CONFIDENTIAL EXAM PAPER This paper is not to be removed from the exam venue Mathematics and Statistics EXAMINATION Semester 1 - Main, 2019 MATH2021 Vector Calculus and Differential Equations (Differential Equations/Vector Calculus) EXAM WRITING TIME: 2 hours READING TIME: 10 minutes EXAM CONDITIONS: This is a CLOSED book examination - no material permitted During reading time - writing is not permitted on the answer material(s), ONLY on the exam paper MATERIALS PERMITTED IN THE EXAM VENUE: (No electronic aids are permitted e.g. laptops, phones) Calculator - non-programmable MATERIALS TO BE SUPPLIED TO STUDENTS: 1 x 16-page answer book INSTRUCTIONS TO STUDENTS: Please tick the box to confirm that your examination paper is complete. ? ? Room Number ________ Seat Number ________ Student Number |__|__|__|__|__|__|__|__|__| ANONYMOUSLY MARKED (Please do not write your name on this exam paper) For Examiner Use Only Q Mark 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Total ________ page 2 of 7 Formula Sheet Most of the formulas and theorems provided are stated without the conditions under which they apply. The notation used is the same as that used in lectures. Curves For a given curve C in R 3 with parametrization ↵ : [a,b] ! R 3 , then the arc-length L(C) = Z b a ||↵ 0 (t)||dt the skalar curvature k(s) = ||v ⇥ v 0 || ||v|| 3 , where v is the velocity of ↵. Line Integrals Z C ?(x,y,z)ds = Z b a ?(x(t),y(t),z(t)) s ✓ dx dt ◆ 2 + ✓ dy dt ◆ 2 + ✓ dz dt ◆ 2 dt. Z C F · dr = Z C F 1 dx + F 2 dy + F 3 dz = Work done by F along C. Grad grad ? = r? = @? @x i + @? @y j + @? @z k. r? is normal to the tangent plane of the surface ?(x,y,z) = k, (k a constant). If F is continuous and equal to r? for some ?, then F is a conservative field, ? is a potential function of F, and Z C F · dr is path-independent. Curl curlF = r ⇥ F = ? i j k @ @x @ @y @ @z F 1 F 2 F 3 ? If the domain of F is simply connected and r ⇥ F = 0 then F is conservative. turn to page 3 page 3 of 7 Double integrals over a region R in R 2 . Area of R = ZZ R dA. Volume of the solid under the surface z = f(x,y) over R = ZZ R f(x,y)dA. In polar coordinates: ZZ R f(x,y)dA = ZZ R f(rcos✓,rsin✓)rd✓dr. Divergence divF = @F 1 @x + @F 2 @y + @F 3 @z . Green’s theorem I C F · dr = I C F 1 dx + F 2 dy = ZZ R ✓ @F 2 @x ? @F 1 @y ◆ dxdy. In vector form: I C F · dr = ZZ R (r ⇥ F) · kdA. In divergence form: I C F · nds = ZZ R r · F dxdy = Flux of F across C. Surface integrals ZZ S ?(x,y,z)dS = ZZ R ?(x,y,f(x,y)) q 1 + f 2 x + f 2 y dxdy In cylindrical coordinates: ZZ ?(x,y,z)dS = ZZ ?(acos✓,asin✓,t)ad✓dt. In spherical coordinates: ZZ f(x,y,z)dS = ZZ f(asin✓cos',asin✓sin',acos✓)a 2 sin✓d✓d'. Flux across S = ZZ S F · ndS. Triple integrals In cylindrical coordinates: ZZZ f(x,y,z)dxdydz = ZZZ f(rcos',rsin',t)rdrd'dt. In spherical coordinates: turn to page 4 page 4 of 7 ZZZ f(x,y,z)dxdy dz = ZZZ f ? rsin✓cos',rsin✓sin',rcos✓ ? r 2 sin✓drd✓d' Divergence theorem ZZ S F · ndS = ZZZ V r · F dV Stokes’ theorem I C F · dr = ZZ S (r ⇥ F) · ndS Formal Fourier series Given a 2L-periodic function f : R ! R, (L > 0), the formal Fourier series of f: F(f)(x) := 1 2 a 0 + 1 X n=1 n a n cos ? n⇡ L x ? + b n sin ? n⇡ L x ? o where the Fourier coe?cients (a n ) n?0 and (b n ) n?1 are given by a n = 1 L Z L ?L f(x) cos ? n⇡ L x ? dx for every n ? 0, b n = 1 L Z L ?L f(x) sin ? n⇡ L x ? dx for every n ? 1, turn to page 5 page 5 of 7 Table of Standard Integrals 1. Z x n dx = x n+1 n + 1 + C (n 6= ?1) 2. Z dx x = ln|x| + C 3. Z e x dx = e x + C 4. Z dx p 1 ? x 2 = arcsin(x) + C 5. Z sinxdx = ?cosx + C 6. Z cosxdx = sinx + C 7. Z sec 2 xdx = tanx + C 8. Z sinhxdx = coshx + C 9. Z coshxdx = sinhx + C 10. Z dx p a 2 ? x 2 = sin ?1 x a + C 11. Z x sinxdx = ?xcosx + sinx + C 12. Z dx a 2 + x 2 = 1 a tan ?1 x a + C 13. Z dx p x 2 + a 2 = sinh ?1 x a + C = ln ⇣ x + p x 2 + a 2 ⌘ + C 0 14. Z dx p x 2 ? a 2 = cosh ?1 x a + C (x > a) = ln ?x + p x 2 ? a 2 ? ? + C 0 (x > a or x < ?a) turn to page 6 page 6 of 7 1. Question (a) Determine, whether the following statement is true or false: For every y 0 2 R, there is an open interval I ✓ R with 0 2 I and a unique solution y : I ! R of initial value problem y 0 = p 1 ? y 2 , y(0) = y 0 . Support your answer with a short argument!! (b) Find the general solution of the homogeneous equation Ly := (x 2 + 1)y 0 + xy = 0 (1) (c) Use the method of variation of constants to find the general solution of the inhomogeneous equation Ly = x with L defined by (1). (d) Find the unique solution y of initial value problem (x 2 + 1)y 0 + xy = x, y(0) = 3 and give the maximal solution interval I max ✓ R. 2. Question (a) Use the method of reduction of order to find the general solution of the 2nd-order di↵erential equations 4t 2 y 00 + 8ty 0 + y = 0 on (0,+1) given that y 1 (t) = t ?1/2 is a solution. (b) Find the unique solution of the initial value problem ( Ly := y 000 + y 00 + y 0 + y = 2(cosx ? sinx) on R, y(0) = 1, y 0 (0) = ?1, y 00 (0) = ?2. Hint: y p (x) := xsinx is a particular solution of Ly = 2(cosx ? sinx). (c) Let ˜ f : R ! R be the odd 2-periodic extension on R of the function f(x) = 1 ? x on [0,1]. (i) Sketch the graph of ˜ f for x 2 [?4,4]. (ii) Find the Fourier series of ˜ f. (iii) Find a solution of y 00 + y = ˜ f(x) on R. (d) Find all ? 2 R and all functions y 6⌘ 0 satisfying the eigenvalue problem y 00 + ?y = 0 on (0,⇡), y 0 (0) = 0, y(⇡) = 0. turn to page 7 page 7 of 7 3. Question (a) Let F be the vector field given by F = e x+y sinz i + e x+y sinzj + e x+y cosz k. (i) Decide whether F is conservative or not. Support your statement with a short proof. (ii) Calculate the work done by the force F along the path C given by the parametrization ↵(t) = sin3ti + tj + cos3tk for t 2 [0,2⇡/3]. (b) After a very tragic burn of a famous cathedral in Paris, the roof of one of the towers of this cathedral needs to be retiled. The roof has the shape of the upper part of the paraboloid x 2 + y 2 + z = 4 which sits on the top of a cylindrical tower of radius r = 1. Thus, the projection of the roof on the (xy)-plane is the unit disc x 2 + y 2  1. The costs of tiling is $2z + 15 per square meter. What is the total cost of retiling the roof of this tower? (c) Calculate the flux of the vector field F = 2xi + xy 2 j outward across the curve C running in anti-clockwise direction along straigtlines from (0,?1) to (1,0), from (1,0) to (0,1), and from (0,1) to (0,?1), forming a triangle. (d) A function f : B ! R defined on the closed unit ball B := {(x,y,z)|x 2 + y 2 + z 2  1} in R 3 satisfies the following properties: (i) f(x,y,z) > 0 for every (x,y,z) 2 B, (ii) ||rf(x,y,z)|| 2 = 8f(x,y,z) for every (x,y,z) 2 B, (iii) r · ⇣ f(x,y,z)rf(x,y,z) ⌘ = 17f(x,y,z) for every (x,y,z) 2 B. Use these three properties to calculate the flux of the vector field F := rf across the boundary of the unit ball B. This is the end of the examination paper of Section A.