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# hw2 - CALIFORNIA INSTITUTE OF TECHNOLOGY Control and...

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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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hw2 - CALIFORNIA INSTITUTE OF TECHNOLOGY Control and...

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