Graphics Reference
In-Depth Information
R
n
h
2
h
2
(B)
U
2
B
q
V
2
V
1
h
1
(p)
R
k
+
M
n
N
k
h
2
(q)
U
1
h
1
h
1
(A)
p
A
(a)
N
1
1
N
1
1
M
2
M
2
N
1
2
N
1
2
good submanifolds
bad “submanifolds”
(b)
(c)
Figure 8.11.
Good and bad “submanifolds”.
A submanifold
N
k
of a manifold
M
n
Definition.
is said to have
codimension
n - k
in
M
n
.
Unless stated otherwise, all manifolds from now on are assumed to be differen-
tiable manifolds.
8.4
Tangent Vectors and Spaces
Curves are basic to understanding our definition of tangent vectors and tangent spaces
of manifolds.
Definition.
A
C
r
parametric curve
is a C
r
function F : [a,b] Æ
R
n
. The space
X
=
F([a,b]) traced out by F will be called the
path
of the parametric curve F. A
differen-
tiable parametric curve
is a C
•
parametric curve. The parametric curve F is said to be
