Graphics Reference
In-Depth Information
8.12.15. Theorem.
(Stokes' Theorem) Let
M
n
be an oriented differentiable mani-
fold with boundary. If w is an (n - 1)-form on
M
with compact support, then
Ú
Ú
dww
=
.
M
∂
M
Proof.
The proof is surprisingly simple using what we know, but we refer the reader
to [Spiv70a] for the details.
The following two well-known theorems are trivial consequences of Stokes's
theorem.
(Green's Theorem) Given a surface
S
Ã
R
2
8.12.16. Theorem.
and differentiable
functions
fg
,
:
SR
Æ
,
then
∂
∂
g
x
∂
∂
f
y
Ê
Ë
ˆ
¯
Ú
ÚÚ
fdx
+
gdy
=
-
dxdy
.
S
S
We assume that the surface has been given the induced orientation from
R
2
and its
boundary the induced counter-clockwise orientation.
Proof.
Exercise 8.12.3.
8.12.17. Theorem.
(The Divergence Theorem) Let
M
3
be a differentiable sub-
manifold of
R
3
that has boundary. Let
n
(
p
) be the outward-pointing unit normal
vector field on ∂
M
. We assume that
M
has been given the induced orientation
and Riemannian metric from
R
3
. Let F be a differentiable vector field on
M
.
Then
Ú
Ú
=< >
div F dV
F
,
n
dA
.
M
∂
M
Proof.
Exercise 8.12.4.
Two more topics related to differential forms on manifold seem worth mention-
ing. The first has to do with volume and volume elements.
Given an oriented Riemannian manifold
M
n
, if dV is the volume element
Definition.
of
M
, then
Ú
dV
is called the
volume
of
M
, assuming that the integral exists.
One can show that this definition of volume agrees with the classical definition of
lengths of curves, areas of regions in the plane, or volumes of solids. We shall show
some of this in Chapter 9.
