This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: Math 260, Spring 2012 Jerry L. Kazdan Problem Set 6 Due: In class Thursday, Feb. 23. Late papers will be accepted until 1:00 PM Friday . Unless otherwise stated use the standard Euclidean norm . 1. Let u ( x,t ) be the temperature at time t at a point x on a homogeneous rod of length , say 0 x . Assume u satisfies the heat equation u t = 2 u x 2 with boundary conditions u (0 ,t ) = 0 , u x vextendsingle vextendsingle vextendsingle vextendsingle x = = 0 (so the right end is insulated) and initial condition u ( x, 0) = sin 5 2 x. a) Find the solution. b) Use Q ( t ) = 1 2 integraltext u 2 ( x,t ) dx to prove there is at most one solution (uniqueness). 2. [Marsden-Tromba Sec. 2.4#1-2] Sketch the curves a ) . x = sin t, y = 4cos t, t 2 b ) . x = 2sin t, y = 4cos t, t 2 3. Consider the curve F ( t ) = (sin2 t, 1 3 t, cos 2 t, 2 t 3 / 2 ) for 0 t 2 . a) Find the equation of the tangent line at t = / 2....
View Full Document
- Spring '12