Tensors and manifolds : with applications to physics

Description

This book is a new edition of "Tensors and Manifolds: With Applications to Mechanics and Relativity" which was published in 1992. It 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 specialized courses which both mathematics and physics students require in their 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. He existing material from the first edition has been reworked and extended in some sections to provide extra clarity, as well as 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 rigor 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. Content: 1. Vector spaces 2. Multilinear mappings and dual spaces 3. Tensor product spaces 4. Tensors 5. Symmetric and skew-symmetric tensors 6. Exterior (Grassmann) algebra 7. The tangent map of real cartesian spaces 8. Topological spaces 9. Differentiable manifolds 10. Submanifolds 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 18. Mechanics 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