CALIFORNIA INSTITUTE OF TECHNOLOGY
Control and Dynamical Systems
CDS 202
R. Murray
Winter 2004
Problem Set #2
Issued:
21 Jan 04
Due:
26 Jan 04
Reading:
Boothby, Chapter I and Sections III.1–III.3
Note:
Most of the problems are taken from the exercises in Guillemin and Pollack. If you
read Guillemin and Pollack, be warned that they treat manifolds slightly differently, using
parameterizations instead of coordinate charts.
Problems:
1. [Guillemin and Pollack, page 5, #3]
Let
M
,
N
, and
P
be smooth manifolds and let
f
:
M
→
N
and
g
:
N
→
P
be smooth maps.
(a) Show that the composite map
g
◦
f
:
M
→
P
is smooth.
(b) Show that if
f
and
g
are diffeomorphisms, so is
g
◦
f
.
(You may use the fact that the composition of smooth functions between open subsets of
Euclidian spaces are smooth.)
2. [Boothby II.1.2] Using stereographic projection from the north pole
N
(0
,
0
,
+1) of all of the
standard unit sphere in
R
3
except (0
,
0
,
+1) determine a coordinate neighborhood
U
N
, φ
N
. In
the same way determine by projection from the south pole
S
(0
,
0
,

1) a neighborhood
U
S
, φ
S
(see figure in Boothby). Show that these two neighborhoods determine a
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 Spring '08
 Marsden,J
 Manifold, Pollack, Smooth function

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