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G12RAN: REAL ANALYSIS
2002/2003
URL:
http://www.maths.nott.ac.uk/personal/jff/G12RAN/index.html
Last modified: December 11 2002
Many documents relevant to this module will be
available on the web
in two formats: ps (postscript) or pdf (portable document format).
Which format is easiest to view will probably depend on the computer that
you are using.
Module information for 2002/2003

Credits: 10

Duration: 22 lectures, two lectures a week plus problem
classes in Autumn Semester, starting Friday 27/9/2002

Lecturer: Dr J. F. Feinstein, room C301, email
Joel.Feinstein@nottingham.ac.uk

Lecture times: Monday 3 in M&P Building
room B1 and Friday 10 in Pope Building
room C14.

Problem classes: problem classes will take place fortnightly, in the
weeks commencing 7/10, 21/10, 4/11, 18/11 and 2/12.
The problem classes will be in room C5. You should attend as follows:
Surnames  Problem Class

AG  Thursday 11.00 AM

HN  Wednesday 11.00 AM

OZ  Wednesday 12.00 noon

For these problem classes, please sit only in rows 1, 2, 4, 5, 7, 8 or 10
counting from the front, so that the people taking the problem class
can get to you.
A register will be taken
at each problem class.
If you have a genuine reason for not being able
to attend at the time you are assigned, you may attend at
one of the other two times:
please let me know in writing your name, the time that you wish to attend, and
the reason that you can not attend at the given time.
The attendance registers can then be updated appropriately.

Office hours (Autumn Semester):
See the web page
http://www.maths.nott.ac.uk/personal/jff/ttnow.html
for details of Dr Feinstein's timetable and office hours.

Brief content description:
This module provides a basic introduction to mathematical analysis.
The main concepts of the subject will be introduced (including limits,
continuous functions, differentiable functions and Riemann integrals).
While studying these concepts, the main notions and methods of proof
in analysis will be presented. The material in this module is of
crucial importance for all later modules in analysis. Rigorous
proof is vital throughout Pure Mathematics.

Prerequisites:
G1ALIM and G1AMSK or equivalent, including in particular an introduction to
complex numbers and an understanding of the limit of a sequence.

Module aims:
To provide a basic introduction to mathematical analysis building on
the experience gained in G1ALIM and G1AMSK. To introduce the main
notions and methods of analysis, to introduce a
mathematically rigorous approach and to lay the foundation for the
subsequent study of complex analysis and functional analysis.

Module objectives:
That the student should:
 Be able to apply the definitions and theorems presented in the module to the
solution of simple unseen problems;

Be able to state and prove the principal theorems proved in the module;

Understand the distinction between countable and uncountable sets and be able
to identify sets with each property;

Understand the concept of a limit of a function, and be able to prove from
first principles that suggested limits do or do not exist;

Understand the definitions of boundedness, continuity, uniform continuity,
differentiability and Riemann integrability, and be able to determine (with
proof) whether suggested functions possess these properties;

Be able to provide examples of functions satisfying conditions phrased in terms
of the five properties just mentioned;

Be able to determine limits using l'Hôpital's rule;

Be able to estimate functions using Taylor's theorem.

Brief Syllabus:

Properties of the real numbers

Functions and sets

Limit values for functions

Sequences and continuous functions

Differentiability

L'Hôpital's rule and Taylor's theorem

Integration

Books: The following books, available in the George Green
science
library,
are all worth looking at.
The book by Haggarty is very good but does not cover countability of sets.
The book by Kopp provides a good overview, but
does not include all of the details.
In addition,
Professor Langley's lecture notes are excellent
on detail and are
available in the short loan section of the
George Green Science Library, and also online,
here: ps, pdf.

K. Binmore,
Mathematical Analysis, a Straightforward Approach

R. Haggarty,
Fundamentals of Mathematical Analysis

P. Kopp,
Analysis

J. Reade,
An Introduction to Mathematical Analysis.

M. Spivak,
Calculus
More advanced:
Handouts and additional documents
The following documents are currently available from the
module web page.

Module information sheet : this page
 20012 blowbyblow for the module:
ps, pdf

20023 blowbyblow for the module (now complete):
ps, pdf
 Comments on student performances and common errors on the 19992000 exam:
ps, pdf
 NOTES/CHAPTER SUMMARIES
 Chapter 1: Properties of the real numbers.
 Part I, Review of notation, definitions and results from earlier modules:
ps, pdf
 Part II, Further properties:
ps, pdf
 Chapter 2, Functions and sets:
ps, pdf
 Chapter 3, Limit values for functions:
ps, pdf
 Density of the rationals and the irrationals in terms of sequences:
ps, pdf
 Chapter 4, Sequences and continuous functions:
ps, pdf
 Chapter 5, Differentiability:
ps, pdf
 Chapter 6, L'Hôpital's rule and Taylor's theorem:
(extract from Professor Langley's lecture notes, very slightly modified by
Dr Feinstein)
ps, pdf
 Chapter 7, Integration:
ps, pdf
 Uniform continuity:
ps, pdf
You should also note that a complete set of notes for a similar module given by Professor Langley in 1993 is available in the usual formats:
ps, pdf.
Course work
Provisionally the dates for handing in coursework will be:
18/10/02, 1/11/02, 15/11/02, 29/11/02 and 11/12/02.
The following question sheets/solutions are currently available from the module web page.
(Solutions to nonprize questions will be available from the web page at the appropriate time.)
 Exercises for the enthusiast (with prizes!):
ps, pdf
Please note that I did
not personally invent these questions.
(They are wellknown, and mostly
rather hard.)
 Question sheet 1:
ps, pdf
 Sheet 1, solutions to questions 15:
ps, pdf
 Sheet 1, solutions to questions 610:
ps, pdf
 Question sheet 2:
ps, pdf
 Sheet 2, solutions to questions 15:
ps, pdf
 Sheet 2, solutions to questions 612:
ps, pdf
 Question sheet 3:
ps, pdf
 Sheet 3, solutions to questions 15:
ps, pdf
 Sheet 3, solutions to questions 610:
ps, pdf
 Question sheet 4:
ps, pdf
 Sheet 4, solutions to questions 15:
ps, pdf
 Sheet 4, solutions to questions 610:
ps, pdf
 Question sheet 5:
ps, pdf
 Sheet 5, solutions to questions 14:
ps, pdf
 Sheet 5, solutions to questions 512:
ps, pdf
Assessment
Assessment will be by means of a two hour written examination in January.
There will be five questions, and your best four answers will count. If you
answer four of the questions perfectly, then you will obtain full marks.
This is the same format as the exams from the 20002001 and 20012 sessions, but is
different from the formats of earlier exams.
Please note that no calculators will be permitted in the examination.
Should a resit examination be required in August/September it will take the same form
as above (five questions, best four answers count, no calculators).
You may find it useful to look at the exam papers from previous years.
I have been the lecturer for this module since Autumn 1998.
Please note that the format of the exam changed after the 19992000
session.
Miscellaneous links which may be of interest:
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