Graphics Reference
In-Depth Information
Definition.
Let
M
n
be an oriented compact n-dimensional submanifold of
R
n+1
that
admits a regular one-to-one parameterization F that induces the same orientation.
Define the
volume
of
M
, denoted by volume (
M
), by
volume
()
=
()
V F .
When n = 1 or 2, one uses the terms
length
and
area
, respectively, instead of the generic
term “volume.”
We need compactness in the definition to guarantee that the volume will be finite.
An immediate consequence of Lemma 9.8.6 is that the definition of volume of a man-
ifold does not depend on the parameterization.
9.8.7. Theorem.
The volume of an oriented compact n-dimensional submanifold
M
n
of
R
n+1
that admits a regular one-to-one parameterization is well defined.
Compare Theorem 9.8.5, the formula for our current definition of volume, with
Theorem 8.12.19, which is the differential form version. Roughly speaking, Theorem
9.8.5 is Theorem 8.12.19 for one coordinate neighborhood (
M
,F
-1
). Volume, as defined
in Section 8.12, was by definition an intrinsic property of a manifold. Our current
approach needed Lemma 9.8.6 to get the same result. Exercise 9.8.2 asks the reader
to show that this new definition again agrees with the definition given in Section 9.2
when n = 1, that is, in the case of curves in the plane.
Specializing to surfaces in
R
3
, we get
9.8.8. Theorem.
Let F :
U
Æ
S
be a regular and one-to-one parameterization of a
surface
S
in
R
3
. Then
()
=
ÚÚ
2
.
area
S
EG
-
F
U
Proof.
This is an immediate consequence of Theorem 9.8.5 and the relation between
integrals over regions in the plane and double integrals.
Note the important identity
2
2
(
)
(
)
-∑
(
)
2
,
F F
¥
=
FFFF
∑
∑
FF
=-
EG
F
(9.40)
u
v
u
u
v
v
u
v
which follows from Proposition 1.10.4(3).
To find the area of a region
S
in the plane
R
2
.
9.8.9. Example.
Solution.
If we use the identity map to parameterize
S
, then it is easy to check that
g = 1 and
()
=
ÚÚ
1,
area
S
S
which is the usual calculus definition of area.
The definitions above are inadequate for finding the area or volume of even simple
manifolds because the natural parameterizations are often neither regular nor one-
