1.3 Fundamentals of the Theory of Field
23
Let a = P(x, y)i + Q(x, y) j , and according to Eq. (1.3.46), then the flux ϕ
through the curve Γ can be written as
ϕ =
Γ
a · ndΓ =
Γ
a n dΓ =
Γ
(Pdy − Qdx)
(1.3.53)
Suppose that there is a planar vector field a = a(M), in a neighborhood of a point
M of the field doing any closed curve Γ containing the point, the area surrounded by
Γ is D, Calculating the flux of the vector a through Γ , when D is contracted
to a point M in any manner, if there is a limit of ratio
lim
D→0
ϕ
D
= lim
D→0
Γ a · ndΓ
D
then the limit is called the divergence of the vector function a at point M, it is written
as div a. When using the Hamiltonian operator, it can be written as ∇ · a, namely
div a = ∇ · a = lim
D→0
ϕ
D
= lim
D→0
Γ a · ndΓ
D
(1.3.54)
Theorem 1.3.3 Suppose that a planar vector a within a closed curve Γ has a continuous first order partial derivative, then the flux of a on Γ equals the integral of the
divergence of the vector with respect to the area D surrounded by Γ , that is
Γ
a · ndΓ =
Γ
a n dΓ =
¨
D
div ad D =
¨
D
∇ · ad D
(1.3.55)
where, n is the unit outward normal vector on Γ ; ∇ is the Hamiltonian operator.
Equation (1.3.55) is called the Green(’s) formula or Green(’s) theorem. It has
established the relationship between the double integral of a continuous function in
planar domain D and the curve integral on the boundary curve Γ . The method of
proof is the same as the method of proof of the Gauss theorem, it need only to change
the closed surface S into the closed curve Γ , and change the volume V surrounded
by S into the area D surrounded by Γ .
Let a = Q(x, y)i − P(x, y) j , and note that n =
dy
dΓ
i −
dx
dΓ
j , then the Green
formula can be written as
Γ
[P(x, y)dx + Q(x, y)dy] =
¨
D
∂ Q
∂ x
−
∂ P
∂ y
dxdy
(1.3.56)
here, the curvilinear integral is along the positive direction of the curve Γ , the double
integral distributes on the domain D. The so-called positive direction of the curve Γ
is such a direction, imagine a person walking around the curve Γ , make the domain
D surrounded by Γ always keep on its left.
23
Let a = P(x, y)i + Q(x, y) j , and according to Eq. (1.3.46), then the flux ϕ
through the curve Γ can be written as
ϕ =
Γ
a · ndΓ =
Γ
a n dΓ =
Γ
(Pdy − Qdx)
(1.3.53)
Suppose that there is a planar vector field a = a(M), in a neighborhood of a point
M of the field doing any closed curve Γ containing the point, the area surrounded by
Γ is D, Calculating the flux of the vector a through Γ , when D is contracted
to a point M in any manner, if there is a limit of ratio
lim
D→0
ϕ
D
= lim
D→0
Γ a · ndΓ
D
then the limit is called the divergence of the vector function a at point M, it is written
as div a. When using the Hamiltonian operator, it can be written as ∇ · a, namely
div a = ∇ · a = lim
D→0
ϕ
D
= lim
D→0
Γ a · ndΓ
D
(1.3.54)
Theorem 1.3.3 Suppose that a planar vector a within a closed curve Γ has a continuous first order partial derivative, then the flux of a on Γ equals the integral of the
divergence of the vector with respect to the area D surrounded by Γ , that is
Γ
a · ndΓ =
Γ
a n dΓ =
¨
D
div ad D =
¨
D
∇ · ad D
(1.3.55)
where, n is the unit outward normal vector on Γ ; ∇ is the Hamiltonian operator.
Equation (1.3.55) is called the Green(’s) formula or Green(’s) theorem. It has
established the relationship between the double integral of a continuous function in
planar domain D and the curve integral on the boundary curve Γ . The method of
proof is the same as the method of proof of the Gauss theorem, it need only to change
the closed surface S into the closed curve Γ , and change the volume V surrounded
by S into the area D surrounded by Γ .
Let a = Q(x, y)i − P(x, y) j , and note that n =
dy
dΓ
i −
dx
dΓ
j , then the Green
formula can be written as
Γ
[P(x, y)dx + Q(x, y)dy] =
¨
D
∂ Q
∂ x
−
∂ P
∂ y
dxdy
(1.3.56)
here, the curvilinear integral is along the positive direction of the curve Γ , the double
integral distributes on the domain D. The so-called positive direction of the curve Γ
is such a direction, imagine a person walking around the curve Γ , make the domain
D surrounded by Γ always keep on its left.