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MAST20009 Vector Calculus

创建日期：2024-04-13 15:43:08

所属分类：数学mathematics

标签： MAST20009

Vector Calculus

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MAST20009 Vector Calculus
Writing time: 3 hours
Reading time: 15 minutes
This is NOT an open book exam
This paper consists of 5 pages (including this page)
Authorised Materials
• Mobile phones, smart watches and internet or communication devices are forbidden.
• No written or printed materials may be brought into the examination.
• No calculators of any kind may be brought into the examination.
Instructions to Students
• You must NOT remove this question paper at the conclusion of the examination.
• There are 12 questions on this exam paper.
• All questions may be attempted.
• Marks for each question are indicated on the exam paper.
• Start each question on a new page.
• Clearly label each page with the number of the question that you are attempting.
• There is a separate 3 page formula sheet accompanying the examination paper, which you
may use in this examination.
• The total number of marks available is 151.
Instructions to Invigilators
• Students must NOT remove this question paper at the conclusion of the examination.
• Please supply graph paper.
• Initially students are to receive the exam paper, the 3 page formula sheet, two 11 page
script books and two sheets of graph paper.
This paper may be held in the Baillieu Library
Blank page (ignored in page numbering)
MAST20009 Semester 2, 2019
Question 1 (12 marks)
Prove that the function
f(x, y) =
 x
4 sin
(
1
x2 + |y|
)
, (x, y) 6= (0, 0)
0, (x, y) = (0, 0)
is continuous on all of R2.
Question 2 (9 marks)
For each of the following statements about a given scalar valued function f in two variables x
and y, decide whether it is true or false. Give brief justifications for your answers.
(a) If the partial derivatives of f both exist and are continuous, then f is continuous.
(b) Wherever the second derivatives of f are defined, we have
∂2f
∂x∂y
=
∂2f
∂y∂x
.
(c) If f is C1 then f is also C2.
Question 3 (12 marks)
(a) State the matrix chain rule. Be careful to include the conditions under which it holds and
to explain each symbol you use.
(b) Let
f(u, v, w) = (eu−w, sin(u+ v + w))
and let
g(x, y) = (ex, cos(y − x)).
Calculate g ◦ f and, using the matrix chain rule, calculate D(g ◦ f)|(0,0,0).
Question 4 (20 marks)
(a) Use Lagrange multipliers to determine the maximum of the function f(x, y) = 4xy + y
restricted to the line segment given by x+ y = 1 and x ≥ 0 and y ≥ 0.
(b) Prove that the answer you found in part (a) is indeed a maximum.
(c) On the graph paper provided, draw a high quality picture of this situation: the constraint,
at least three level curves of f , the gradient of f at three points and the gradient of the
constraint function g at three points. Pay attention to labels and scale.
Page 2 of 5 pages
MAST20009 Semester 2, 2019
Question 5 (16 marks)
(a) Sketch the path
γ(t) =
t cos(tpi)t sin(tpi)
t
 t ∈ [0, 6]
in R3 and describe the journey of a particle along this path (interpreting t as time variable)
in words.
(b) Compute the tangent, normal and binormal vectors of the path
c(t) =
 2 sin(t)2−√2 cos(t)√
2 cos(t)
 t ∈ R.
(c) Compute the curvature and the torsion of c.
(d) Interpret your answers to (b) and (c) geometrically.
Question 6 (12 marks)
Consider the vector field
~F (x, y) =
[
2y
−2x
]
.
(a) Use the graph paper provided to draw the vector field at the points (1, 1), (−1, 2), (2,−1),
(−3,−1) and (0,−1).
(b) Make an educated guess about the flow lines of ~F .
(c) Determine the equation for the flow line of ~F passing through the point (1, 1) in terms of
x and y. Show your working.
Question 7 (6 marks)
(a) Let f be a scalar valued function in three variables x, y and z. Under which condition on
f does the gradient of f satisfy the vector identity
∇× (∇f) = ~0?
[Hint: It is possible to figure this out.]
(b) Prove the identity in part (a), assuming the condition you found for f .
Page 3 of 5 pages
MAST20009 Semester 2, 2019
Question 8 (8 marks)
Consider the double integral ∫ 2
0
∫ x
0
x2y dy dx.
(a) Sketch the region of integration.
(b) Evaluate the integral in the form given.
(c) Change the order of integration and evaluate the integral again.
Question 9 (6 marks)
(a) Use polar coordinates to parametrize the interior of the ellipse with major axis of radius
a along the x-axis and minor axis of radius b along the y-axis.
a
b
y
x
(b) Using double integrals and your parametrization from part (a), prove that the volume
of the elliptical cylinder with semi-major axis a, semi-minor axis b, and height h equals
piabh.
Question 10 (10 marks)
Let D be the region bounded by the paraboloids z = x2 + y2 − 4 and z = 8− 2x2 − 2y2.
(a) Determine where the paraboloids intersect.
(b) Sketch the region D.
(c) Using cylindrical coordinates, evaluate the triple integral∫∫∫
D
x2 dV.
Page 4 of 5 pages
MAST20009 Semester 2, 2019
Question 11 (20 marks)
Let E ⊂ R2 be the ellipse with main radius equal to 5 and minor radius equal to 3, centred at
the origin with the main axis aligned with the x axis.
(a) Parametrize E using the counter-clockwise orientation and starting at the point (5, 0).
(b) Parametrize the rays from (−4, 0) to (0, 3) and from (−4, 0) to
(
5√
2
, 3√
2
)
.
(c) Using Green’s theorem in the plane, prove that the area of a closed bounded domain D
with boundary ∂D is given by
A(D) =
∫
∂D
x dy.
(d) Use your result from part (c) to calculate the area of the domain bounded by E and the
two rays in part (b).
(e) Using the graph paper provided, draw a high quality picture of the domain D and its
boundary, indicating orientations and paying attention to scale and labels.
Question 12 (20 marks)
Let S be the sphere of radius 6, centred at the origin.
(a) Give a parametrization
Ψ : [0, pi]× [0, 2pi] −→ S.
(b) Calculate the tangent vectors and the outward normal vector of S.
(c) Let S+ be the upper hemisphere, i.e., the part of S with z ≥ 0, oriented by the outward
normal vector. Let ~F be the vector field
~F (x, y, z) =
yz
1

on S+. Verify Stokes’ theorem for ~F and S+.
End of Exam—Total Available Marks = 151

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25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 MATH20602 COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 CIVL2010 ECMT2150 CS 333 CP1406 KIT308 Economics 701 ECON3740 ECE 5759 BFC3340 FINC2012 COMP4500 MATH3871 FIT3080 18-819C Economic ECO202Y5Y PHY100 FINC-780 Econ 100B EC602 IE582 EE450 COMP9517 WRDS 150B ECON 570 LECTURE 2 ECA5101 ECE 4420 ADAD9311 FIT3031 CS 544 MF 703 CS 6140 KIT501 ENGSCI 344 CSC171 IE 332 FINM8007 21S2 COSC 1114 FE545 E471 SIT102 CSC108 MAT301 IAE 101 COP3503 STAT2004 EC 303 CS1026 STAT1203 MATH3171 ISOM5160 IFN657 FIT2014 ACCT2102 DSCI550 STATS 330 ELEC421 COMP3670 QBUS6840 MEES:698B MAST30027 MIET1077 Economics IFT 6758 FIT5122 BUSN5101 CptS 111 HRMT5502 Math2018 PP 312 INFS3200 CZ3005 CSSE1001 SENG 2011 MKTG 553Q BUFN 640 COMP507 ETF5952 DATA 311 ECE4043 BANK3011 finance M3H623544 HW1 INFS 7410 AI6103 STAT3888 INFS3208 ACCT7103 ECOS3013 PSYCH 20B F79MA STAT3006 ENGG952 2X MGSC 7000 ECE4132 MSF 526 MATH 3033 CDS 511 EEEE4120 EEET2465 ECA5103 MATH2070 ELEC3104 BU51019 EEE8147 ENGR20003 COSC 1107 ETF3200 FNCE 435 MSFI 435 ISyE6420 GENG4405 HW ETC5523 IE 7315 MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 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