1.3 Fundamentals of the Theory of Field
25
V
∇ · (ϕ∇ψ)dV =
V
(ϕ∇ · ∇ψ + ∇ϕ · ∇ψ)dV =
V
(ϕϕψ + ∇ϕ · ∇ψ)dV
=
S
ϕ
∂ψ
∂n
dS =
S
ϕ|∇ψ|dS =
S
ϕn · ∇ψdS
(1.3.63)
Transposing ϕ and ψ in Eq. (1.3.63), we obtain
V
∇ · (ψ∇ϕ)dV =
V
(ψψϕ + ∇ψ · ∇ϕ)dV =
S
ψ
∂ϕ
∂n
dS
=
S
ψ|∇ϕ|dS =
S
ψ∇ϕ · ndS
(1.3.64)
Equation (1.3.64) is another form of the Green’s first formula.
Subtracting the two forms of the Green’s first formula from each other, the
Green(’s) second formula is obtained, it is also called the Green(’s) second
theorem, that is
V
(ϕϕψ − ψψϕ)dV =
S
ϕ
∂ψ
∂n
− ψ
∂ϕ
∂n
dS =
S
(ϕ∇ψ − ψ∇ϕ) · ndS
(1.3.65)
When ϕ = ψ, the Green(’s) third formula is obtained by Eq. (1.3.59), it is also
called the Green(’s) third theorem, that is
V
[ϕϕϕ + (∇ϕ)
2
]dV =
S
ϕ
∂ϕ
∂n
dS =
S
ϕ∇ϕ · ndS
(1.3.66)
In Eq. (1.3.64), let ψ = 1, then there is
V
ϕdV =
S
∂ϕ
∂n
dS =
S
∇ϕ · ndS
(1.3.67)
In a rectangular coordinate system, Eq. (1.3.67) can be written as
V
∂
2
ϕ
∂ x 2 +
∂
2
ϕ
∂ y 2 +
∂
2
ϕ
∂z 2
dV =
S
∂ϕ
∂ x
dydz +
∂ϕ
∂ y
dzdx +
∂ϕ
∂z
dxdy
(1.3.68)
If ϕ = 0, from Eq. (1.3.66) to (1.3.67), we obtain respectively
V
(∇ϕ)
2 dV =
S
ϕ
∂ϕ
∂n
dS =
S
ϕ∇ϕ · ndS
(1.3.69)
S
∂ϕ
∂n
dS =
S
∇ϕ · ndS = 0
(1.3.70)