Page 519 - Electromagnetics

− jk·r − jk·r
∇× Ee =− jk × Ee (B.78)
2
− jk·r 2 − jk·r
∇ Ee =−k Ee (B.79)

Identities involving the transverse/longitudinal decomposition

Note: ˆ u is a constant unit vector, A u ≡ ˆ u · A, ∂/∂u ≡ ˆ u ·∇, A t ≡ A − ˆ uA u , ∇ t ≡
∇− ˆ u∂/∂u.

A = A t + ˆ uA u (B.80)
∂
∇= ∇ t + ˆ u (B.81)
∂u
ˆ u · A t = 0 (B.82)
(ˆ u ·∇ t ) φ = 0 (B.83)
∂φ
∇ t φ =∇φ − ˆ u (B.84)
∂u
∂φ
ˆ u · (∇φ) = (ˆ u ·∇)φ = (B.85)
∂u
ˆ u · (∇ t φ) = 0 (B.86)
∇ t · (ˆ uφ) = 0 (B.87)
∇ t × (ˆ uφ) =−ˆ u ×∇ t φ (B.88)
(B.89)
∇ t × (ˆ u × A) = ˆ u∇ t · A t
ˆ u × (∇ t × A) =∇ t A u (B.90)
ˆ u × (∇ t × A t ) = 0 (B.91)
ˆ u · (ˆ u × A) = 0 (B.92)
(B.93)
ˆ u × (ˆ u × A) =−A t
∂φ
∇φ =∇ t φ + ˆ u (B.94)
∂u
∂ A u
∇· A =∇ t · A t + (B.95)
∂u

∂A t
∇× A =∇ t × A t + ˆ u × −∇ t A u (B.96)
∂u
2
∂ φ
2 2
∇ φ =∇ φ + (B.97)
t
∂u 2
2
∂ A t ∂ A u ∂ 2
∇× ∇× A = ∇ t ×∇ t × A t − +∇ t + ˆ u (∇ t · A t ) −∇ A u (B.98)
t
∂u 2 ∂u ∂u
2
2 ∂ A t 2
∇ A = ∇ t (∇ t · A t ) + −∇ t ×∇ t × A t + ˆ u∇ A u (B.99)
∂u 2

© 2001 by CRC Press LLC