E-Book Overview
The notion of symmetry is important in many disciplines, including physics, art, and music. The modern mathematical way of treating symmetry is through transformation groups. This book offers an easy introduction to these ideas for the relative novice, such as undergraduates in mathematics or even advanced undergraduates in physics and chemistry. The first two chapters provide a warm-up to the material with, for example, a discussion of algebraic operations on the points in the plane and rigid motions in the Euclidean plane. The notions of a transformation group and of an abstract group are then introduced. Group actions, orbits, and invariants are covered in the next chapter. The final chapter gives an elementary exposition of the basic ideas of Sophus Lie about symmetries of differential equations. Throughout the text, examples are drawn from many different areas of mathematics. Plenty of figures are included, and many exercises with hints and solutions will help readers master the material.
E-Book Content
Transformation Groups for Beginners S. V. Duzhin B. D. Tchebotarevsky
Contents
Preface
5
Introduction
6
Chapter 1. Algebra of points
11
§1. Checkered plane
11
§3. Multiplying points by numbers
17
§2. Point addition
13
§4. Centre of gravity
19
§6. Point multiplication
24
Chapter 2. Plane Movements
37
§1. Parallel translations
37
§3. Rotations
41
§5. Coordinates
§7. Complex numbers
§2. Reflections
21 28
39
§4. Functions of a complex variable
44
§6. Glide reflections
52
§5. Composition of movements
47
§7. Classification of movements
53
§9. Calcul