7Integral theorems
IA Vector Calculus
7.1 Statement and examples
There are three big integral theorems, known as Green’s theorem, Stoke’s theorem
and Gauss’ theorem. There are all generalizations of the fundamental theorem of
calculus in some sense. In particular, they all say that an
n
dimensional integral
of a derivative is equivalent to an
n −
1 dimensional integral of the original
function.
We will first state all three theorems with some simple applications. In the
next section, we will see that the three integral theorems are so closely related
that it’s easiest to show their equivalence first, and then prove just one of them.
7.1.1 Green’s theorem (in the plane)
Theorem
(Green’s theorem)
.
For smooth functions
P
(
x, y
),
Q
(
x, y
) and
A
a
bounded region in the (x, y) plane with boundary ∂A = C,
Z
A
∂Q
∂x
−
∂P
∂y
dA =
Z
C
(P dx + Q dy).
Here
C
is assumed to be piecewise smooth, non-intersecting closed curve, tra-
versed anti-clockwise.
Example.
Let
Q
=
xy
2
and
P
=
x
2
y
. If
C
is the parabola
y
2
= 4
ax
and the
line x = a, both with −2a ≤ y ≤ 2a, then Green’s theorem says
Z
A
(y
2
− x
2
) dA =
Z
C
x
2
dx + xy
2
dy.
From example sheet 1, each side gives
104
105
a
4
.
Example. Let A be a rectangle confined by 0 ≤ x ≤ a and 0 ≤ y ≤ b.
x
y
a
b
A
Then Green’s theorem follows directly from the fundamental theorem of calculus
in 1D. We first consider the first term of Green’s theorem:
Z
−
∂P
∂y
dA =
Z
a
0
Z
b
0
−
∂P
∂y
dy dx
=
Z
a
0
[−P (x, b) + P (x, 0)] dx
=
Z
C
P dx
Note that we can convert the 1D integral in the second-to-last line to a line integral
around the curve
C
, since the
P
(
x,
0) and
P
(
x, b
) terms give the horizontal part
of
C
, and the lack of d
y
term means that the integral is nil when integrating the
vertical parts.
Similarly,
Z
A
∂Q
∂x
dA =
Z
C
Q dy.
Combining them gives Green’s theorem.
Green’s theorem also holds for a bounded region
A
, where the boundary
∂A
consists of disconnected components (each piecewise smooth, non-intersecting
and closed) with anti-clockwise orientation on the exterior, and clockwise on the
interior boundary, e.g.
The orientation of the curve comes from imagining the surface as:
and take the limit as the gap shrinks to 0.
7.1.2 Stokes’ theorem
Theorem (Stokes’ theorem). For a smooth vector field F(r),
Z
S
∇ × F · dS =
Z
∂S
F · dr,
where
S
is a smooth, bounded surface and
∂S
is a piecewise smooth boundary
of S.
The direction of the line integral is as follows: If we walk along
C
with
n
facing up, then the surface is on your left.
It also holds if
∂S
is a collection of disconnected piecewise smooth closed
curves, with the orientation determined in the same way as Green’s theorem.
Example.
Let
S
be the section of a sphere of radius
a
with 0
≤ θ ≤ α
. In
spherical coordinates,
dS = a
2
sin θe
r
dθ dϕ.
Let F = (0, xz, 0). Then ∇ × F = (−x, 0, z). We have previously shown that
Z
S
∇ × F · dS = πa
3
cos α sin
2
α.
Our boundary ∂C is
r(ϕ) = a(sin α cos ϕ, sin α sin ϕ, cos α).
The right hand side of Stokes’ is
Z
C
F · dr =
Z
2π
0
a sin α cos ϕ
| {z }
x
a cos α
| {z }
z
a sin α cos ϕ dϕ
| {z }
dy
= a
3
sin
2
α cos α
Z
2π
0
cos
2
ϕ dϕ
= πa
3
sin
2
α cos α.
So they agree.
7.1.3 Divergence/Gauss theorem
Theorem (Divergence/Gauss theorem). For a smooth vector field F(r),
Z
V
∇ · F dV =
Z
∂V
F · dS,
where
V
is a bounded volume with boundary
∂V
, a piecewise smooth, closed
surface, with outward normal n.
Example. Consider a hemisphere.
S
2
S
1
V is a solid hemisphere
x
2
+ y
2
+ z
2
≤ a
2
, z ≥ 0,
and ∂V = S
1
+ S
2
, the hemisphere and the disc at the bottom.
Take F = (0, 0, z + a) and ∇ · F = 1. Then
Z
V
∇ · F dV =
2
3
πa
3
,
the volume of the hemisphere.
On S
1
,
dS = n dS =
1
a
(x, y, z) dS.
Then
F · dS =
1
a
z(z + a) dS = cos θa(cos θ + 1) a
2
sin θ dθ dϕ
| {z }
dS
.
Then
Z
S
1
F · dS = a
3
Z
2π
0
dϕ
Z
π/2
0
sin θ(cos
2
θ + cos θ) dθ
= 2πa
3
−1
3
cos
3
θ −
1
2
cos
2
θ
π/2
0
=
5
3
πa
3
.
On S
2
, dS = n dS = −(0, 0, 1) dS. Then F · dS = −a dS. So
Z
S
2
F · dS = −πa
3
.
So
Z
S
1
F · dS +
Z
S
2
F · dS =
5
3
− 1
πa
3
=
2
3
πa
3
,
in accordance with Gauss’ theorem.