1. The Proof of Green’s Theorem
Theorem 1.1. (Fubini’s Theorem) Let R = [a, b] × [c, d] be a closed rectangle and f : R → R be a continuous function. Then
Z Z
R
f (x, y)dA = Z b
a
Z d c
f (x, y)dy
! dx =
Z d c
Z b a
f (x, y)dx
! dy.
Let F = P i + Qj be a smooth vector field on a domain D containing R. Equip the boundary C = ∂R of R with the positive orientation. We write C = C1+ C2− C3− C4 where C1, C2, C3, C4
are line segments parametrized as follows:
C1: r1(x) = (x, c), a ≤ x ≤ b C2: r2(y) = (b, y), c ≤ y ≤ d C3: r3(x) = (x, d), a ≤ x ≤ b C4: r4(y) = (a, y), c ≤ y ≤ d.
We rewrite the line integral I
C
F · dr = I
C
P dx + Qdy as I
C
P dx + Qdy = Z
C1
P dx + Qdy − Z
C3
P dx + Qdy +
Z
C2
P dx + Qdy − Z
C4
P dx + Qdy.
We computeR
C1P dx + Qdy andR
C3P dx + Qdy : Z
C1
P dx + Qdy = Z b
a
P (x, c)dx Z
C3
P dx + Qdy = Z b
a
P (x, d)dx which give
Z
C1
P dx + Qdy − Z
C3
P dx + Qdy = Z b
a
[P (x, c) − P (x, d)]dx = − Z b
a
[P (x, d) − P (x, c)]dx.
By fundamental Theorem of calculus,
P (x, d) − P (x, c) = Z d
c
Py(x, y)dy.
We discover that Z
C1
P dx + Qdy − Z
C3
P dx + Qdy = Z b
a
Z d c
−Pydy
! dx =
Z Z
R
−PydA by Fubini’s Theorem. Similarly, one can show that
Z
C2
P dx + Qdy − Z
C4
P dx + Qdy = Z Z
R
QxdA.
Combining all the above results, we obtain the Green’s Theorem I
∂R
P dx + Qdy = Z Z
R
(Qx− Py)dA for rectangle.
1
2
Theorem 1.2. (Green’s Theorem) Let D be a simply connected domain in R2. Assume that the boundary ∂D of D is a piecewise smooth curve. Then
I
∂D
P dx + Qdy = Z Z
D
(Qx− Py)dA.
The notationH
means that we equip the curve ∂D with the positive orientation.
From here, we can derive the Stoke’s Theorem in the following non-rigorous way. We observe that
curl F = (Qx− Py)k
and hence Qx− Py= curl F · k. Since k is a unit normal vector of the xy-plane, we can rewrite the Green theorem as
I
∂D
F · dr = Z Z
D
curl F · ndA.
In fact, the region D can be replaced by an orientable surface S and k by the unit outer normal vector field n to the surface S. This leads to Stoke’s Theorem:
Theorem 1.3. (Stoke’s Theorem) Let S be an orientable surface with boundary C = ∂S. Let F be a vector field in R3 on a region containing S. Then
I
C
F · dr = Z Z
S
curl F · ndA where n is the unit outer normal vector field on S.
It is also common to denote ndA by dA. The Stoke’s Theorem can be rewritten as the following vector form:
I
C
F · dr = Z Z
S
curl F · dA By Green theorem, the flux
Z
C
F · nds can be calculated by the following double integral:
Z
C
−Qdx + P dy = Z Z
D
(Px+ Qy)dA
where n is the unit outernormal of the curve C and D is the simply connected region whose boundary is C. Note that Px+ Qy is exactly the divergence of the vector field F, i.e.
Px+ Qy= div F.
This observation leads to the Gauss Theorem.
Theorem 1.4. (Gauss divergence Theorem) Let D be a simply connected solid region in R3whose boundary S = ∂D is an orientable surface. Let F be a vector field on a region containing D. Then
Z Z
S
F · dA = Z Z Z
D
div FdV.
