Differential geometry and Lie groups for physicists

Marián Fecko

Differential geometry plays an increasingly important role in modern theoretical physics and applied mathematics. This 2006 textbook gives an introduction to geometrical topics useful in theoretical physics and applied mathematics, covering: manifolds, tensor fields, differential forms, connections, symplectic geometry, actions of Lie groups, bundles, spinors, and so on. Written in an informal style, the author places a strong emphasis on developing the understanding of the general theory through more than 1000 simple exercises, with complete solutions or detailed hints. The book will prepare readers for studying modern treatments of Lagrangian and Hamiltonian mechanics, electromagnetism, gauge fields, relativity and gravitation. Differential Geometry and Lie Groups for Physicists is well suited for courses in physics, mathematics and engineering for advanced undergraduate or graduate students, and can also be used for active self-study. The required mathematical background knowledge does not go beyond the level of standard introductory undergraduate mathematics courses.

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[目次]

  • Introduction
  • 1. The concept of a manifold
  • 2. Vector and tensor fields
  • 3. Mappings of tensors induced by mappings of manifolds
  • 4. Lie derivative
  • 5. Exterior algebra
  • 6. Differential calculus of forms
  • 7. Integral calculus of forms
  • 8. Particular cases and applications of Stoke's Theorem
  • 9. Poincare Lemma and cohomologies
  • 10. Lie Groups - basic facts
  • 11. Differential geometry of Lie Groups
  • 12. Representations of Lie Groups and Lie Algebras
  • 13. Actions of Lie Groups and Lie Algebras on manifolds
  • 14. Hamiltonian mechanics and symplectic manifolds
  • 15. Parallel transport and linear connection on M
  • 16. Field theory and the language of forms
  • 17. Differential geometry on TM and T*M
  • 18. Hamiltonian and Lagrangian equations
  • 19. Linear connection and the frame bundle
  • 20. Connection on a principal G-bundle
  • 21. Gauge theories and connections
  • 22. Spinor fields and Dirac operator
  • Appendices
  • Bibliography
  • Index.

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この本の情報

書名 Differential geometry and Lie groups for physicists
著作者等 Fecko, Marián
Fecko Marian
出版元 Cambridge University Press
刊行年月 2011, c2006
ページ数 xv, 697 p.
大きさ 25 cm
ISBN 9780521187961
NCID BB07415173
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言語 英語
出版国 イギリス
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