Graphics Reference
In-Depth Information
k
∂
∂
v
u
Â
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
ζ
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
Œ
M
k
, then define a map
(
)
Æ
(
)
()
k
m
Df
pM
:
T
T
W
(8.15a)
(
)
p
f
p
by