THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS Term 1, 2019 MATH2011 Several Variable Calculus (1) TIME ALLOWED – Two (2) hours (2) TOTAL NUMBER OF QUESTIONS – 4 (3) ANSWER ALL QUESTIONS (4) THE QUESTIONS ARE OF EQUAL VALUE (5) ANSWER EACH QUESTION IN A SEPARATE BOOK (6) THIS PAPER MAY BE RETAINED BY THE CANDIDATE (7) ONLY CALCULATORS WITH AN AFFIXED “UNSW APPROVED” STICKER MAY BE USED All answers must be written in ink. Except where they are expressly required pencils may only be used for drawing, sketching or graphical work. Term 1, 2019 MATH2011 Page 2 USEFUL INFORMATION The following information may be used without proof: sin2 t = 1 2 − 1 2 cos 2t, cos2 t = 1 2 + 1 2 cos 2t The Fourier series F of a function f , where f is defined on [−L,L], is given by the formula F (x) = a0 2 + ∞∑ n=1 { an cos (npix L ) + bn sin (npix L )} , with the Fourier coefficients an and bn defined by an = 1 L L∫ −L f(x) cos (npix L ) dx, n = 0, 1, 2, . . . , bn = 1 L L∫ −L f(x) sin (npix L ) dx, n = 1, 2, . . . . Please see over . . . Term 1, 2019 MATH2011 Page 3 Use a separate booklet clearly marked question 1 1. a) [3 marks] Suppose S is the surface given x2 + 3y2 − z2 = 0. Sketch the intersection of S with the xz and yz planes. Hence sketch the surface S. b) [3 marks] If f is a function of u and v, where u = 2r − s and v = r + s2, use the chain rule to find ∂f ∂r , ∂f ∂s and ∂2f ∂s ∂r in terms of ∂f ∂u , ∂f ∂v , ∂2f ∂u2 , ∂2f ∂v2 and ∂2f ∂u ∂v . c) [4 marks] The luminosity, L, at a point (x, y, z) in a galaxy is given by L(x, y, z) = 1 1 + x2 + y2 + z4 . i) Find the rate of change of luminosity at (1,−1, 1) in the direction of the vector ˜ v = ˜ i + ˜ j + ˜ k. ii) In which direction does L change most rapidly at the point (1,−1, 1), and what is the maximum rate of change of L at (1,−1, 1)? d) [3 marks] Do the surface 2×2 − xy − xz − y2 + 4z2 + 4x− 4z = 30 and the sphere x2 + y2 + z2 + x− y − 3z = 12 touch tangentially at the point (1, 1,−2)? Prove your answer. e) [7 marks] Let f be the function f(x, y) = x3 + 3×2 + 24xy + 12y2 + 15x− 2. i) Show that (1,−1) is a saddle point for f . ii) Show that f has one other critical point and find out what type is it. iii) Write down the Taylor series expansion of f about the critical point (1,−1). Please see over . . . Term 1, 2019 MATH2011 Page 4 Use a separate booklet clearly marked question 2 2. a) [4 marks] Let R be the region in the first quadrant bounded by y = x, y = 3x, x = 1 and x = 2. i) Sketch the region R. ii) If f(x, y) = xy, find ∫∫ R f(x, y) dA . b) [5 marks] Let g(x, y) = 1 1 + x2 + y2 . Using polar coordinates, find the integral of g over the region bounded by the circles x2 + y2 = 1 and x2 + y2 = 3. c) [8 marks] i) Let x(r, t) = r cos4 t, y(r, t) = r sin4 t. Calculate the Jacobian det ( ∂(x, y) ∂(r, t) ) . ii) Let P be the region of the two-dimensional plane bounded by the inequali- ties x ≥ 0, y ≥ 0 and √x +√y ≤ 1. Express the region P in terms of the new coordinates (r, t). iii) Using the change of variables from (x, y) to (r, t), compute the integral K = ∫∫ P (√ x+ √ y )3 dxdy. d) [3 marks] Let A be the region defined by the inequalities x ≥ 0, y ≥ 0, y ≤ 1− x2. i) Sketch the region A. ii) Find the centroid of the region A (assuming that the density is constant). Please see over . . . Term 1, 2019 MATH2011 Page 5 Use a separate book clearly marked Question 3 3. a) [10 marks] Let ˜ F(x, y, z) = xy ˜ i + z ˜ j + x2 ˜ k. Evaluate the line integral ∫ C ˜ F · d ˜ s, from the point point A = (0, 0, 0) to the point B = (1, 1, 1) along the curve C i) when C is the straight line segment connecting A and B; ii) when C is the portion of the twisted cubic y = x2, z = x3 connecting A and B. iii) Is ˜ F conservative? Explain why. b) [10 marks] i) Find a parametrisation for the surface region S of the plane x + y + z = 4 bounded by x2 + y2 ≤ 1. ii) Find the area of the region S. iii) Let ˜ F = y3 ˜ i− x3 ˜ j + z3 ˜ k. Verify that curl ˜ F = −3(x2 + y2) ˜ k. iv) Let C be a bounding curve of the region S. Find ∫ C ˜ F · d ˜ s using Stokes’ theorem. Please see over . . . Term 1, 2019 MATH2011 Page 6 Use a separate booklet clearly marked question 4 4. a) [10 marks] The solid V is the cube −1 ≤ x ≤ 1, −1 ≤ y ≤ 1, −1 ≤ z ≤ 1. The closed surface S is the boundary of V . The vector field F is defined by ˜ F(x, y, z) = yx2 ˜ i + y3 ˜ j + z ˜ k i) State the Divergence Theorem. ii) Apply the divergence theorem to calculate the flux∫∫ S ˜ F · ˆ ˜ n · dS of the field ˜ F through the surface S. iii) Calculate the flux of the field ˜ F through the surface S ′ obtained by removing the top side of the cube from the surface S. b) [10 marks] Let f be such that f(x+ 2pi) = f(x) and f(x) = { 1, if 0 ≤ |x| ≤ d, 0, if d < |x| ≤ pi, where d is some constant such that 0 < d < pi. i) Sketch the function f for −2pi ≤ x ≤ 2pi. ii) Verify that F (x) = d pi + 2 pi ∞∑ k=1 sin(kd) cos(kx) k is the Fourier series for the function f . iii) Using Parseval’s identity and the Fourier series for f , find ∞∑ k=1 sin2 kd k2 . END OF EXAMINATION
