This book introduces the tools of modern differential geometry, exterior calculus, manifolds, vector bundles and connections, to advanced undergraduate and beginning graduate students in mathematics, physics and engineering. The book covers both classical surface theory and the modern theory of connections and curvature, and includes a chapter on applications to theoretical physics. The only prerequisites are multivariate calculus and linear algebra; no knowledge of topology is assumed. The powerful and concise calculus of differential forms is used throughout. Through the use of numerous concrete examples, the author develops computational skills in the familiar Euclidean context before exposing the reader to the more abstract setting of manifolds. There are nearly 200 exercises, making the book ideal for both classroom use and self-study.
"...Darling's exegesis is clear and easy to understand, and his frequent use of examples is beneficial to the reader. There are many exercises that serve to reinforce the concepts." D.P. Turner, Choice "...easy on the eyes; some nice exercises..." American Mathematical Monthly "The exposition is clear and, in the American textbook style, has many exercises, both theoretical and computational. In summary, this text provides a worthwhile elementary introduction to anyone who wants to understand the basic mathematical ingredients of Differential Geometry and its interactions with Physics." F.E. Burstall, Contemporary Physics "...a good introduction to differential geometry and its applications to physics by using the calculus of differential forms...Nearly 200 exercises and many examples will help the reader's understanding...this book can be recommended as a good textbook for advanced undergraduate and beginning graduate students in mathematics, physics, and engineering." Akira Asada, Mathematical Reviews
Preface; 1. Exterior algebra; 2. Exterior calculus on Euclidean space; 3. Submanifolds of Euclidean spaces; 4. Surface theory using moving frames; 5. Differential manifolds; 6. Vector bundles; 7. Frame fields, forms and metrics; 8. Integration on oriented manifolds; 9. Connections on vector bundles; 10. Applications to gauge field theory; Bibliography; Index.