a5 - 2014 Fall Math 5041 Assignment 5 Due Name 1 Let f M M...

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2014 Fall Math 5041 Assignment 5. Due: Nov 24, 3014 Name: 1. Let f : M M be a diffeomorphism. For vector fields X and Y on M , show that f * ([ X, Y ]) = [ f * ( X ) , f * ( Y )] 2. Consider two vector fields X = ∂y and Y = y ∂x - ∂z on R 3 , where we use the coordinates ( x, y, z ). Is it possible to find a 2-dimensional submanifold of R 3 with the property that both X and Y are tangent to it at all its points? Justify your answer. 3. Let V be a finite dimensional vector space. Then TV = V × V . a) The trivial map E : x 7→ ( x, x ) defines a section of the tangent bundle, i.e. a vector field. Compute the time- t flow of this vector field and determine whether it is complete or not. b) If A : V V is a linear map, then the map A : x 7→ ( x, Ax ) defines a vector field on V . Compute its flow and determine if it is complete. c) If A and B are two linear maps, compute the Lie derivative of the vector fields determined by A and B , i.e. compute their bracket. Verify that if the vector fields commute, then the flows commute.
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