Cds202-hw4_wi09

Cds202-hw4_wi09 - ) and show that it is invariant. 6. Let...

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CALIFORNIA INSTITUTE OF TECHNOLOGY Control and Dynamical Systems CDS 202 R. Murray Winter 2009 Problem Set #4 Issued: 30 Jan 09 (Fri) Due: 6 Feb 09 (Fri) Reading: Abraham, Marsden, and Ratiu (MTA), sections 3.3, 4.1, 4.2 Problems: 1. MTA 3.3-1: tangent spaces/maps for graphs 2. If F ( x 1 ,x 2 ,x 3 ) = 0 is a submersion defining a 2-dimensional manifold in R 3 , under what conditions is X = v 1 ∂x 1 + v 2 ∂x 2 + v 3 ∂x ∂x 3 (evaluated at a point where F ( x 1 ,x 2 ,x 3 ) = 0) a tangent vector to M ? 3. [Boothby, page 119, #12] Show that any smooth vector field Y on S n - 1 R n is the restriction of a smooth vector field X on R n . 4. MTA 4.1-5: convolution equation 5. [Boothby, page 126, #6] Show that φ t ( x,y ) defined by φ t ( x,y ) = ( xe 2 t ,ye - 3 t ) defines a C flow on M = R 2 . Determine the vector field that generates this flow (called the infinitesimal generator of the flow
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Unformatted text preview: ) and show that it is invariant. 6. Let SO (3) be the set of 3 3 orthogonal matrices with determinant +1. The tangent space of SO (3) at the identity is given by the set of skew-symmetric matrices of the form b = ( ) = - 3 2 3- 1- 2 1 (well show this later in the course). (a) Show that if v R 3 , b v = v , where is the cross product in R 3 . (b) Show that the tangent space T R SO (3) consists of matrices of the form b R where b is skew-symmetric. (c) Show that the ow of a vector eld g ( R ) = b R is given by t ( R ) = exp( b t ) R where exp is the matrix exponential....
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