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Note that EG > 0 because g > 0.
Next, we show how to define and compute area for surfaces. Section 8.12
already gave an intrinsic definition of volume for an abstract oriented Riemannian
manifold based on the theory of differential forms. We could build on that but
differential forms are rather abstract. This chapter has mostly been following the
more classical approach and we shall continue that to get some well-known
formulas. We shall stick with manifolds defined by parameterizations and only
require the reader to be familiar with basic advanced calculus. When appropriate
we shall make some comments to tie what we do here to what was done in Section
8.12.
We indulge in one bit of generalization. Rather than restricting ourselves to para-
meterized surfaces in
R
3
we shall consider parameterized n-dimensional manifolds
M
n
in
R
n+1
. Let F :
U
Æ
M
be a regular parameterization. The parameterization F(u
1
,
u
2
,...,u
n
) induces an orientation of
M
. Assume that
n
(
p
) is the standard unit normal
vector field on
M
determined by this orientation of
M
. See Section 8.5.
Definition.
The
volume
of the parameterization F, denoted by V(F), is defined
by
∂
∂
F
Ê
ˆ
Á
Á
Á
Á
Á
u
˜
˜
˜
˜
˜
1
M
Ú
det
()
=
V
F
.
∂
∂
F
U
u
n
Ë
¯
n
When n = 1, volume is called
length
. When n = 2, volume is called
area
.
Note that V(F) is positive, since the normal vector was chosen in such a way as
to make the determinant positive. Also, the volume might be infinite.
It is not hard to show that if n = 1, then this definition agrees with the definition
of length of a curve given earlier. See Exercise 9.8.1 To justify this definition for arbi-
trary n, observe that the determinant is just the volume of the (n + 1)-dimensional
parallelotope spanned by the ∂F/∂u
i
and
n
(Corollary 4.8.9). Since the normal vector
n
is orthogonal to the tangent plane and has length 1, the determinant is also the
volume of the n-dimensional base of this parallelotope, namely, the parallelotope in
the tangent plane spanned by the tangent vectors ∂F/∂u
i
. See Figure 9.17. Therefore,
to take the case of a surface as an example, the integral is just the limit of Riemann
sums of areas of parallelograms that approximate the manifold.
The definition of volume is not very satisfactory as it stands because the formula
also involves the normal vector
n
.
Let F :
U
Æ
M
n
be a regular parameterization of manifold
M
n
9.8.5. Theorem.
in
R
n+1
. Then
()
=
Ú
()
=
Ú
V
det g
ij
g
.
U
U
Proof.
We have
