620-295 |

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

Time and Location:

Lecture Monday 12:00pm - 1:00pm Old Geology-Theatre 1

Lecture Tuesday 10:00am - 11:00am Old Geology-Theatre 1

Lecture Thursday 10:00am - 11:00am Old Geology-Theatre 1

Practical Monday 1:00pm - 2:00pm Richard Berry-G03

Practical Monday 2:15pm - 3:15pm Richard Berry-G03

Practical Tuesday 9:00am - 10:00am Richard Berry-G03

Practical Tuesday 11:00am - 12:00pm Richard Berry-G03

Practical Wednesday 9:00am - 10:00am Richard Berry-G03

Practical Wednesday 10:00am - 11:00am Richard Berry-G03

Laboratory Wednesday 1:00pm - 2:00pm Richard Berry-G70 [Wilson Laboratory]

Laboratory Thursday 9:00am - 10:00am Richard Berry-G70 [Wilson Laboratory]

Laboratory Thursday 12:00pm - 1:00pm Richard Berry-G70 [Wilson Laboratory]

- SWOT-VAC consultation hours for Prof. Ram will be 4 Nov. from 10-1 in Old Geology 2.

The handbook entry for this course is at https://app.portal.unimelb.edu.au/CSCApplication/view/2009/620-295. 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.

- 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

- 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:

Various lectures 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
- Derivations and Taylor's theorem
- Sequences and series
- Sets and functions
- 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
- 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

- Lecture 1, 27 July 2009: Numbers, sets and functions(pdf file)
- Lecture 2, 28 July 2009: Operations, monoids, groups, rings, fields (pdf file)
- Lecture 3, 30 July 2009: The Pythagorean theorem, rationals, and √2 (pdf file)
- Lecture 4, 3 August 2009: Real and complex numbers (pdf file)
- Lecture 5, 4 August 2009: Functions, injectivity, surjectivity, composition, inverse functions (pdf file)
- Lecture 6, 6 August 2009: Polynomials, derivations and Taylor's theorem (pdf file)

- Lecture 7, 10 August 2009: Proofs by induction (pdf file)
- Lecture 8, 11 August 2009: Binomial theorem, exponential functions, trig functions, inverse functions (pdf file)
- Lecture 9, 13 August 2009: Orders, Ordered groups and ordered fields (pdf file)
- Lecture 10, 17 August 2009: Order properties of the real numbers (pdf file)
- Lecture 11, 18 August 2009: Cardinality (pdf file)
- Lecture 12, 20 August 2009: Triangle inequality (pdf file)

- Lecture 13, 24 August 2009: Sequences, sup, lim sup, convergence and divergence (pdf file)
- Lecture 14, 25 August 2009: Series, geometric series, harmonic series, Riemann zeta function (pdf file)
- Lecture 15, 27 August 2009: The interest sequence, Picard iteration, Newton iteration (pdf file)
- Lecture 16, 31 August 2009: Root, ratio and integral tests (pdf file)
- Lecture 17, 1 September 2009: Radius of convergence, conditional convergence (pdf file)
- Lecture 18, 3 September 2009: Metric spaces, Cauchy sequences and completeness (pdf file)

- Lecture 19, 7 September 2009: Metric spaces, continuity, uniform continuity (pdf file)
- Lecture 20, 8 September 2009: Topological spaces and continuity (pdf file)
- Lecture 21, 10 September 2009: Derivatives and the Intermediate Value Theorem (pdf file)
- Lecture 22, 14 September 2009: Connected and Compact Sets (pdf file)
- Lecture 23, 15 September 2009: Mean value theorems (pdf file)
- Lecture 24, 17 September 2009: Mean value theorems and discussion of L'Hopital's rule (pdf file)

- Lecture 25, 5 October 2009: Integrals and Fundamental Theorem of Calculus (pdf file)
- Lecture 26, 6 October 2009: Fundamental theorem of calculus and Improper integrals (pdf file)
- Lecture 27, 8 October 2009: Trapezoidal and Simpson integrals and Pointwise and uniform convergence (pdf file)
- Lecture 28, 12 October 2009: Error estimates (pdf file)
- Lecture 29, 13 October 2009: Taylor series, Fourier series, and Stone-Weierstrass (pdf file)
- Lecture 30, 15 October 2009: Limits, revision and examples (pdf file)

- Lecture 31, 19 October 2009: Derivatives, revision and examples (pdf file)
- Lecture 32, 20 October 2009: Taylor series and Mean Value Theorem, revision (pdf file)
- Lecture 33, 22 October 2009: Topology, revision (pdf file)
- Lecture 34, 26 October 2009: Numbers, revision (pdf file)
- Lecture 35, 27 October 2009: Sequences and Series, revision (pdf file)
- Lecture 36, 29 October 2009: Assorted examples (pdf file)

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%).

Problem Sheets:

- 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

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

- Assignment 1: Due 7 August pdf file The first assignment is due at 5pm on 7 August in the appropriate assignment box on the ground floor of Richard Berry. (SOLUTIONS: pdf file)
- Assignment 2: Due 21 August pdf file The second assignment is due at 5pm on 21 August in the appropriate assignment box on the ground floor of Richard Berry. (SOLUTIONS: pdf file)
- Lab 1: to be completed during the week of 17-21 August (pdf file)
- Assignment 3: Due 4 September (pdf file) (SOLUTIONS:pdf file)
- Lab 2: to be completed during the week of 31 Aug-4 Sep (pdf file)
- Assignment 4: Due 18 September (pdf file) (SOLUTIONS: pdf file)
- Lab 3: to be completed during the week of 12-16 October (pdf file)
- Assignment 5: Due 16 October (pdf file) (SOLUTIONS:pdf file)
- Lab 4: to be completed during the week of 19-23 October (pdf file)
- Assignment 6: Due 30 October (pdf file) (SOLUTIONS:pdf file)