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Math 2XX3 - Assignment

创建日期：2024-05-02 10:10:51

所属分类：数学mathematics

标签： Math 2XX3

Assignment

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Math 2XX3 - Assignment 4

Due: April 8, 11:59 pm (EST)
Please submit scans / photographs of your full solutions (if handwritten) or pdfs (if typed) to
the Crowdmark dropbox by the above date. You are welcome to discuss these problems with your
classmates, but your submission must be your own work and written in your own words. Also
please include an acknowledgement naming any classmate who helped you and citing any outside
sources you consulted. You are not required to use exactly the same notations that are used in the
lectures and course notes, but if you wish to modify or introduce your own notation you must first
clearly define it.
1. Draw pictures to illustrate the case n = 1 and m = 2 of Exercise 8 on page 118 of the Lecture
Notes, and explain how your pictures relate to the four hints given there.
2. Let L(u) =
∫ 1
0 F (u
′(s), u(s), s) ds for u ∈ C2([0, 1]), where F (v, w, s) =
√
1+v2
w . Find the
Euler-Lagrange equation, and solve it to find a general solution for the local extremals of
L(u). Hint: note that F is independent of the variable x.
3. Justify in detail each line of the tricky application of the chain rule in display (2.2) on page
126 of the Lecture Notes.
4. Let f(u) =
√
2u4
1+u4
in Subsection 3.2 on page 130 of the Lecture Notes. Use an online graphing
program (such as Desmos) to help sketch a variety of geodesics with differing parameters λ as
in the picture on page 132 of the Lecture Notes. Extend your sketches to show the geodesics
returning to the y-axis as well (when applicable). Make sure your examples demonstrate
each of behaviours (2), (3), and (4) in Summary 1 on page 131 of the Lecture Notes. You
may submit either hand-drawn sketches or printouts from your graphing calculator for this
problem.
5. Let Yn(x) =
√
2 cosnx and Y0(x) = 1, so that {Yn}∞n=1 is an orthonormal basis for L2([0, pi])
with respect to inner product 〈P,Q〉 = 1pi
∫ pi
0 P (x)Q(x) dx. Then for φ ∈ L2([0, pi]), the cosine
series of φ is given by
φ(x) =
∞∑
n=0
anYn(x) = a0 +
∞∑
n=0
an
√
2 cosnx, 0 ≤ x ≤ pi,
where equality in the display above holds in the sense of mean square convergence (you may
simply assume this as it is analogous to the situation for sine series). Calculate the Fourier
cosine series of x2 on the interval [0, pi], and then use Parseval’s identity to prove that
∞∑
n=1
1
n4
=
pi4
90
.

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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 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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 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ECON7310 FIT 1006 MANAGEMENT MATH3821 MTHM504 CS975 COMM1100 ELEC2133 MMAI 5500 MIS611 ELEC3114 CS 173 ELEC 3105 ISO 9001 IDSC10 Rural livelihoods and globalisati IDSC10 K17 202 CS823 MAST90007 SPG8024 MAT246 EG-264 CVEN2303 INFS1603 3PR3 MN-2508 4CCE1MCP SPG8030 CMT224 Inflation consumption Applying MAT137 CSE 111 DPBS1110 FINS5530 ELEC4613 GSBS6200 COMP4336 Math W54 DIFFERENTIAL 7PAVMALM ACC 1110 ME 3210 MATH2801 MGMT20001 ECON 6202 PSTAT 120B COMP9024 ENG5027 comp3219 ELE00107M ECON20003 ECON2103 FINA2209 MAST20018 LUBS5308M01 MATH2023 Computation COMP1600 CHEM 3X23 BBMM506 FNCE20003 BMET2922 BISM7216 financial ECMT5001 ACST3059 MAST30012 COMP3511 ECOS 2201 MKTG7512 ECON2540 MTH1010 MAT1841 COMP4600 ECON7740 PAPER C331 CAP 6617 ECON2420 ACCT7107 HLTH 7001 ECOS3002 CVEN30010 MAST30021 MATH1014 BIOL1007 ECOS 3010 SWEN90016 BISM7202 ACCT90012 INFO3315 DSCI 553 INFO90002 COMP3419 CITS 3001 EE5731 FINA3326 NFS 284 COMM2501 FIT2104 PHYS3035 RSM260 MAN3080 CHEM30028 MATH2088 ELEC0144 MECM10006 2D AE 202 ENG5022 CHEM3121 FINM7006 CS5340 CSE 594 FIT9136 FINS3633 RSM222 STAT3017 CS 8395 MAST10005 AMA546 ETC2410 FINM 7406 MSBA 7004 BUSS6002 CHEN 90038 PSYC20007 ACTL 3162 DIGITAL COSC 2670 PROGRAMMING ECON 5007 FNCE90045 PSTAT 126 MATH2307 STAT7055 SOC 360 BINF2010 WSTA01 AMME2200 ECO 204 BIO773P BU 312 AERO2598 ENGG 5103 ECON1010 ECON3031 MATH3078 HPS319H1 COMP26020 COMP42215 ACCT7106 COMP3271 MKTG10001 ECOS2025 MATH36031 CSE 402 ACCT1110 ECON 20005 MTH319 COMP 3711 PHYS1160 COMP3601 MSIN0026 ECON1202 ARCL0103 PHYS5033 PSYC 171 QBUS1040 ATOC10001 1ENVS222 COMP9032 CIVL2812 JURD7152 CS 449 ECON 7073 INFS6015 INFS7203 MATH2521 ECON7073 CSE101 COMP2121A DSC 40A MGT3009 COMP4161 FINM8006 BUSM054 COM6503 ST443 P 500 DPBS1190 ELEC5101 STA457H1 MATH3411 COMP5216 PLAN30081 STAT7203 MTHS3003 37T37T BLAW30002 GY7702 MATH580 ME 3057 PHYS 2012 ACCT13 ECON 3210 EE7207 ECOS3005 EC 502 MATH1004 CC0675 PHYS 4315 FINS 3636 COMP1012 economy ISE 562 MACS 30121 ML4ENG ECOS2201 ECOS3008 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