholtz equation ay(r,6,φ) ay(r,6,φ) sin 0 a2 y(r,6,φ)=0 y(r,6,φ)=R(r)e(6)Φ(φ) D.11 (D.114) n+ 1-n2 d-e(n de(n dn2 ≤n≤ @2001 by CRC Press LLC
Differential operations dl = rˆ dr + θˆr dθ + φˆr sin θ dφ (D.102) dV = r 2 sin θ dr dθ dφ (D.103) d Sr = r 2 sin θ dθ dφ (D.104) d Sθ = r sin θ dr dφ (D.105) d Sφ = r dr dθ (D.106) ∇ f = rˆ ∂ f ∂r + θˆ 1 r ∂ f ∂θ + φˆ 1 r sin θ ∂ f ∂φ (D.107) ∇ · F = 1 r 2 ∂ ∂r r 2Fr + 1 r sin θ ∂ ∂θ (sin θ Fθ ) + 1 r sin θ ∂Fφ ∂φ (D.108) ∇ × F = 1 r 2 sin θ rˆ rθˆ r sin θφˆ ∂ ∂r ∂ ∂θ ∂ ∂φ Fr r Fθ r sin θ Fφ (D.109) ∇2 f = 1 r 2 ∂ ∂r r 2 ∂ f ∂r + 1 r 2 sin θ ∂ ∂θ sin θ ∂ f ∂θ + 1 r 2 sin2 θ ∂2 f ∂φ2 (D.110) ∇2 F = rˆ ∇2Fr − 2 r 2 Fr + cos θ sin θ Fθ + 1 sin θ ∂Fφ ∂φ + ∂Fθ ∂θ + + θˆ ∇2Fθ − 1 r 2 1 sin2 θ Fθ − 2 ∂Fr ∂θ + 2 cos θ sin2 θ ∂Fφ ∂φ + + φˆ ∇2Fφ − 1 r 2 1 sin2 θ Fφ − 2 1 sin θ ∂Fr ∂φ − 2 cos θ sin2 θ ∂Fθ ∂φ (D.111) Separation of the Helmholtz equation 1 r 2 ∂ ∂r r 2 ∂ψ(r,θ,φ) ∂r + 1 r 2 sin θ ∂ ∂θ sin θ ∂ψ(r,θ,φ) ∂θ + + 1 r 2 sin2 θ ∂2ψ(r,θ,φ) ∂φ2 + k2 ψ(r,θ,φ) = 0 (D.112) ψ(r,θ,φ) = R(r)(θ)(φ) (D.113) η = cos θ (D.114) 1 R(r) d dr r 2 d R(r) dr + k2 r 2 = n(n + 1) (D.115) (1 − η2 ) d2(η) dη2 − 2η d(η) dη + n(n + 1) − µ2 1 − η2 (η) = 0, −1 ≤ η ≤ 1 (D.116)
d2d(φ) dφ Φ(φ) Asin(p)+BC0s(φ),μ≠0 118) aφ+b e(0)=Ag PI(cos 0)+ BeO(cos 8) (D.119) IR()=A, r+B, r-(n+, k=0, Ar Fi(kr)+B, f2(kr) (D120) F1(),F21)=mn) (D.121) h2)() @2001 by CRC Press LLC
d2(φ) dφ2 + µ2 (φ) = 0 (D.117) (φ) = Aφ sin(µφ) + Bφ cos(µφ), µ = 0, aφφ + bφ, µ = 0. (D.118) (θ) = Aθ Pµ n (cos θ) + Bθ Qµ n (cos θ) (D.119) R(r) = R(r) = Arr n + Brr−(n+1) , k = 0, Ar F1(kr) + Br F2(kr), otherwise. (D.120) F1(ξ), F2(ξ) = jn(ξ) nn(ξ) h(1) n (ξ) h(2) n (ξ) (D.121)