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 LLCDifferential 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)