Graphics Reference
In-Depth Information
k
v
u
 1
i
j
b
=
a
.
(8.13)
i
j
j
=
Definition. An equivalence class of tuples ( U ,j, a ) with respect to the relation ~,
denoted by [ U ,j, a ] or [ U ,j, a ] p , is called a tangent vector of M k at p . One calls a i the
ith component of the tangent vector [ U ,j, a ] in the coordinate neighborhood ( U ,j). The
tangent space of M k
at p , T p ( M k ), is the set of all tangent vectors to M k
at p . The set
T p ( M k ) is made into a vector space by defining
[
] + [
] =
[
]
Ua U Uab
Ua U a
,,
j
,,
j
b
,,
j
+
[
] = [
]
c
,,
j
,,
j
c
.
One can show that T p ( M k ) is in fact a k-dimensional vector space. Furthermore,
we can tie our new definition to the earlier one where we defined tangent vectors in
terms of curves.
Definition.
If g : (c -e,c +e) Æ U (or g : [c,c +e) Æ U if p Œ∂ M ) is a curve with
g(c) = p , then
d
dt
uc d
dt
È
Í
Ê
Ë
ˆ
¯
˘
˙
[
] =
(
)( )( )
(
)( )
(
)( )
UD
, ,
jjg
o
c
1
U
, ,
j
o
g
,...,
uc
o
g
(8.14)
1
k
is called the tangent vector of g(t) at c.
Tangent vectors at points of curves in a manifold are well-defined tangent vectors
of the manifold that depend only on the curve near the points and not on the choice
of coordinate neighborhood ( U ,j). They can be used to define standard bases for the
tangent spaces. Define curves
(
) Æ U
g
i :
-
e e
,
by
() =
-1
(
() +
)
g
t
j
j
pe .
t
i
i
Notation.
Denote the tangent vector to g i (t) at 0 by e i, U .
It is easy to show that the vectors e 1, U , e 2, U ,..., e k, U form a basis for T p ( M k ) (Exer-
cise 8.8.2). They are the natural basis of T p ( M k ) with respect to the coordinate neigh-
borhood ( U ,j) and the current definition of tangent vectors.
Definition. Let M k and W m be differentiable manifolds and let f : M k Æ W m be a dif-
ferentiable map. If p ΠM k , then define a map
(
) Æ
(
)
()
k
m
Df
pM
:
T
T
W
(8.15a)
(
)
p
f
p
by
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