Page 523 - Electromagnetics

Differential operations

dl = ˆ x dx + ˆ y dy + ˆ z dz (D.15)

dV = dx dy dz (D.16)

dS x = dy dz (D.17)
dS y = dx dz (D.18)
dS z = dx dy (D.19)

∂ f ∂ f ∂ f
∇ f = ˆ x + ˆ y + ˆ z (D.20)
∂x ∂y ∂z

∂ F x ∂ F y ∂ F z
∇· F = + + (D.21)
∂x ∂y ∂z

ˆ x ˆ y ˆ z
∂ ∂ ∂

∇× F = (D.22)
∂x ∂y ∂z
F x F y F z

2
2
2
∂ f ∂ f ∂ f
2
∇ f = + + (D.23)
∂x 2 ∂y 2 ∂z 2
2 2 2 2
∇ F = ˆ x∇ F x + ˆ y∇ F y + ˆ z∇ F z (D.24)
Separation of the Helmholtz equation
2
2
2
∂ ψ(x, y, z) ∂ ψ(x, y, z) ∂ ψ(x, y, z) 2
+ + + k ψ(x, y, z) = 0 (D.25)
∂x 2 ∂y 2 ∂z 2
ψ(x, y, z) = X(x)Y(y)Z(z) (D.26)

2
2
2
k + k + k = k 2 (D.27)
x y z
2
d X(x) 2
+ k X(x) = 0 (D.28)
x
dx 2
2
d Y(y) 2
+ k Y(y) = 0 (D.29)
y
dy 2
2
d Z(z) 2
+ k Z(z) = 0 (D.30)
z
dz 2

A x F 1 (k x x) + B x F 2 (k x x), k x
= 0,
X(x) = (D.31)
a x x + b x , k x = 0.

A y F 1 (k y y) + B y F 2 (k y y), k y
= 0,
Y(y) = (D.32)
a y y + b y , k y = 0.
© 2001 by CRC Press LLC