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Cds202-hw3_wi09 - n is a matrix of dimension n n − 1...

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CALIFORNIA INSTITUTE OF TECHNOLOGY Control and Dynamical Systems CDS 202 R. Murray Winter 2009 Problem Set #3 Issued: 22 Jan 09 Due: 29 Jan 09 Reading: Abraham, Marsden, and Ratiu (MTA), sections 2.5 and 3.5 Problems: 1. MTA 2.5-3 (i), (ii) and (iv): exponential maps. You can assume (iii), which is a bit tricky to prove. 2. MTA 2.5-4: Equivalence of implicit and inverse function theorems. 3. MTA 2.5-12: Roots of polynomials are smooth functions of polynomial coefficients. 4. MTA 3.5-1 (i)–(ii): matrix manifolds. For (ii), you focus on showing that O
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Unformatted text preview: n ) is a matrix of dimension n ( n − 1) / 2 (you already showed it has two components in HW #1). 5. [Guillemin and Pollack, page 18, #6; MTA 3.5-5] (a) If f and g are submersions/immersions, show that f × g is. (b) If f and g are submersions/immersions, show that g ◦ f is. (c) If f is an immersion, show that its restriction to any submanifold of its domain is an immersion. (d) When dim M = dim N , show that submersions/immersions f : M → N are the same as local di²eomorphisms. 6. MTA 3.5-11: covering maps....
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