Preliminary Remarks

Preliminary Remarks
These Notes provide an introduction to 20th century mathematics, and in particular to Mathematical Analysis, which roughly speaking is the \in depth" study of Calculus.All of the Analysis material from B21H and some of the material from B30H is included here.
Some of the motivation for the later material in the course will come from B23H, which isnormally a co-requisite for the course. We will not however formally require this material.
The material we will cover is basic to most of your subsequent mathematics courses (e.g. differentialequations, differential geometry, measure theory, numerical analysis, to name a few), as well as to much of theoretical physics, engineering, probability theory and statistics. Variousinteresting applications are included; in particular to fractals and to differential and integral
equations.
There are also a few remarks of a general nature concerning logic and the nature ofmathematical proof, and some discussion of set theory.
There are a number of Exercises scattered throughout the text. The Exercises are usually simple results, and you should do them all asan aid to your understanding of the material.
Sections, Remarks, etc. marked with a * are non-examinable material, but you should read them anyway. They often help to set the other materialin context as well as indicating further interesting directions.
There is a list of related books in the Bibliography.
The way to learn mathematics is by doing problems and by thinkingvery carefully about the material as you read it. Always ask yourself why the various assumptions in a theorem are made. It is almost always the case that if any particular assumption isdropped, then the conclusion of the theorem will no longer be true. Try to think of examples where the conclusion of the theorem is no longer valid if the various assumptions are changed.