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Course: MATH 428, Fall 2008
School: Cornell
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428 Math Ram rez Final Exam: Part 1 Due May 15, 2003 SHOW ALL OF YOUR WORK. You may only discuss this exam with the lecturer or the TA. Using sources other than the textbook is allowed and encouraged but please include references to any books you use. 1. Suppose a particle moves in R3 . Its position at time t is denoted by x(t) = (x(t), y(t), z(t)). Let T = m x2 be the kinetic energy of the particle and 2 V = V...

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428 Math Ram rez Final Exam: Part 1 Due May 15, 2003 SHOW ALL OF YOUR WORK. You may only discuss this exam with the lecturer or the TA. Using sources other than the textbook is allowed and encouraged but please include references to any books you use. 1. Suppose a particle moves in R3 . Its position at time t is denoted by x(t) = (x(t), y(t), z(t)). Let T = m x2 be the kinetic energy of the particle and 2 V = V (x) be its potential energy. The principle of least action states that the particle will move from time t1 to t2 from the point x1 to the point x2 in such a way as to minimize the action integral t2 A= t1 T V dt. Show that the Euler-Lagrange equations for this problem are Newtons equations of motion. 2. A wave guide is a long, hollow tube that is used to conne electromagnetic (or acoustic) waves with the purpose of propagate them with minimal decay. Assume that we have an innite wave guide ( x ) with rectangular cross section of width (0 y ) and height h(0 z h). Assume that the electromagnetic waves inside the guide are modeled by the wave equation with speed light of c and that the tube shell is grounded so that we have Dirichlet boundary conditions. By using separation of variables show that the potential inside the wave guide at time t and position x = (x, y, z) is given by u(x, t) = dk fn,m (k)ei(kxt) sin[ny/ ] sin[mz/h] 2 n,m=1 where 2 = c2 [k 2 + (ny/ )2 + (mz/h)2 ]. f is the Fourier transform of the initial condition in the variable x of the Fourier coecients in y and z, that is 4 fm,n (k) = h h 0 0 sin[ny/ ] sin[mz/h]eikx u(x, y, z, 0) dx dy dz. 3. Use the method of reection to nd the Greens function of for the strip < x < , 0 < y < a. 4. Solve problems 21 and 22...

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