Preface to the Third Edition; Preface to the Second Edition; Preface to the revised printing; Preface to the First Edition; Overview; Part I. Manifolds, Tensors, and Exterior Forms: 1. Manifolds and vector fields; 2. Tensors and exterior forms; 3. Integration of differential forms; 4. The Lie derivative; 5. The Poincaré Lemma and potentials; 6. Holonomic and nonholonomic constraints; Part II. Geometry and Topology: 7. R3 and Minkowski space; 8. The geometry of surfaces in R3; 9. Covariant differentiation and curvature; 10. Geodesics; 11. Relativity, tensors, and curvature; 12. Curvature and topology: Synge's theorem; 13. Betti numbers and De Rham's theorem; 14. Harmonic forms; Part III. Lie Groups, Bundles, and Chern Forms: 15. Lie groups; 16. Vector bundles in geometry and physics; 17. Fiber bundles, Gauss–Bonnet, and topological quantization; 18. Connections and associated bundles; 19. The Dirac equation; 20. Yang–Mills fields; 21. Betti numbers and covering spaces; 22. Chern forms and homotopy groups; Appendix A. Forms in continuum mechanics; Appendix B. Harmonic chains and Kirchhoff's circuit laws; Appendix C. Symmetries, quarks, and Meson masses; Appendix D. Representations and hyperelastic bodies; Appendix E. Orbits and Morse–Bott theory in compact Lie groups.
Provides a working knowledge of tools that are of great value in geometry and physics and in engineering.
Theodore Frankel received his PhD from the University of California, Berkeley. He is currently Emeritus Professor of Mathematics at the University of California, San Diego.
Review of previous edition: '… highly readable and enjoyable … The
book will make an excellent course text or self-study manual for
this interesting subject.' Physics Today
Review of previous edition: 'This book provides a highly detailed
account of the intricacies involved in considering geometrical
concepts.' Contemporary Physics
'If you're looking for a well-written and well-motivated
introduction to differential geometry, this one looks hard to
beat.' Fernando Q. Gouvêa, MAA Online
'… a first rate introductory textbook … the style is lively and
exposition is clear which make the text easy to read … This book
will be beneficial to students and scientists wishing to learn the
foundations of differential geometry and algebraic topology as well
as geometric formulations of modern physical theories.' Pure and
Applied Geophysics
'… this book should not be missing in any physics or mathematics
library.' European Mathematical Society
'This book is a great read and has a lot to offer to graduate
students in both mathematics and physics. I wish I had had it on my
desk when I began studying geometry.' AMS Review
Review of previous edition: 'The layout, the typography and the
illustrations of this advanced textbook on modern mathematical
methods are all very impressive and so are the topics covered in
the text.' Zentralblatt für Mathematik und ihre Grenzgebiete
'… contains a wealth of interesting material for both the beginning
and the advanced levels. The writing may feel informal but it is
precise - a masterful exposition. Users of this 'introduction' will
be well prepared for further study of differential geometry and its
use in physics and engineering … As did earlier editions, this
third edition will continue to promote the language with which
mathematicians and scientists can communicate.' Jay P. Fillmore,
SIAM Review
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