Page 517 - Electromagnetics

Helmholtztheorem

∇ · A(r ) A(r ) · ˆ n

A(r) =−∇ dV − dS +

V 4π|r − r | S 4π|r − r |

∇ × A(r ) A(r ) × ˆ n

+∇ × dV + dS (B.33)

V 4π|r − r | S 4π|r − r |
Miscellaneous identities

dS = 0 (B.34)
S

ˆ n × (∇a) dS = adl (B.35)
S

(∇a ×∇b) · dS = a∇b · dl =− b∇a · dl (B.36)
S

dl A = ˆ n × (∇A) dS (B.37)
S
Derivative identities

∇ (a + b) =∇a +∇b (B.38)
∇· (A + B) =∇ · A +∇ · B (B.39)
∇× (A + B) =∇ × A +∇ × B (B.40)
∇(ab) = a∇b + b∇a (B.41)
∇· (aB) = a∇· B + B ·∇a (B.42)
∇× (aB) = a∇× B − B ×∇a (B.43)
∇· (A × B) = B ·∇ × A − A ·∇ × B (B.44)
∇× (A × B) = A(∇· B) − B(∇· A) + (B ·∇)A − (A ·∇)B (B.45)
∇(A · B) = A × (∇× B) + B × (∇× A) + (A ·∇)B + (B ·∇)A (B.46)
2
∇× (∇× A) =∇(∇· A) −∇ A (B.47)
2
∇· (∇a) =∇ a (B.48)
∇· (∇× A) = 0 (B.49)
∇× (∇a) = 0 (B.50)
∇× (a∇b) =∇a ×∇b (B.51)
2
2
2
∇ (ab) = a∇ b + 2(∇a) · (∇b) + b∇ a (B.52)
2
2
2
∇ (aB) = a∇ B + B∇ a + 2(∇a ·∇)B (B.53)
2
∇ ¯ a =∇(∇· ¯ a) −∇ × (∇× ¯ a) (B.54)
∇· (AB) = (∇· A)B + A · (∇B) = (∇· A)B + (A ·∇)B (B.55)
∇× (AB) = (∇× A)B − A × (∇B) (B.56)
∇· (∇× ¯ a) = 0 (B.57)
© 2001 by CRC Press LLC