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MATH2021 Vector Calculus and Differential Equations
创建日期：2024-04-13 17:38:46
所属分类：数学mathematics
标签： MATH2021
Vector Calculus and Differential Equations

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 Mathematics and Statistics EXAMINATION 

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.


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MAS316/414 ACCT2522 MSIN0181 ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 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