ps11 - u ( x, t ): u t + 5 u x = 0 u ( x, 0) = cos(3 x ) 4....

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MATH 405: Problem Set 11 due Friday 12/15/2006 1. Using Faraday’s Law of Induction ∇× E = B /∂t between the electric field E and the magnetic field B , find the line integral of the electric field Z C E · d~ r around a closed current loop C with area A due to a spatially homogeneous time- varying magnetic field B = B 0 sin ωt which is oriented orthogonally to the plane of the loop C . 2. Consider a harmonic function φ , which means that φ is a solution to Laplace’s equation, 2 φ = 0. Show that for a region B with boundary ∂B : Z ∂B g ∂g ∂n dA = Z B |∇ g | 2 dV 3. Using the method of characteristics (change of variables into a new frame, as done in class) solve the following PDE for
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Unformatted text preview: u ( x, t ): u t + 5 u x = 0 u ( x, 0) = cos(3 x ) 4. Using the method of characteristics solve the following PDE for u ( x, t ): u t-16 u x =-5 u u ( x, 0) = e-2 x 2 5. For the following one-dimensional wave equation 2 u t 2 = 7 2 u x 2 show explicitly that u 1 = cos(2 x-2 7 t ) is a solution, and so is u 2 = ( x + 7 t ) 3 . Is u 1 + u 2 also a solution? 6. For the same wave equation, given the Gaussian initial condition u ( x, 0) = 3 exp(-x 2 ), what do you expect will happen at later times? Sketch u ( x, t )....
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This note was uploaded on 07/23/2008 for the course MATH 405 taught by Professor Belmonte during the Fall '06 term at Pennsylvania State University, University Park.

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