Differential Forms and Connections [Paperback]

R. W. R. Darling (Author)

Book Description

Publication Date: October 28, 1994 | ISBN-10: 0521468000 | ISBN-13: 978-0521468008

This book introduces the tools of modern differential geometry--exterior calculus, manifolds, vector bundles, connections--and covers both classical surface theory, the modern theory of connections, and curvature. Also included is a chapter on applications to theoretical physics. The author uses the powerful and concise calculus of differential forms 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. The only prerequisites are multivariate calculus and linear algebra; no knowledge of topology is assumed. Nearly 200 exercises make the book ideal for both classroom use and self-study for advanced undergraduate and beginning graduate students in mathematics, physics, and engineering.

Editorial Reviews

Review

"...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

Product Details

• Paperback: 268 pages
• Publisher: Cambridge University Press (October 28, 1994)
• Language: English
• ISBN-10: 0521468000
• ISBN-13: 978-0521468008
• Product Dimensions: 10 x 7 x 0.6 inches
• Shipping Weight: 1.2 pounds (View shipping rates and policies)
• Average Customer Review: 4.0 out of 5 stars  See all reviews (9 customer reviews)
• Amazon Best Sellers Rank: #1,419,164 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

44 of 46 people found the following review helpful:

5.0 out of 5 stars The ONLY book really suited for self study, March 4, 2002

By 

Assaf Tal (Israel) - See all my reviews

This review is from: Differential Forms and Connections (Paperback)

I would just like to point out that Darling's book is the only book I've encountered which is suited for self study. It resembles someone's classroom notes - i.e., nothing fancy, no glossy color 3-d graphics or such - but it is very neatly organized, with many examples and helpful problems, and it is really, really suited for someone trying to study the subject by him/herself (me ... ). It is not very physically oriented - not many physical examples are provided throughout the text, and it is mathematical in nature, but don't let that deter you! In fact, the sharp distinction between mathematics and physics is pedagogically wise.
Another good thing about this book is that it does not begin with completely abstract definitions. First of all it develops exterior calculus and diff. manifolds in ordinary Euclidian space. This is a must for anyone studying on their own, believe me! No matter how mathematically mature you are, those things just don't make sense unless you've seen how they work in familiar settings. You don't have to worry, though - Darling keeps his notation clean; Darling tries as hard as he can to keep everything in pure geometrical language, referring to a specific basis only when absolutely necessary (or when it helps one understand).
I cannot say how good a classroom text this is, but do yourself a favor and check it out if you're thinking of studying this on your own! Darling is a clear and (equally important!) responsible teacher.

39 of 41 people found the following review helpful:

5.0 out of 5 stars A must for both the physicist and mathematician, January 8, 1999

By A Customer

This review is from: Differential Forms and Connections (Paperback)

RWR Darling should be the first and foremost book for learning about differential geometry both for physicists and mathematicians. I have learned from numerous books on this subject, and while I can't say Darling includes everything one could want (I can't say anyone ever does), his text explains some very esoteric ideas in terms of linear algebra and vector calculus.

A notable departure this book makes is dispensing with the usual coordinate basis for tangent spaces which is commonly used by physicists. To the experienced physics reader, this may seem daunting, and unnecessarily abstract at first. However, the pay-off in the ability later on to discuss gauge theories and fiber bundles is huge.

This book is also suited for mathematicians less interested in physics. Darling does not always assume that a manifold has some metric, and discusses the subtle differences between vectors and co-vectors in modern mathematical language. Secondly, he provides a lot of motivation for the mathematical constructions and takes great care to present key definitions in extremely coordinate free ways.

34 of 36 people found the following review helpful:

5.0 out of 5 stars Gauge theories in the mathematical way, August 3, 1998

By A Customer

This review is from: Differential Forms and Connections (Paperback)

The main difficulty found by physicists in the learning of modern differential geometry is topology. The various constructions introduced by Cartan and others, differential forms, connections, even fiber bundles, on the contrary, pose no difficulties: it is only a question of developing the appropriate muscles and reflexes. R. Darling wrote the ideal book to teach connections on a G-bundle (gauge theories, in the nomenclature of physicists), by refraining, as much as possible, to use explicit topology. As physicists are not a special kind of human beings, I believe what I said above is also true of (beginning) mathematicians. Otherwise, why would Darling choose such course (in the navigational sense). The book starts with Cartan calculus in Euclidean space, continues there up to surface theory, then introduces (intrinsic) manifolds. Perhaps the key concept of the book comes next: Vector Bundles. All previous constructions are extended to bundles, and the concept of conn! ections on vector bundles deserves a special chapter. The book ends with Applications to Gauge Field Theory (mathematics-wise, but quite accessible). There are many pedagogical virtues in this much welcome book. Finally a good alternative to Bishop-Goldberg`s "Tensor Calculus on Manifolds".