Graphics Reference
In-Depth Information
be the orthogonal projection of
R
n+k
onto
V
and let
O
be the open neighborhood of
V
in G
n
(
R
n+k
) consisting of all n-dimensional linear subspace
W
of
R
n+k
that p
V
proj-
ects
onto V
. The map
nk
j:
OR
Æ
defined by
j
()
=
(
)
nn
,
,...,
nnn
,
,...,
nn
,...,
n
,
11
12
1
k
,
21
22
n
1
,
n
2
nk
where (n
ij
) is the n ¥ k matrix N
W
as defined in equation (8.47) with respect to
W
, is
a homeomorphism and (
O
,j) serves as a coordinate neighborhood for G
n
(
R
n+k
).
In the general case all that we can assert is that some n columns of M
B
will be
linearly independent. In this case we can find a unique basis B so that these columns
will define an n ¥ n identity matrix and the remaining k columns form a unique n ¥
k matrix which we again denote by N
V
. There will be a corresponding coordinate
neighborhood (
O
,j). It is not hard to show that all these coordinate neighborhood
define a C
•
structure on G
n
(
R
n+k
) and we are done with the proof of Theorem 8.14.1.
8.15
E
XERCISES
Section 8.2
8.2.1.
Prove that every regular differentiable map f :
R
Æ
R
is one-to-one and onto an open
interval. Show by example, that a regular differentiable map f :
R
n
Æ
R
n
, n > 1, need not
be one-to-one.
Section 8.4
8.4.1.
Redo Example 8.4.5 but use the spherical coordinate parameterization
)
=
(
)
Fq f
(
,
cos
f
sin
f
,
sin
q
sin
f
,
cos
f
,
of the sphere, where (q,f) Œ [0,2p] ¥ [0,p].
The
spherical coordinates
(r,q,f) of a point
p
in
R
3
Note:
are related to the Cartesian
coordinates (x,y,z) of
p
via the equations
xr
yr n
zr s
=
=
=
cos
qf
qf
f
sin
,
,
s
sin
c
,
2
2
2
where . See Figure 8.37. In other words, q is the polar coordinate angle
of the point (x,y) in the plane and f is the angle between the z-axis and the vector
p
.
The angle q is sometimes called the
azimuth
of
p
and the angle f, the
colatitude
of
p
.
Unfortunately, the notation for spherical coordinates is not as standardized as that for
r xyz
=++
