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¢ () =
() ¢ ()
(
)
m
0
DF q
g
0
.
(8.4)
This shows that DF maps tangent vectors to R k at q to tangent vectors to M at p , that
is,
(
) () Õ
(
(
) =
)
()
k
k
n
n
DT
F q
:
RR
T
M
T
RR
.
q
p
p
Since F is a local parameterization, DF( q ) is a one-to-one linear map from R k to R n .
It remains to show that DF( q ) is onto T p ( M ).
Let v Œ T p ( M ) and let m(t) be a curve in M with m(0) = p and m¢(0) = v . We need to
find a curve g(t) in R k with g(0) = q and m(t) =F(g(t)), because then the chain rule
(equation (8.4)) implies that
() ¢ ()
(
) =
DF q
g
0
v
.
But F -1 (m(t)) is such a curve (there is no loss in generality in assuming that m lies in
V ) and (2) is proved.
Part (3) clearly follows from (2). Finally, we prove (4). Using the g i defined above,
define curves m i through p by m i (t) =F(g i (t)). The m i can be thought of as defining a
local curvilinear coordinate system for M at p . See Figure 8.13. Furthermore, the curves
m i (t) are often called the u i -parameter curves for the parameterization F(u 1 ,u 2 , . . . ,u k ).
In the special case of a surface, one would refer to m 1 (t) and m 2 (t) as the u- and v-
parameter curves at p in M , respectively.
Recalling how partial derivatives are defined it is easy to see that
F
F
F
Ê
Ë
ˆ
¯
n
i
¢ () =
()
()
()
m i
0
q
,...,
q
=
q
.
u
u
u
i
i
On the other hand, the chain rule shown in equation (8.4) shows that
¢ () =
() ¢ ()
(
) =
()( )
m
0
D
F
q
g
0
D
F
q e
.
i
i
i
This proves (4).
8.4.4. Corollary. If M is a surface in R 3 , then the cross product
is a normal vector for T p ( M ).
∂∂ () ¥∂
()
F
x
q
F
y
q
Note: The fact that (4) holds in Theorem 8.4.3 was built into our definition of local
parameterizations of a manifold because they are assumed to be regular. However, as
mentioned earlier, it is sometimes natural to use parameterizations that are not
regular. For example, consider the parameterization of the surface of revolution
obtained by rotating the parabolic arc y = 1 - x 2 , -1 £ x £ 1, about the x-axis. One can
parameterize this surface via the map
(
(
)
(
)
)
(
) =
2
2
F x
,
q
x
,
1
-
x
cos ,
q
1
-
x
sin
q
,
-£ £
1
x
1 0
,
£ £
q
p
.
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