Page 518 - Electromagnetics

∇× (∇A) = 0 (B.58)
∇(A × B) = (∇A) × B − (∇B) × A (B.59)
∇(aB) = (∇a)B + a(∇B) (B.60)
¯
¯
¯
∇· (ab) = (∇a) · b + a(∇· b) (B.61)
¯
¯
¯
∇× (ab) = (∇a) × b + a(∇× b) (B.62)
¯
∇· (aI) =∇a (B.63)
¯
∇× (aI) =∇a × I ¯ (B.64)

Identities involving the displacement vector
ˆ

Note: R = r − r , R =|R|, R = R/R, f (x) = df (x)/dx.
ˆ

∇ f (R) =−∇ f (R) = R f (R) (B.65)
ˆ
∇ R = R (B.66)
1 R
ˆ
∇ =− 2 (B.67)
R R
− jkR − jkR
e 1 e
ˆ
∇ =−R + jk (B.68)
R R R
f (R)
ˆ
ˆ

∇· f (R)R =−∇ · f (R)R = 2 + f (R) (B.69)

R
∇· R = 3 (B.70)
2
ˆ
∇· R = (B.71)
R
e − jkR 1 e − jkR
ˆ
∇· R = − jk (B.72)
R R R
ˆ

∇× f (R)R = 0 (B.73)
1

2
∇ =−4πδ(R) (B.74)
R
e − jkR
2
2
(∇ + k ) =−4πδ(R) (B.75)
R
Identities involving the plane-wave function
Note: E is a constant vector, k =|k|.

− jk·r − jk·r
∇ e =− jke (B.76)

− jk·r − jk·r
∇· Ee =− jk · Ee (B.77)
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