26
1 Preliminaries
Similarly, if substituting Eqs. (1.3.60) and (1.3.61) into the Green Formula
(1.3.55), the formulae similar to Eqs. (1.3.62)–(1.3.70) can be obtained, it need
only to change the integral of the closed surface S into the integral of the closed
curve Γ , change integral of the volume V surrounded by S into the integral of the
area D surrounded by Γ , and change the three-dimensional Hamiltonian operator
and Laplace operator into the two-dimensional ones.
Historically, the Green formula was independently proposed, thus it is an original
formula. But because it is closely related to Gauss formula, it can be thought that
the Gauss formula is a generalization of the planar Green formula in space, and the
Green formula is a special case of the Gauss formula on the plane.
If ϕ in the Green’s first formula is changed into ϕ, then there is
V
∇ · ((ϕ∇ψ)dV =
V
((ϕ∇ · ∇ψ + ∇ϕ · ∇ψ)dV
=
V
((ϕϕψ + ∇ϕ · ∇ψ)dV
=
S
ϕ
∂ψ
∂n
dS =
S
ϕn · ∇ψdS
(1.3.71)
If ψ in the Green’s first formula is changed into ψ, then there is
V
∇ · (ϕ∇ =
V
(ϕ∇ · ∇ψ + ∇ϕ · ∇ψ)dV
=
V
(ϕϕ
2
ψ + ∇ϕ · ∇ψ)dV
=
S
ϕ
∂∂ψ
∂n
dS =
S
ϕn · ∇ψdS
(1.3.72)
Transposing ϕ and ψ in Eq. (1.3.72), we obtain
V
∇ · (ψ∇ϕ)dV =
V
(ψ∇ · ∇ϕ + ∇ψ · ∇ϕ)dV
=
V
(ψψ
2
ϕ + ∇ψ · ∇ϕ)dV
=
S
ψ
∂∂ϕ
∂n
dS =
S
ψ n · ∇ϕdS
(1.3.73)
Comparing the both ends of the third equal sign in Eq. (1.3.73) with the one in
Eq. (1.3.71), we obtain
V
ψψ
2
ϕdV = −
V
∇ψ · ∇ϕdV +
S
ψ
∂∂ϕ
∂n
dS
=
V
ϕϕψdV +
S
ψ
∂∂ϕ
∂n
− ϕ
∂ψ
∂n
dS
(1.3.74)