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Course: MAT 208, Fall 2009School: Paul Quinn
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208 Math Homework 3 due on Thursday 3/1/07 Problem 1. Evaluate the following triple integrals: (i) D zex+y dx dy dz, where D is the box [0, 1] [0, 1] [0, 1]. xy 2 z 3 dx dy dz, where D is the solid in R3 bounded by the surface z = xy and the planes x = y, x = 1, and z = 0. 1 1x2 0 1y D (ii) Problem 2. The iterated integral 1 0 f (x, y, z) dz dy dx is given. If you were to reverse the order of integrations...
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208 Math Homework 3
due on Thursday 3/1/07 Problem 1. Evaluate the following triple integrals: (i)
D
zex+y dx dy dz, where D is the box [0, 1] [0, 1] [0, 1]. xy 2 z 3 dx dy dz, where D is the solid in R3 bounded by the surface z = xy and the planes x = y, x = 1, and z = 0.
1 1x2 0 1y D
(ii)
Problem 2. The iterated integral
1 0
f (x, y, z) dz dy dx is given. If you
were to reverse the order of integrations to dx dy dz, what would be the appropriate limits of the three Problem integrals? 3. Suppose D1 and D2 are disjoint thin plates in the plane with masses m1 and m2 and centers of mass c1 and c2 . Show that the center of mass c of the union D1 D2 is given by m1 m2 c= c1 + c2 . m1 + m2 m1 + m2 Thus c is the point on the line segment from c1 to c2 whi...
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Paul Quinn - MAT - 208
Math 208 Homework 1due on Thursday 2/8/07 Problem 1. Use Cavalieris principle to prove the well-known formula V = 4 r3 for 3 the volume of a solid sphere of radius r. Problem 2. Compute the iterated integrals1 0 0 /2 /2 1(y cos x + 2) dx dy and v
Paul Quinn - MAT - 208
Math 208 Homework 4due on Thursday 3/22/07 Problem 1. Evaluate the following line integrals: (i)c(x2 2xy) dx + (y 2 2xy) dy, where c is the part of the parabola y = x2 from (1, 1) to (1, 1).(ii)c(2xy) dx + (x2 + z) dy + y dz, where c is th
Paul Quinn - MAT - 208
Math 208 Homework 5due on Thursday 3/29/07 Problem 1. Find the normal vector of the surface described parametrically as x = u2 v 2 y =u+v z = u2 + 4v at an arbitrary point. At what point(s) is this surface regular? Problem 2. A surface S is describ
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Paul Quinn - MAT - 328
Math 328 Homework 5due on Thursday 3/15/07 Problem 1. Find the (formal) solution of the ut = 4 uxx u(0, t) = 1, ux (, t) = 0 u(x, 0) = x heat equation 0 < x < , t > 0 t>0 0<x<Problem 2. Show that the (formal) solution of the heat equation
Paul Quinn - MAT - 328
Math 328 Homework 10due on Thursday 5/12/05 Problem 1. Recall that the solution to the one-dimensional heat equation ut = kuxx < x < , t > 0 u(x, 0) = f (x) is given by (xy)2 1 u(x, t) = f (y)e 4kt dy. 4kt Compute u(x, t) when the initial condi
Paul Quinn - MAT - 328
Paul Quinn - MAT - 328
Math 328 Homework 10due on Tuesday 5/8/07 Problem 1. Suppose f (x) has the Fourier transform F (). If a = 0, show that 1 f (ax) has the Fourier transform |a| F ( ). a Problem 2. Consider the function f (x) = ex 0 x0 . x<0 (i) Find the Fourier tr
Paul Quinn - MAT - 328
Paul Quinn - MAT - 328
Math 328 Homework 9due on Thursday 4/26/07 Problem 1. Find the Fourier integral of the functions f (x) = |x| + 1 |x| 1 0 |x| > 1 and 1 0<x<1 g(x) = 1 1 < x < 0 0 |x| > 1For what values of x does each of the equalities F I(f )(x) = f (x) hold?
Paul Quinn - MAT - 328
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Paul Quinn - MAT - 328
Math 328 Homework 6due on Thursday 3/17/05 Problem 1. Find the (formal) solution of the ut = 4uxx u(0, t) = 1, ux (, t) = 0 u(x, 0) = x heat equation 0 < x < , t > 0 t>0 0<x<Problem 2. Show that the (formal) solution of the heat equation 0
Paul Quinn - MAT - 328
Math 328 Homework 2due on Tuesday 2/20/07 Problem 1. Find the Fourier series of the following functions: (i) f (x) = cos x, < x < +. (ii) f (x) = cos x, < x < , f (x + ) = f (x) for all x. 2 2 (iii) f (x) = 0 1 < x < 0 , f (x + 2) = f (x) for al
Paul Quinn - MAT - 328
Math 328 Homework 6due on Thursday 3/29/07 Problem 1. Consider the wave equation 0 < x < , t > 0 utt = uxx u(0, t) = u(, t) = 0 t>0 u(x, 0) = sin(3x), u (x, 0) = sin(2x) 0 < x < t Verify directly that the solution u(x, t) given by the separa
Paul Quinn - MAT - 328
Math 328 Homework 4due on Thursday 3/8/07 Problem 1. Consider the function f (x) = C 0 0<x<L 2 L 2 x < L, that the (formal) solution of the heat 0 < x < L, t > 0 t>0 0<x<L nx . Lwhere C and L > 0 are constants. Show equation ut = k uxx u(0,
Paul Quinn - MAT - 328
Paul Quinn - MAT - 328
Paul Quinn - MAT - 328
Paul Quinn - MAT - 328
Paul Quinn - MAT - 328
Paul Quinn - MAT - 328
Math 328 Homework 3due on Thursday 2/24/05 Problem 1. Let f be the 2-periodic function dened by f (x) = 2x + 2 1 < x < 0 x 0<x<1 f (x + 2) = f (x).Sketch the graph of f as well as its Fourier series over 3 x 3. (No need to compute the Fourier se
Paul Quinn - MAT - 328
Paul Quinn - MAT - 328
Math 328 Homework 2due on Thursday 2/17/05 Problem 1. Find the Fourier series of the following functions: (i) f (x) = cos x, < x < +. (ii) f (x) = cos x, < x < , extended as a -periodic function on the real line. 2 2 (iii) f (x) = 0 1 < x < 0 ,
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Virginia Tech - CS - 4234
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Virginia Tech - CS - 4234
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Virginia Tech - CS - 4234
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Virginia Tech - CS - 4234
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