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

This preview shows pages 1–2. Sign up to view the full content.

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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern