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# Fndns Of Analysis I MATH 3210

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This 8 page Class Notes was uploaded by Miss Noel Mertz on Monday October 26, 2015. The Class Notes belongs to MATH 3210 at University of Utah taught by R. Brooks in Fall. Since its upload, it has received 185 views. For similar materials see /class/229926/math-3210-university-of-utah in Mathematics (M) at University of Utah.

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Date Created: 10/26/15

Syllabus for Math 3210 3220 Foundations of Analysis I II Texts Joe Taylor Foundations of Analysis Available for download at http Wwwmath utah edutaylorfoundations html Anne Roberts Basic Logic Concepts Available for download at http Wwwmath utah edu7Earoberts M321 O ld pdf Math 3210 20 has two objectives to cover the theory of one and several variable calculus and to train the student in essentials of the professional mathematician logic proof and how to write a mathematical argument Prerequisites are three semesters of calculus Starting fall 2008 Math 2270 linear algebra will be a prerequisite for math 3220 and Math 2200 finite math will be strongly recommended for Math 3210 Students often take differential equations concurrently So students have had very little exposure to proofs programming or writing logically Students who will be able to take finite math M2200 beginning fall 2008 should be better able to cope with the high demands of M3210 Still the pacing of lectures should be slow and methodical at first building the student s ability to handle proofs Because some mathematics courses require Math 3210 20 as prerequisites you are expected to cover all of the material listed for each semester M3210 deals exclusively with one variable calculus M3220 deals with several variables This necessitates repetition of some material although often in a more abstract version eg utilizing notions of Euclidean topology I consider this pedagogically advantageous Proofs that are patterned after the one variable case are often not repeated in the text although these can be repeated as well Taylor s notes give the student a clear and concise account of the material They succeed and are organized similarly to the text by W Wade An Introduction to Analysis 3rd ed which is a useful reference Homework should be assigned regularly and graded conscientiously giving students plenty of feedback I suggest two or three midterms and final given in class each semester The instructor should grade all exams Students found optional extra problem sessions outside of regular class time in which they are given the chance to present solutions helpful It is not recommended that students take M3210 and M3220 concurrently Math 3210 should cover Taylor s chapters 1 6 This schedule is based on class meeting four days a week for 50 minTopics to be covered Ch 0 Introduction to logic sets functions 6 lectures From Anne Roberts notes and Taylor 11 Logic quantifiers sets Alternatively you could hand out introductory material such as Ch 11 27 of S Lay Analysis with an Introduction to Proof 3rd ed or the appendix of A Mattuck Introduction to Analysis Prentice Hall 1999 Real Numbers 10 lectures From Taylor Ch 12 13 Omit the construction of real numbers via Dedekind Cuts of Ch 14 and verification of properties of real numbers Real numbers are constructed in Math 5210 Assume that the reals exist Th 145 and discuss ordered field properties Archimidean property well ordering and induction completeness CF Wade Ch 1 Ch 15 supinf Ch 2 Real Sequences 8 lectures 0 r Taylor Ch 21 26 Limits of sequences applications of limits theorems about limits monotone sequences Cauchy Sequences Bolzano Weierstrass Theorem lim inf lim sup Ch 3 Continuous Functions in R 8 lectures Continuity composition properties of continuous functions Intermediate Value Theorem inverse functions uniform continuity sequences of functions uniform convergence uniformly Cauchy sequences Ch 4 Derivative of Real Functions 9 lectures Taylor Ch 41 40ne sided limits infinite limits difference quotient differentiation theorems chain rule inverse function mean value theorem monotone functions uniform continuity Cauchy s mean value theorem L Hopital s Rule Ch 5 Riemann Integral in R 8 lectures Taylor Ch 51 54 Upper and lower sums partitions integral existence of integrsl properties of integral Fundamental theorem of Calculus substitution integration by parts log exponential improper integrals Ch 6 Infinite series 7 lectures Taylor Ch 61 65 Infinite series geometric series series with nonnegative terms comparison tests integral test root test absolute and conditional convergence products of series power series taylor s formula lagrange s remainder Math 3220 should cover Taylor s chapters 7 10 Topics to be covered Ch 7 Convergence of sequences in Rquot 17 lectures Taylor Ch 71 5 Inner product norm distance triangle inequality in Rquot Vector space inner product space Schwarz Inequality normed vector spaces e g CI metric spaces Convergence of sequences in a metric space Limit theorems Cauchy sequences Rquot topology openclosed sets interior closure boundary Compact sets heine Borel theorem in Rquot Connected sets It is better to limit the time spent on topology which is covered in math 4510 anyway to make sure you have enough time to cover integration theory of Ch 10 thoroughly Ch 8 Functions of Euclidean Space 8 lectures Taylor Ch 81 3 Continuous fuctions of several variables Sequences and continuity Parameterized surfaces Continuous functions and open sets Uniform continuity Sequences and series of functions Uniform convergence Uniformly Cauchy Weierstrass M test Sections 84 85 matrix theory and linear algebra can be skipped or be done quickly as it is a review of Math 2280 material I found reviewing concepts frome linear algebra as they are needed to be adequate Ch 9 Differentiation in Several Variables 17 lectures Taylor Ch 91 7 Fitzpatrick 174 Partial derivatives Higher derivatives Equality of mixed partials Differential and linearization Jacobian matrix Chain rule Change of variables Directional derivatives Tangent space of parameterized surface Taylor s formula mean value theorem Necessary conditions for the maximum of a function Inverse function theorem Implicit function theorem Level set as a parameterized surface Cover also the necessary condition for the maximum of a function under equality constraint and Lagrange Multipliers I covered section 174 in Advanced Calculus 2nd ed by Patrick Fitzpatrick BrooksCole Pub 2006 Alternatively do section 117 of Wade Because Taylor s notes have discussed the constraint set as a parameterized surface it is possible to make a far more elegant presentation of this material than given in Fitzpatrick or Wade The Lagrange multiplier formula follows directly from the fundamental theorem of linear algebra Ch 10 Integration in Several Variables 14 lectures Taylor Ch 101 105 Integration over a rectangle Upper and lower sums Upper and lower integrals Properties of the integral Jordan Regions Taylor defines a Jordan region as one whose characteristic function is integrable Hence he does not need a separate discussion as does Wade Properties of volume Characterization of Jordan Regions Integration over Jordan regions Integrals of sequences Iterated integrals Fubini s Theorem Iterated integrals over non rectangular regions The change of variables formula Integral over a smooth image of a Jordan Region Sample Syllabus based on 3 lectures per week Math 3210 Foundations of Analysis I Fall Semester 2006 Instructor David C Dobson Of ce LCB 210 Of ce hours MTF 1250 140 or by appointment Prerequisites Math 2210 or consent of instructor Text Foundations ofAnalysis by Joseph L TaylorSupplemental notes Basic Logic Concepts by Anne Roberts Homework Problems will be assigned weekly and will generally be due in class on Wednesday of each week Late homework cannot be accepted A random subset of problems from each assignment will be graded To succeed in the course it is necessary to do and understand all of the homework Exercise Set 11 Exercise Set 21 on Set 22 1 11 Exercise Set 24 3 12 Exercise Set 25 5 7 8 Exercise Set 26 6 Exercise Set 32 on Set 42 7 11 Exercise Set 44 6 8 ll Exercise Set 53 9 Exercise Set 61 2 5 9 12 Exercise Set 62 11 Exercise Set 64 1 4 6 Exams Two midterm exams and a comprehensive final exam will be given in class Exam I is closed book closed notes For Exam II a single 8 12 x 11 sheet of notes is allowed For the Final Exam two 8 12 x 11 sheets of notes are allowed No electronic devices are allowed Grades Your grade will be determined by your scores on the three exams and the total of your homework scores The dates and weights of these are as follows Homework weekly 40 Exam I Monday October 2 15 Exam 11 Monday November 6 20 Final Exam Tuesday December 12 1030 am 1230 pm 25 There will be no opportunities for extra credit Makeups for exams and homework will only be given in case of University excused absences Tutoring is available from the Math Tutoring Center Students with disabilities may contact the instructor at the beginning of the semester to discuss special accomodations for the course Copyright notice All printed and electronic material provided to you in this course are protected by copyright laws Sample Syllabus based on four lectures per week Math 3220 1 Foundations of Analysis 11 August 15 2007 MTWF 940 7 1030 in JTB 110 Instructor A Treibergs JWB 224 58178350 Office Hours 10451 1 45 MWF tent amp by appt Eimail treibergmathutahedu Homepage http wwwmath utah edutreibergM322 3 html Text Joseph L Taylor Foundations ofAnalysLs 2007 PDF Notes available for download from http WWWmath utah edutaylorfoundations html Grading Homework To be assigned weekly Midterrns There will be three iniclass oneihour midterm exams on Wednesdays Sept 5 Oct 3 and Nov 7 Final Exam Tue Dec 11 8 001000 AM Half of the final will be devoted to material covered after the third midterm exam The other half will be comprehensive Students must take the final to pass the course Course grade Best two of three midterrns 36 homework 37 final 27 Withdrawals Last day to drop a class is Aug 29 Last day to add a class is Sept 4 Until Oct 19 you can withdraw from the class with no approval at all After that date you must petition your dean39s office to be allowed to withdraw ADA The Americans with Disability Act requires that reasonable accommodations be provided for students with cognitive systemic learning and psychiatric disabilities Please contact me at the beginning of the quarter to discuss any such accommodations you may require for this course Objectives To refine our skill at proof and facility with computation to gain an appreciation for abstraction from the concepts of topology and metiic spaces and to learn the theory behind multidimensional calculus Topics We shall try to cover the following chapters Chapter 7 Convergence in Euclidean Space Chapter 8 Functions on Euclidean Space Chapter 9 Differentiation in Several Variables Chapter 10 Integration in Several Variables Math 3220 1 Homework Problems Fall 2007 Treibergs You are responsible for knowing how to solve the following exercises from the text quotFoundations of Analysisquot by Joseph L Taylor Please hand in the starred quotquot problems from the text and the additional exercises FIRST HOMEWORK ASSIGNMENT Due Monday August 27 2007 711 2 3 4 5 6 7 9 12 SECOND HOMEWORK ASSIGNMENT Due Tuesday September 4 2007 72 1 2 3 4 5 8 11 A Let X11 be ameuic space Suppose x y are points in X and 16 and yn are sequencesin X suchthat x gt16 and yn7gty as n7gt 00 Show that dxny 7gt dxy as n 7gt 00 B Let 147 be asequence in Ed andlet u be apoint in Ed Suppose that for all vectors v in Ed we have vounigtv u as n7gt 00 Showthat un7gtu as 11gt 00 THIRD HOMEWORK ASSIGNMENT Due Monday September 10 2007 72121316 7312 A Let a b and c be real numbers suchthat a2 b2 I Let L xy inR239wc by c be the points on a line in the plane Show that L is a closed set B Let S Una 16 be the set in R2 consisting of points from the sequence x In In Determine whether S is open closed or neither Prove your answer FOURTH HOMEWORK ASSIGNMENT Due Monday September 17 2007 73 4 6 8 9 10 12 74 1 2 3 4 5 A Let E be a subset of R How many different sets can be obtained from E by taking closure or complementation Prove your answer If we denote closure by E and complement by B then the question is at most how many different sets can occur in the sequence E E E E E E E 7 B Show that every open set G in Rd is the union of at most countany many open balls G 11 r1 Hint Consider balls whose centers have rational coordinates and whose radii are rational FIFTH HOMEWORK ASSIGNMENT Due Friday September 28 2007 75 1 2 3 4 6 11 12 81 1 3 5 A Prove that the quottopologist39s sine curvequot E in the plane is connected E0y3971 ltylt1 Ux sinIx 390ltxlt1 B Suppose I and J are open intervalsin the line and a is in I and b is in J Suppose that f39IxJ7 ab 7gt R is a function such that for all x in I a the limit exists gx limyrgt Z7fcy and that for all y in J A b the limit exists My limxrm xy i Show that even though the quotiterated limitsll may exist L limx gt5 gx M limyrgtb My it may be the case that L does not equal M Show in thaqt case the twoidimensional limit limim 7gt yw xy fails to exist ii Suppose in addition to the existence of the iterated limits one knows that the two dimensional limit exists fxy 7gt N as xy 7gt 11 Show that then L M N SIXTH HOMEWORK ASSIGNMENT Due Friday October 5 2007 81 8 10 11 82 1 2 4 5 6 7 8 10 11 83 1 2 3 5 7 8 11 SEVENTH HOMEWORK ASSIGNMENT Due Friday October 19 2007 Read the review sections 83 and 84 about linear algebra Do any problem whose solution isn39t immediately clear 8415 8510 91135678 92134589 EIGHTH HOMEWORK ASSIGNMENT Due Friday October 26 2007 931359 94148111213 Problem 7 is not due until next week A See 938 Suppose that xyz are the Cartesian coordinates of a point in R3 and the spherical coordinates of the same point is given by xrcos9 sing yrsin9 sinq zrcos Let u fxyz be a twice continuously differentiable function on R3 Find a formula for the partial derivatives of u with respect to xyz in terms of partial derivatives with respect to r 94 Find a formula for the Laplacian of u in terms of partial derivatives with respect to r 9 q where the Laplacian is given by Au02uax202uay2a2uaz2 NINTH HOMEWORK ASSIGNMENT Due Friday November 2 2007 947 This problem is postponed from last week 952 3 4 6 9 12 The problems from section 96 are postponed until next week A Find the critical point 0130 in the set st in R2 39 s gt 0 for the function with any real A and B fst logs tiA2 32 s Find the second order Taylor39s expansion for f about the point 0130 Prove that f has a local minimum at swig B Letp gt I Find all extrema of the function fx 16 x22 x2 subjectto the constraint lxlf lxnP I If 51 52 show for any x and n that np39Z2p lxllp lxnlp1quot x12 x22 192 Z lxllp lxnlp1 p TENTH HOMEWORK ASSIGNMENT Due Friday November 9 2007 961 2 3 7 8 9 A Let F 39R2 7gtR2 be givenby x 112 7 v2 y 2 v Findan open set U in R2 suchthat 34 in U and V FU is anopen set and find a C1 function G 39 V 7gt U such that G 0 Fuv uv for all uv in U and F0 G xy xy for all xy in V Find the differential dGF34 G is a local inverse Solve for G and check its properties Do not use the Inverse Function Theorem which guarantees the existence of local inverse near 34 assuming F is continuously differentiable near 34 and dF34 is invertible ELEVENTH HOMEWORK ASSIGNMENT Due Friday November 16 2007 96101112 97 1 3 5 7 8 A In section 96 the Implicit Function Theorem was deduced from the Inverse Function Theorem Show that the Inverse Function Theorem can be deduced from the Implicit Function Theorem TWELVTH HOMEWORK ASSIGNMENT Due Monday November 26 2007 101 2 3 4 5 6 8 9 10 102 1 2 3 4 5 6 7 8 12 A Let g 39 R 7gt R2 be a continuously differentiable curve Show that the image of a compact interval gab has Jordan content zero Called quot volume zeroll in the notes Note that this would be false under the hypothesis of continuity only This is because there exist quot space filling curves quot See e g http wwwmath Ohio state edufiedoroWmath655 Peano html THIRTEENTH HOMEWORK ASSIGNMENT Due Monday December 3 2007 103 1 2 3 4 5 6 7 9 10 11 1042910 FOURTEENTH ASSIGNMENT Due Friday December 7 2007 104 5 6 7 8 12 105278111213

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