DEPARTMENT OF MATHEMATICS MATH2000 Assignment 4 Semester 2 2020 Submit your answers - along with this cover sheet - by 5:00pm on Friday October 9. Note that you may find some of these problems challenging. Attendance at weekly tutorials is assumed. Family name: Given names: Student number: Marker’s use only Each question marked out of 3. • Mark of 0: You have not submitted a relevant answer, or you have no strategy present in your submission. • Mark of 1: Your submission has some relevance, but does not demonstrate deep under- standing or sound mathematical technique. This topic needs more attention! • Mark of 2: You have the right approach, but need to fine tune some aspects of your calculations. • Mark of 3: You have demonstrated a good understanding of the topic and techniques involved, with well-executed calculations. A mathematician in the making? 1a 2 3 4a 1b 4b 1c 1d Total (out of 24): 1 Consider the vector field F (x, y) = (y2 − x2)i + xyj and let C be the closed curve consisting of three segments: the straight line from (0, 0) to (1, 0) followed by the straight line from (1, 0) to (1, 1) followed by the curve y = √ 2x− x2 from (1, 1) to (0, 0). Let n denote the unit normal vector to C directed out of the region D bounded by C. (a) Calculate the line integral ∮ C F · dr directly without using Green’s theorem. (b) Calculate the double integral x D ( ∂F2 ∂x − ∂F1 ∂y ) dA without using Green’s theorem and compare your answer with part (a). (c) Calculate the flux out of D by evaluating the line integral ∮ C F · n ds directly without using the flux form of Green’s theorem. (d) Calculate the flux out of D by applying Green’s Theorem and evaluating the double integral x D ∇ · F dA. 2 Calculate the flux of the vector field F (x, y, z) = xyi+ yzj + xzk out of the solid region enclosed by the surface x2 + y2 + z2 = 2y + 3. Make sure that you specify the direction of positive flux. Show all working. 3 A solid with thermal conductivity k = 2 occupies the region R in R3 bounded by the surfaces z = 1 − x2, y = 0, y = 1 and z = 0. The temperature at the point (x, y, z) is T (x, y, z) = xz2 + x2 + xy2 + 3z2 and the heat-flow vector is given by F = −k∇T . Calculate the heat flux (flux of F ) out of the closed surface S of R. 4 Let a, b ∈ R with 0 < a < b and let S be the surface(√ x2 + y2 − b)2 + z2 = a2 which is obtained by rotating the circle (y − b)2 + z2 = a2 in the plane x = 0 about the z-axis. (a) Parametrise the surface S and hence compute its surface area. (b) Find the average value of the function f(x, y, z) = x2 + z2 over S.
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