IV - MATHEMATICAL TOOLS AND THEIR CONNECTION TO PHYSICS

Summary

In this last part of the book we collect some mathematical background material which is heavily used in the physics part of the book. There are several reasons for doing this: First of all, it makes the book almost self-contained. Secondly, some of this material is not covered by the obligatory courses in mathematics for physicists. Thirdly, while the material is covered in some mathematics courses, it is often presented in such a way that a physicist does not recognise it any more or it is not given sufficient attention. Clearly we can mostly give definitions and state theorems, proofs are often omitted for reasons of space. However, we try to motivate the mathematical theory from a physicists’ point of view, explain how the various theorems fit together and indicate their various applications. We thus hope that the ambitious reader feels encouraged to study the mathematical theory in appropriate depth, going through the proofs by himself.

The material is presented in logical order, not in the order as it is applied in the physics part of the book. For instance, topology is needed before one speaks about differential geometry, measure theory and (functional) analysis.