This is an introduction to geometrical topics that are useful in applied mathematics and theoretical physics, including manifolds, metrics, connections, Lie groups, spinors and bundles, preparing readers for the study of modern treatments of mechanics, gauge fields theories, relativity and gravitation. The order of presentation corresponds to that used for the relevant material in theoretical physics: the geometry of affine spaces, which is appropriate to special relativity theory, as well as to Newtonian mechanics, is developed in the first half of the book, and the geometry of manifolds, which is needed for general relativity and gauge field theory, in the second half. Analysis is included not for its own sake, but only where it illuminates geometrical ideas. The style is informal and clear yet rigorous; each chapter ends with a summary of important concepts and results. In addition there are over 650 exercises, making this a book which is valuable as a text for advanced undergraduate and postgraduate students.
Author(s): M. Crampin, F. A. E. Pirani
Series: London Mathematical Society Lecture Note Series
Publisher: Cambridge University Press
Year: 1987
Language: English
Pages: 402
Cover ......Page 1
Title Page ......Page 4
Copyright Page ......Page 5
Contents ......Page 6
0. The background: vector calculus ......Page 8
1. Affine spaces ......Page 15
2. Curves, functions and derivatives ......Page 36
3. Vector fields and flows ......Page 60
4. Volumes and subspaces: exterior algebra ......Page 92
5. Calculus of forms ......Page 124
6. Frobenius's theorem ......Page 147
7. Metrics on affine spaces ......Page 171
8. Isometries ......Page 195
9. Geometry of surfaces ......Page 223
10. Manifolds ......Page 243
11. Connections ......Page 275
12. Lie groups ......Page 305
13. The tangent and cotangent bundles ......Page 334
14. Fibre bundles ......Page 360
15. Connections revisited ......Page 378
Bibliography......Page 390
Index......Page 393
Back Cover......Page 402
