464 pages, 46 line drawings
This second edition of Tensors and Manifolds is based on courses taken by advanced undergraduate and beginning graduate students in mathematics and physics, giving an introduction to the expanse of modern mathematics and its application in modern physics. It aims to fill the gap between the basic courses and the highly technical and specialised courses which both mathematics and physics students require in ther advanced training, while simultaneously trying to promote, at an early stage, a better appreciation and understanding of each other's discipline.
The book sets forth the basic principles of tensors and manifolds, describing how the mathematics underlies elegant geometrical models of classical mechanics, relativity and elementary particle physics. The existing material from the first edition has been reworked and extended in some sections to provide extra clarity, with additional problems. Four new chapters on Lie groups and fibre bundles have been included, leading to an exposition of gauge theory and the standard model of elementary particle physics.
Mathematical rigour combined with an informal style makes this a very accessible book and will provide the reader with an enjoyable panorama of interesting mathematics and physics.
1. Vector spaces;
2. Multilinear mappings and dual spaces;
3. Tensor product spaces;
5. Symmetric and skew-symmetric tensors;
6. Exterior (Grassmann) algebra;
7. The tangent map of real cartesian spaces;
8. Topological spaces;
9. Differentiable manifolds;
11. Vector fields, 1-forms and other tensor fields;
12. Differentiation and integration of differential forms;
13. The flow and the Lie derivative of a vector field;
14. Integrability conditions for distributions and for pfaffian systems;
15. Pseudo-Riemannian manifolds;
16. Connection 1-forms;
17. Connection on manifolds;
19. Additional topics in mechanics;
20. A spacetime;
21. Some physics on Minkowski spacetime;
22. Einstein spacetimes;
23. Spacetimes near an isolated star;
24. Nonempty spacetimes;
25. Lie groups;
26. Fiber bundles;
27. Connections on fiber bundles;
28. Gauge theory
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