Contact | Description | Topics | Texts | Notes | Lectures | Assessment and Assignments |

mulogo

 

620-295
Real Analysis with applications

Semester I 2010

Lecturer: Arun Ram, 174 Richard Berry, phone: 8344 6953, email: aram@unimelb.edu.au

Time and Location:
       Lecture Monday 10:00 - 11:00 Lowe Theatre - Redmond Barry
       Lecture Wednesday 16:15 - 17:15 Latham Theatre - Redmond Barry
       Lecture Friday 12:00 - 13:00 Cuming Theatre - Chemistry
       Practical Monday 11:00am - 12:00pm David Caro Podium 203
       Practical Monday 1:00pm - 2:00pm David Caro Podium 205
       Practical Monday 2:15pm - 3:15pm David Caro Podium 205
       Practical Tuesday 9:00am - 10:00am David Caro Podium 205
       Practical Tuesday 10:00am - 11:00pm David Caro Podium 203
       Practical Wednesday 10:00am - 11:00am David Caro Podium 205
       Practical Wednesday 11:00am - 12:00am David Caro Podium 205
       Practical Thursday 1:00pm - 2:00pm David Caro Podium 205
       Laboratory Tuesday 2:15pm - 3:15pm Richard Berry-212 [Nanson Laboratory]
       Laboratory Wednesday 9:00am - 10:00am Richard Berry-212 [Nanson Laboratory]
       Laboratory Thursday 9:00am - 10:00am Richard Berry-212 [Nanson Laboratory]
       Laboratory Friday 10:00am - 11:00am Richard Berry-G70 [Wilson Laboratory]

Announcements

  • No books, notes, calculators, ipods, ipads, phones, etc at the exam.
  • The pre-exam consultation for Prof. Ram on 14 June (Queen's birthday) WILL be held 14 June 2-4:30 in Old Geology 2 and security has promised to open the building for us. (It was not possible to arrange an alternate time before the exam).
  • Extra consultation hours for Prof. Ram will be
    • Friday 4 June, 2-4:30 in Old Geology 2
    • Monday 14 June, 2-4:30 in Old Geology 2
    SSLC raw data (pdf file, tutor codes-pdf file), SSLC raw data comments (pdf file), SSLC representative analysis (pdf file)
  • Tips to avoid freaking out:
    • The assignments are designed to take "an average of 12 hours per assignment". This is an average.
    • The assignments can be reformatted to reduce the freak factor: See Assignment 1 (pdf file)
    • The assignments and the course are designed to make you know exactly what is on the exam, practice what is on the exam and do well on the exam.
    • The assignments are worth 20% of the total mark. If you skip a few questions it will affect your total mark very little.
    • Thousands of students have made it through this course format with Professor Ram in the past (and are proud to tell the tale). You can do it too.
  • Tips for time management:
    • It is much easier (and safer) to run 45 min per day to attain 12 hours in 2 weeks, than to run for 12 hours solid every other week on Sunday.
    • To actually run 45 min, it takes me at least 15 min to psyche myself up and convince myself that it is actually not raining and so therefore I should go running, and after a 45 min run I always walk for 5 min and I always go home and have a glass of milk and tell my wife (at length) how cool I am for running 45 min per day. All in all, I waste a good 40 min when I go running for 45 min. If I were more efficient (and every so often, but rarely, I am) then it would only takes me 50 min.
    • Measurement of time is a tricky thing and requires real discipline. Teaching and research faculty at University of Melbourne recently had to complete a survey on distribution of their time on the various activities of the job: Do I count the 6 times I had to go check my email and the weather and my iPhone in the time that I spend preparing my Real analysis lecture?
  • Tips for exam preparation:
    • The time that a 100m olympic runner (who wins a medal) is actually competing at the olympics is say (5 heats, 7sec each) 40 seconds. Successful performance during these 40 sec is impossible without adequate preparation.
    • The time that a Real analysis student spend on the final exam is 3 hours. Sucessful performance during these 3hours is .....
  • As announced in lectures on 24 March, the second assignment is now due on Thursday 1st April at 12pm.
  • There has been a room change for Friday lectures. For the course of semester 1 2010, Monday lectures will be held in Chemistry - Cuming Theatre, 12-1pm Fridays.
  • Due the numbers in 620295 still increasing an extra tutorial is now available Monday at 12.
  • There has been a room change for Monday lectures. For the course of semester 1 2010, Monday lectures will be held in Redmond Barry - Lowe Theatre, 10-11am Mondays.
  • Consultation hours for Prof. Ram will be Fridays from 2:00-4:30 in the following locations:
    • Old Geology 1: 5, 12, 19, 26 March and 2, 9, 16, 23, 30 April and 7, 14, 21, 28 May,
    • Richard Berry 213: 4, 11, 18 June.
  • Prof. Ram reads email but generally does not respond.
  • The start of semester pack includes: Plagiarism (pdf file), Plagiarism declaration (pdf file), Academic Misconduct (pdf file), SSLC (pdf file), SSLC responsibilities (pdf file), SSLC timelines (pdf file), Beyond third year (pdf file), Vacation scholarships (pdf file), Local third ann (pdf file).
  • It is University Policy that:           

    “a further component of assessment, oral, written or practical, may be administered by the examiners in any subject at short notice and before the publication of results.  Students must therefore ensure that they are able to be in Melbourne at short notice, at any time before the publication of results” (Source: Student Diary).

    Students who make arrangements that make them unavailable for examination or further assessment, as outlined above, are therefore not entitled to an alternative opportunity to present for the assessment concerned (i.e. a ‘make-up’ examination).
  • Printing arrangements from computer Lab G70: Students must use their Unicards to print documents. Locations for Unicard uploaders can be found at: http://www.studentit.unimelb.edu.au/images/facilities/autoloaders.gif

Subject Outline

The handbook entry for this course is at https://app.portal.unimelb.edu.au/CSCApplication/view/2010/620-295. The subject overview that one finds there:

This subject introduces the field of mathematical analysis both with a careful theoretical framework and its application in numerical approximation. A review of number systems; the fundamentals of topology of the real line; continuity and differentiability of functions of one and several variables; sequences and series including the concepts of convergence and divergence, absolute and conditional, and tests for convergence; Taylor’s theorem and series representation of elementary functions with application to Fourier series. The subject will introduce methods of proof such as induction and also introduce the use of rigorous numerical approximations.
Topics include the definition of limits, lim sup, lim inf; Rolle's Theorem, Mean Value Theorem, Intermediate Value Theorem, monotonicity, boundedness, and the definition of the Riemann integral.

Main Topics

  • Numbers: Integers, rationals, reals, complexes
  • The binomial theorem and the exponential function
  • Sequences and Series
  • sets and functions: relations and cardinality
  • operations, fields, orders and ordered fields
  • topological spaces, continuity, and the limit definition of continuity
  • differentiation and integration

Assessment

Assessment will be based on six assignments to be handed in during semester (worth 20%) and a final 3-hour exam at the end of semester (worth 80%).

The plagiarism declaration is available here. The homework assignments will soon appear below:

  • Assignment 1: Due 15 March: Do problems with problem number divisible by 5 on Problem Sheet - Expressions (pdf file) and Problem Sheet - Graphing (pdf file). The first assignment is due at 10am on 15 March in the appropriate assignment box on the ground floor of Richard Berry.
  • Assignment 2: Due 29 March: Do problems with problem number divisible by 5 on Problem Sheet - Limits (pdf file) and Problem Sheet - Sequences (pdf file). The second assignment is due at 10am on 29 March in the appropriate assignment box on the ground floor of Richard Berry.
  • Assignment 3: Due 19 April: Do problems with problem number divisible by 5 on Problem Sheet -Series (pdf file) and Problem Sheet - Improper integrals (pdf file). The third assignment is due at 10am on 19 April in the appropriate assignment box on the ground floor of Richard Berry.
  • Assignment 4: Due 3 May: Do problems with problem number divisible by 5 on Problem Sheet -Derivatives and Taylor approximations (pdf file) and problems with problem number divisible by 5 from sections 1 and 2 of Problem Sheet - Approximating integrals (pdf file).
  • Assignment 5: Due 17 May: Do problems with problem number divisible by 5 on Problem Sheet - Numbers and Ordered fields (pdf file) and problems with problem number divisible by 5 from ALL of Problem Sheet - Sets, ordered and functions (pdf file).
  • Assignment 6: Due 31 May. Do problems with problem number divisible by 5 on Problem Sheet - Topology and definitions (pdf file) and problems with problem number divisible by 5 from Problem Sheet - Theorems and revision (pdf file).

The laboratories are as follows:

  • Lab 1: to be completed during the week of 8-12 March (pdf file)
  • Lab 2: to be completed during the week of 15-19 March (pdf file)
  • Lab 3: to be completed during the week of 22-26 March (pdf file)
  • Lab 4: to be completed during the week of 12-16 April (pdf file)

Resources part I: recommended Texts

  • Jerry Koliha, 620-695 Real Analysis with Applications Workbook for Semester I 2009. Download from http://www.ms.unimelb.edu/~ram/Teaching/Analysis2009/KolihaWORKBOOK_295.pdf
  • William Chen, Fundamentals of Analysis, 100 pp. (web edition, 2008). Download from http://rutherglen.ics.mq.edu.au/wchen/lnfafolder/lnfa.html
  • William Trench, Introduction to Real Analysis, (web release 2009). Download from http://ramanujan.math.trinity.edu/wtrench/misc/index.shtml.
  • W. Rudin, Principles of Mathematical Analysis, 3rd edition ISBN: 978-0070542358McGraw Hill
  • Wikipedia

The following problems page may have helpful examples:

  • http://www.exampleproblems.com/wiki/index.php/Main_Page

Resources part II: Lectures and lecture notes

  • Lecture 1, 1 March 2010: Math Grammar: Definitions, Theorems and How to do Proofs (pdf file) and handwritten lecture notes-pdf file, html file

    Examples of proofs written in proof machine (pdf file)

  • Lecture 2, 3 March 2010: Polynomials and rational functions (pdf file), exponential, trigonometric, and hyperbolic expressions (pdf file) and handwritten lecture notes - pdf file, html file
  • Lecture 3, 5 March 2010: Log and inverse trig and hyperbolic functions and handwritten lecture notes - pdf file, html file

  • Lecture 4, 8 March 2010: Induction, bounds, sup, inf and absolute value in R - hand written lecture notes (pdf file), html file
  • Lecture 5, 10 March 2010: Graphing (pdf file), html file
  • Lecture 6, 12 March 2010: Graphing, continuity, and existence of limits (pdf file), html file
  • Lecture 7, 15 March 2010: The defnition of limits and continuity at a point (pdf file), html file
  • Lecture 8, 17 March 2010: Limits, limit theorems, and examples (pdf file), html file
  • Lecture 9, 19 March 2010: Examples of limits (pdf file), html file

  • Lecture 10, 22 March 2010: Sequences, bounds, limsup, liminf, Cauchy and contractive definitions (pdf file) (also see the notes at Sequences-Definitions (pdf file)), html file
  • Lecture 11, 24 March 2010: Sequences in applications: the interest sequence, Picard iteration, Newton iteration (pdf file). Some sequences examples (pdf file), html file
  • Lecture 12, 26 March 2010: Series -- fundamental examples: geometric series, harmonic series/Riemann zeta function and integral tests (pdf file), html file

  • Lecture 13, 29 March 2010: Ratio test, conditional convergence and alternating series (pdf file), html file
  • Lecture 14, 31 March 2010: Radius of convergence (pdf file), html file
  • 2 April 2010: Good Friday: NO lecture

  • Lecture 15, 12 April 2010: Improper integrals (pdf file) html file
    • Additional stuff A (pdf file) Additional stuff B (pdf file),
  • Lecture 16, 14 April 2010: Series and Improper integral examples (pdf file) additional radius of convergence examples (pdf file)
    • html file
  • Lecture 17, 16 April 2010: Proofs of sequence theorems - Cauchy sequences (pdf file), html file

  • Lecture 18, 19 April 2010: Derivatives, Taylor's theorem, and the mean value theorem (pdf file), html file
  • Lecture 19, 21 April 2010: Integrals and approximations (pdf file), html file
  • Lecture 20, 23 April 2010: Fundamental theorem of calculus (pdf file), html file

  • Lecture 20, 26 April 2010: Anzac Day: NO lecture
  • Lecture 21, 28 April 2010: More Integrals examples
  • Lecture 22, 30 April 2010: Fourier series and ζ(2) (pdf file) html file

  • Lecture 23, 3 May 2010: Integers, rationals, reals and complexes (pdf file), html file
  • Lecture 24, 5 May 2010: Ordered fields (pdf file), html file
  • Lecture 25, 7 May 2010: Metric spaces, Absolute value on Rn and the triangle inequality, (pdf file), html file

  • Lecture 26, 10 May 2010: Sets, functions and cardinality (pdf file), html file
  • Lecture 27, 12 May 2010: Relations, partial orders and total orders (pdf file), html file
  • Lecture 28, 14 May 2010: Open sets in R and Rn (pdf file), html file

  • Lecture 29, 17 May 2010: Topology: open, closed, connected and compact sets (pdf file), html file
  • Lecture 30, 29 May 2010: Topology: limit continuity and topological continuity (pdf file), html file
  • Lecture 31, 21 May 2010: Topology: compact sets and the max-min existence theorem (pdf file), html file
  • Lecture 32, 24 May 2010: Revision: Main theorems and ideas (pdf file), html file
  • Lecture 33, 26 May 2010: Revision: definitions and proofs -- MVT and Taylor's theorem
  • Lecture 34, 28 May 2010: Revision: definitions and proofs

Resources part III: past problem sheets

Homework assignments from 2009 Semester 2

  • Assignment 1: Due 7 August 2009 pdf file (SOLUTIONS: pdf file)
  • Assignment 2: Due 21 August 2009 pdf file (SOLUTIONS: pdf file)
  • Assignment 3: Due 4 September 2009 (pdf file) (SOLUTIONS:pdf file)
  • Assignment 4: Due 18 September 2009 (pdf file) (SOLUTIONS: pdf file)
  • Assignment 5: Due 16 October 2009 (pdf file) (SOLUTIONS:pdf file)
  • Assignment 6: Due 30 October 2009 (pdf file) (SOLUTIONS:pdf file)

Problem Sheets from 2009 Semester 2

  • Sheet 1 (pdf file): Numbers, sets, functions and operations
  • Sheet 2 (pdf file): Structures and orders
  • Sheet 3: (pdf file): Sequences and series
  • Sheet 4: (pdf file): Topology and continuity
  • Sheet 5: (pdf file) Continuity and integrals
  • Sheet 6: (pdf file) Approximations and Fourier series
  • Exam preparation problems more problems added at end of list on 28.10.2009 (pdf file)
  • More exam preparation problems (pdf file) (sets and functions problems in this sheet!)
  • Homework assignments from Calc 1 in 2004: pdf files
    • Assignment 1, Assignment 2, Assignment 3, Assignment 4, Assignment 5, Assignment 6, Assignment 7 (mean value theorem problems in here!), Assignment 8, Assignment 9, Assignment 10, Assignment 11, Assignment 12, Assignment 13, Assignment 14.
  • Look at the notes here

Resources part IV: Other notes

Various lecture notes from the past that will be useful and supplemented during the term.

  • Mathematical grammar and How to do proofs (pdf file)
  • Numbers: integers, rationals, reals, complexes,
    • Numbers-- 2003 notes pdf file
    • Angles -- 2003 notes pdf file
    • Numbers -- 2004 notes pdf file
    • Numbers -- 2009 xml version
  • The binomial theorem
  • Polynomials and formal power series
    • Polynomials -- 2004 notes pdf file
  • The exponential function
    • The exponential function -- 2003 notes pdf file
    • The exponential function -- 2004 notes pdf file
    • Binomial series
    • Basic trig identities -- 2003 notes pdf file
    • Inverse functions and their derivatives -- 2003 notes pdf file
    • The genesis lecture -- 2003 notes pdf file
  • Derivations and Taylor's theorem
    • Derivatives -- 2003 notes pdf file
    • Chain rule -- 2003 notes pdf file
    • Derivatives of trig functions -- 2003 notes pdf file
    • Derivations -- 2004 notes pdf file
  • Sequences and series
    • Sequences and series -- 2004 notes pdf file
    • Sequences and series -- 2009 notes xml file
    • The interest sequence and the sequence xn
    • Geometric series, harmonic series and the Riemann zeta function
    • Root test, ratio test, integral test and radius of convergence
  • Sets and functions
    • Sets and functions and cardinality -- 1994 notes pdf file
    • Sets and functions examples and proofs -- 1994 notes pdf file
    • Sets and functions -- 2004 notes pdf file
  • Cardinality (2004 notes) pdf file
  • Relations and Equivalence relations (2004 notes) pdf file
  • Partially ordered sets (2004 notes) pdf file
  • Operations, Groups, rings and fields (2004 notes) pdf file
  • Fields of fractions (2004 notes) pdf file
  • Ordered fields (2004 notes) pdf file
  • Limits and Calculus
    • Limits -- 2006 calculus notes pdf file
    • sequences and series
    • Continuity and derivatives and integrals
    • The limit definition of continuous functions -- 2004 notes pdf file
  • Topological spaces -- 2004 notes pdf file
    • Compact sets -- 2004 notes pdf file
    • Metric spaces
    • Complete metric spaces
    • Completions
  • convergence of functions and Stone Weierstrass theorem (2004 notes) pdf file

Feedback

Every subject at the University of Melbourne uses a student questionnnaire to let teaching staff know what students think about the quality of teaching in that subject. This is usually administered in the last week or two of semester. As such, it is too late to affect the teaching for the cohort of students that answer the questionnaire.

Feedback to students based on 2009 questionnaires:

  • The student survey last year showed high student satisfaction with the course. Most elements of last year's course are being retained.
  • The topic of Fourier series is being treated much earlier in the semester to coordinate better with 2nd year physics courses.
  • Exam performance demonstrated that students had learned concepts and the general framework well, but were weak on skill (they knew what a hammer is for but were unable to use it to hammer in a nail effectively). Skill level is an important goal for this course and this semester there will be a determined effort to get the skill level of all students to a high level:
    • The tutorials will be more structured, with vocabulary drill, and practice marking.
    • The problem sheets will be more organised and directed towards the final exam.