We can directly integrate the left integral as a double integral
ZZ
∂M
dA
−
∂y
D
Z b Z δ(x)
∂M
−
=
dy dx
∂y
a
γ(x)
δ(x)
Z b
dx
−M (x, y)
=
Proof of Green’s theorem
Math 131 Multivariate Calculus
D Joyce, Spring 2014
Summary of the discussion so far.
I
ZZ ∂M
∂N
−
dA.
M dx + N dy =
∂y
∂D
D ∂x
a
Z
y=γ(x)
b
M (x, γ(x)) − M (x, δ(x)) dx
a
Z b
Z b
=
M (t, γ(t)) dt −
(t, δ(t)) dt
=
Green’s theorem can be interpreted as a planer
a
a
case of Stokes’ theorem
We’re almost done. The first integral
I
ZZ
Z b
F · ds =
(∇ × F) · k dA.
M (t, γ(t)) dt
∂D
D
a
In words, that says the integral of the vector field
F around the boundary ∂D equals the integral of
the curl of F over the region D. In the next chapter
we’ll study Stokes’ theorem in 3-space.
Green’s theorem implies the divergence theorem
in the plane.
ZZ
I
∇ · F dA.
F · n ds =
is the path integral along Z
the curve y = γ(x) from
M (x, y) dx. Likewise,
left to right, that is, it is
γ
Z b
the second integral
(t, δ(t)) dt is the parametera
ization along the curve y = δ(x) from left to right,
but that portion of the boundary ∂D should go
from right to left, and the minus sign reverses the
D
∂D
orientation. The two vertical sides x = a and x = b
dx
It says that the integral around the boundary ∂D of D form the other two parts of ∂D. Since dt is
of the the normal component of the vector field F 0 on those vertical paths, therefore
Z
Z
Z
equals the double integral over the region D of the
dx
M (x, y) dx = M (x, y) dt = 0 dt = 0
divergence of F.
dt
along them. Therefore,
Proof of Green’s theorem. We’ll show why
Z b
Z b
I
Green’s theorem is true for elementary regions D.
M (t, γ(t)) dt − (t, δ(t)) dt =
M (x, y) dx
These regions can be patched together to give more
a
a
∂D
general regions.
Likewise, if D is a region of “type 2,” that is, one
First, suppose that D is a region of “type 1,” that
is, it can be described by inequalities a ≤ x ≤ b and bounded between horizontal lines, then
ZZ
I
γ(x) ≤ y ≤ δ(x) where γ and δ are C 1 functions.
∂N
dA =
N (x, y) dy
First, we’ll show that
∂D
D ∂x
ZZ
I
where the minus sign is dropped because the sym∂M
−
dA =
M (x, y) dx
metry exchanging y for x reverses orientation.
∂y
D
∂D
1
Adding these two equations gives Green’s theorem
for D
ZZ I
∂N
∂M
−
dA =
M (x, y) dx+N (x, y) dy
∂x
∂y
D
∂D
That takes care of the case when the region is of
both type 1 and type 2. But regions that can be
decomposed into a finite number of these can be
patched together to take care of the general case.
q.e.d.
There are other proofs that are more inclusive
to show that some regions that are a union of an
infinite number of these regions also satisfy Green’s
theorem.
Math 131 Home Page at
http://math.clarku.edu/~djoyce/ma131/
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