Page 516 - Electromagnetics

∇· ¯ a dV = ˆ n · ¯ a dS (B.19)
V S

∇ s · A dS = ˆ m · A dl (B.20)
S
Gradient theorem

∇adV = adS (B.21)
V S

∇A dV = ˆ nA dS (B.22)
V S

∇ s adS = ˆ madl (B.23)
V
Curl theorem

(∇× A) dV =− A × dS (B.24)
V S

(∇× ¯ a) dV = ˆ n × ¯ a dS (B.25)
V S

∇ s × A dS = ˆ m × A dl (B.26)
S
Stokes’s theorem

(∇× A) · dS = A · dl (B.27)
S

ˆ n · (∇× ¯ a) dS = dl · ¯ a (B.28)
S
Green’s first identity for scalar fields
∂b

2
(∇a ·∇b + a∇ b) dV = a dS (B.29)
V S ∂n
Green’s second identity for scalar fields (Green’s theorem)

∂b ∂a
2 2
(a∇ b − b∇ a) dV = a − b dS (B.30)
V S ∂n ∂n
Green’s first identity for vector fields

{(∇× A) · (∇× B) − A · [∇× (∇× B)]} dV =
V

∇· [A × (∇× B)] dV = [A × (∇× B)] · dS (B.31)
V S

Green’s second identity for vector fields

{B · [∇× (∇× A)] − A · [∇× (∇× B)]} dV =
V

[A × (∇× B) − B × (∇× A)] · dS (B.32)
S

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