Cds202-hw2_wi09 - manifold Is it connected • Repeat the...

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CALIFORNIA INSTITUTE OF TECHNOLOGY Control and Dynamical Systems CDS 202 R. Murray Winter 2009 Problem Set #2 Issued: 15 Jan 09 Due: 22 Jan 09 Reading: Abraham, Marsden, and Ratiu (MTA): Review Sections 2.1 and 2.2 (covered in CDS 201) as needed Read Sections 2.3–2.4 Read Section 3.1–3.3 Problems: 1. MTA 2.3-1: Derivative of bilinear maps 2. MTA 2.3-4: Composition of a nonlinear and linear maps 3. MTA 3.1-4 (i) and (ii): Manifold structure of the M¨ obius band. Optional: Try to use your intuition about M¨ obius band to answer the following questions (then try them to see if you are right): Consider a a M¨ obius band of ±nite width, like the one shown in Figure 3.4.4. What happens is you cut it down the center with a pair of scissors? Is the resulting set a
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Unformatted text preview: manifold? Is it connected? • Repeat the experiment, but this time cutting the M¨ obius band one third of the distance from one of the edges. 4. MTA 3.1-5: Compacti±cation of R n . 5. [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 di²eomorphisms, so is g ◦ f . (You may use the fact that the composition of smooth functions between open subsets of Euclidian spaces are smooth.) 6. MTA 3.3-1: Graphs of manifolds...
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