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 Poincare 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. Gouvea, 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 fur 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
"It 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, University of California, San Diego for SIAM Review