Graphics Reference
In-Depth Information
Solution.
Since
264
001
xyz
)
=
Ê
Ë
ˆ
¯
¢
(
fxyz
,,
and f¢ has rank 2 on the zero set V(f), Theorem 8.3.4 implies that V(f) is a
smooth curve. This is easily verified because V(f), the set of common zeros of the
functions
(
)
=+ + -
2
2
2
gxyz
,,
,,
x
321
y
z
(
)
=
hxy zz
,
is just the ellipse in the plane defined by the equation x
2
+ 3y
2
= 1. See Figure 8.9(b).
The following theorem is a local version of Theorem 8.3.4.
8.3.7. Theorem.
Let f :
R
n
Æ
R
m
and assume that f(
p
) =
0
. If the rank of Df is k in
a neighborhood of
p
, then there is a neighborhood
U
of
p
with
U
«
V
(f) an (n - k)-
dimensional manifold.
Proof.
Since being a manifold is a local property, one can use the same argument
as in Theorem 8.3.4.
Next we define what it means for a map between manifolds to be differentiable.
It might seem as if there is nothing to do since Section 4.3 already defined a notion
of differentiability for functions defined on subsets of
R
m
. However, since our mani-
folds are not necessarily open subsets of
R
m
, it is the definition given at the end of
that section that would have to be used. Unfortunately, using that definition for the
differentiability of a function on an arbitrary set one would not able to get a well-
defined derivative of the function. Therefore, we shall use a definition based on the
parameterizations of a manifold. After all, a parameterization corresponds to a coor-
dinate system for a neighborhood of a point in the manifold and it makes sense to
define differentiability with respect to such local coordinates.
Definition.
Let
M
n
and
N
k
be C
r
manifolds in
R
m
. A map f :
M
n
Æ
N
k
is said to be
of class C
r
or a
C
r
map at a point
p
in
M
n
if there is an open set
U
in
R
n
, an open
neighborhood
V
of
p
in
M
n
, and a local C
r
parameterization F
U
,
V
:
U
Æ
V
, so that
m
f
o
F
U,V
:
UNR
ÆÕ
is a C
r
map. The
rank of f at the point
p
is the rank of D(f F
U
,
V
) at
u
=F
U
,
V
-1
(
p
). The map
f is a
C
r
map
if it is of class C
r
at every point
p
in
M
. A
differentiable map
is a C
•
map.
o
See Figure 8.10. Notice that this definition does not yet define a derivative of the
map f. We will do that in the next section because we need to define tangent vectors
first. Right now we only have a notion of differentiability and rank. In this topic we
shall be mostly concerned with C
•
maps and not get involved in the fine points of C
r
maps, n π •.
