文档介绍：Department of
Mathematical Sciences
Advanced Calculus and Analysis
MA1002
Ian Craw
ii
April 13, 2000, Version
Copyright 2000 by Ian Craw and the University of Aberdeen
All rights reserved.
Additional copies may be obtained from:
Department of Mathematical Sciences
University of Aberdeen
Aberdeen AB9 2TY
DSN: mth200-101982-8
Foreword
These Notes
The notes contain the material that I use when preparing lectures for a course I gave from
the mid 1980’s until 1994; in that sense they are my lecture notes.
”Lectures were once useful, but now when all can read, and books are so nu-
merous, lectures are unnecessary.” Samuel Johnson, 1799.
Lecture notes have been around for centuries, either informally, as handwritten notes,
or formally as textbooks. Recently improvements in typesetting have made it easier to
produce “personalised” printed notes as here, but there has been no fundamental change.
Experience shows that very few people are able to use lecture notes as a substitute for
lectures; if it were otherwise, lecturing, as a profession would have died out by now.
These notes have a long history; a “first course in analysis” rather like this has been
given within the Mathematics Department for at least 30 years. During that time many
people have taught the course and all have left their mark on it; clarifying points that have
proved difficult, selecting the “right” examples and so on. I certainly benefited from the
notes that Dr Stuart Dagger had written, when I took over the course from him and this
version builds on that foundation, itslef heavily influenced by (Spivak 1967) which was the
mended textbook for most of the time these notes were used.
The notes are written in LATEX which allows a higher level view of the text, and simplifies
the preparation of such things as the index on page 101 and numbered equations. You
will find that most equations are not numbered, or are numbered symbolically. However
sometimes I want to refe