MATH264_april2006_final

# MATH264_april2006_final - McGILL UNIVERSITY FACULTY OF...

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Unformatted text preview: McGILL UNIVERSITY FACULTY OF ENGINEERING FINAL EXAMINATION MATH 264 Advanced Calculus “f.- Examiner: Professor N. Kamran - . -.- Date: Thursday April 20, 2006 Associate Examiner: Denis Serbin (I, Time: 9:00 AM- 12:00 PM INSTRUCTIONS 1. Please answer questions in the exam booklets provided 2. Questions 1 to 6 are worth 10 marks each. Question 7 is worth 20 marks 3. This is a closed book examination. No books, crib sheets or lecture notes permitted. 4. Calculators are not permitted. 5. Use of a regular or translation dictionary is not permitted This exam comprises the cover page, and 1 page of of 7 questions. MATH 264 Final examination This is a closed book examination. No calculators are permitted. Questions 1 to 6 are worth 10 marks each. Question 7 is worth 20 marks //D(w — y) dwdy, where D is the interior of the triangle with vertices at (0, 0), (1, 0)and (2, 1). 1. Compute the double integral 2. Compute the volume of the region above the X Y plane, bounded by the surfaces 22 = :32 + y2 and 3:2 + y2 = (12, where a > 0 is a positive constant. 3. Compute the line integral ‘3] . 11' . £(x2 + yz — ysm(a:y)) dm + (\$2 + yz — ats1n(my)) dy where c denotes the triangle with vertices at (1,0), (0,2) and (2, 2), oriented counter? clockwise. 4. Compute the area of the portion of the surface z = 2 — m2 — y2 which lies above the X Y plane. 5. Use Stokes’ formula to compute the line integral f2zdm + (230 + z)dy + (3x + 2y)dz, C where c is the curve deﬁned-by the intersection of the cylinder .732 + 22 = 1 with the plane m + 2y + z = 1, oriented counterclockwise when viewed from (0, 1,0). 6. Compute the outward ﬂux of the vector ﬁeld 1 F=V K+x2+ 2+16z2 < w2+<y—1)2+z2- y across the cube centered at (1, 1, 1) whose edges are of length 4. 7. Use the method of separation of variables and Fourier series to solve the diffusion equation ut : “9:11) with the boundary conditions u(0,t) = 10, um(2,t) = —5, t > 0 and the initial condition u(a:,0) =10, 0 < w < 2. ...
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## This note was uploaded on 04/20/2010 for the course MATH 264 taught by Professor Johnson during the Spring '10 term at McGill.

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MATH264_april2006_final - McGILL UNIVERSITY FACULTY OF...

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