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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 π , say 0 ≤ x ≤ π . 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 = π = 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 π 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 π b ) . x = 2sin t, y = 4cos t, ≤ t ≤ 2 π 3. Consider the curve F ( t ) = (sin2 t, 1 − 3 t, cos 2 t, 2 t 3 / 2 ) for 0 ≤ t ≤ 2 π . a) Find the equation of the tangent line at t = π/ 2....
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- Spring '12