hw8 - Math508 Fall 2010 Jerry L Kazdan Problem Set 8 D UE...

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Math508, Fall 2010 Jerry L. Kazdan Problem Set 8 D UE : Thurs. Nov. 11, 2010. Late papers will be accepted until 1:00 PM Friday . Note: We say a function is smooth if its derivatives of all orders exist and are continuous. 1. a) Let A ( t ) and B ( t ) be n × n matrices whose elements depend smoothly on the real variable t . Use the definition of the derivative (as a limit) to show that their product, G ( t ) = A ( t ) B ( t ) , is differentiable. What is the derivative of A 2 ( t ) ? b) Give an example of a 2 × 2 matrix A ( t ) that depends smoothly on the real variable t with dA 2 ( t ) dt = 2 A ( t ) A ( t ) . 2. Consider two smooth plane curves γ 1 , γ 2 : ( 0 , 1 ) R 2 that do not intersect. Suppose P 1 and P 2 are interior points on γ 1 and γ 2 , respectively, such that the distance | P 1 P 2 | is minimal. Prove that the straight line P 1 P 2 is perpendicular to both curves. 3. Let A ( t ) be a square matrix that depends continuously on t for all t R and let the vector u ( t ) be a solution of the differential equation du ( t ) dt = A ( t ) u ( t ) with u ( 0 ) = 0 .
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