24
1 Preliminaries
Let P = −y, Q = x, then there is
D =
¨
D
dxdy =
1
2
Γ
(xdy − ydx)
(1.3.57)
Equation (1.3.57) is a formula using curvilinear integral to calculate the area of
the planar domain D.
If x = x(t), y = y(t), then Eq. (1.3.57) can be written as
D =
¨
D
dxdy =
1
2
Γ
(x ˙
y − y ˙
x)dt
(1.3.58)
Let P = u
∂u
∂ y
, Q = −u
∂u
∂ x
, then there is
¨
D
∂ 2 u
∂ x 2 +
∂ 2 u
∂ y 2
udxdy +
¨
D
∂u
∂ x
2
+
∂u
∂ y
2
dxdy = −
Γ
u
∂u
∂ y
dx − u
∂u
∂ x
dy
(1.3.59)
Suppose that the functions ϕ, ψ and λ have continuous second derivative in a
domain V , let a = ϕλ∇ψ, by the property of divergence, there is the following
formula
∇ · a = ∇ · (ϕλ∇ψ) = ϕ∇ · (λ∇ψ) + λ∇ϕ · ∇ψ
(1.3.60)
According to Eq. (1.3.24), there is
a · n = ϕλ∇ψ · n = ϕλ|∇ψ|n · n = ϕλ
∂ψ
∂n
(1.3.61)
Substituting Eqs. (1.3.60) and (1.3.61) into the Gauss Formula (1.3.45) and notice
that Eq. (1.3.24), we obtain
V
∇ · (ϕλ∇ψ)dV =
V
[ϕ∇ · (λ∇ψ) + λ∇ϕ · ∇ψ]dV
=
S
ϕλ
∂ψ
∂n
dS =
S
ϕλ|∇ψ|dS
=
S
ϕλ∇ψ · ndS
(1.3.62)
In Eq. (1.3.62), if let λ = 1, then the Green(’s) first formula or Ostrogradsky
formula can be obtained, it is also called the Green(’s) first theorem, namely