Differential Forms and Connections 1st Edition by R. W. R. Darling (PDF)

Description

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.

User’s Reviews

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 Book Description This 1994 book introduces the tools of modern differential geometry, exterior calculus, manifolds, vector bundles and connections.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is an interesting book for those interested in certain topics in differential geometry. It discusses subjects not often seen in a book that is purportedly for those beginning to learn the subject –e.g. Kozul connections. On the other hand, while it claims to be for those with little background, and does not assume familiarity with any topology, it jumps directly into exterior algebras and the exterior calculus and only much later introduces the notion of a manifold. This strikes me as a rather odd way to introduce differential geometry in which the central object of interest is a manifold (a fundamentally topologic construct) and the notions of differential forms and connections are the means by which one carries the ideas of elementary calculus into the setting of a smooth manifold of some generality. Overall, the book is interesting for those who wish to understand the application of calculus on manifolds to physics, but it is not a book that I would recommend as a starting point to learn that theory. I would perhaps recommend it as supplementary materil for someone who already has some knowledge of the subject, at least at the level presented In “Calculus on Manifolds” by Michael Spivak which I consider to be a much better introductory text. In addition to Spivak’s book one seeking an introduction to differential geometry might be better seved by the books by do Carmo (Differential Geometry of Curves and Surfaces followed by Riemannian Geomtery) or by Sternberg (Lectures on Differential Geometry) which arm one to read more advanced texts. If one is truly a novice and seeking something oriented towards physics Sternbergs A Course in Mathematics for Students of Physics provides an excellent text at the undergraduate level,but certainly not a presentation of either general manifold, differential forms in the absract or connections.If one is indeed interest in connections, a somewhat advanced notion, then one can look in Foundations of Differential Geometry by Kobayashi and Nomizu or in Differential Geometry, Lie Groups and Symmetric Spaces by Helgason. Both of these books are directed towards advanced graduate students and are not for the fain of heart. But there is no “royal road to understanding” for any subject and most certainly not for advanced differential geometry.

⭐Others have discussed the merits and defects of the book. The defect (and hence minus 1 star) is that the book is written as an introduction and indeed all the material are introductory but it will be quite difficult for someone who has not learned about differential forms elsewhere to read this book as self study. The defect is that simpler concepts are described in details but transitions to more abstract ideas where crucially more detailed discussion is needed are too brief.On the other hand the book is on mission to introduce differential forms in a way that their advanced application to physics is quickly reached. The use of exterior derivative as a unifying concept is brilliant and elegant. I would say the book is a gem if you need differential forms for physics.

⭐perfect, thank you.

⭐I think the bad reviews on this page are unjustified. I have read the book at a time when I was learning about manifolds and (modern) differential geometry. The textbook we used was Spivak’s Comprehensive Intro. Spivak is undoubtedly excellent, but probably a bit too advanced for most people just wanting an introduction to the tools needs to study advanced physics. This is where Darling’s textbook comes into play. He uses a lot of examples from physics (electromagnetic theory, relativity) to motivate the presentation, so that it doesn’t sound so dry to the newcomer. And yes, the requirements for reading the book are quite modest (calculus, linear algebra). I don’t see why the reader below finds the book difficult to read. Certainly you need mathematical maturity, but there are no other formal prerequisites. I found the presentation to be quite friendly and easy to follow. For example, this book really gave me a great intuitive understanding of the definition of submanifolds, immersions, submersions, the inverse (and implicit) function theorem, etc. When you read a book on manifolds, they always assume you know it so well, and never provide any motivation for it. Darling’s book does a wonderful job at making the transition between dry mathematical subjects and definitions to the more intuitive physical understanding. He uses a lot of pictures, which really help a lot. You rarely find this sort of presentation from a math book. There are a few typos, but nothing to be overly concerned about. Darling used to have a PDF on his web site listing the typos, but I cannot find it anymore. Anyhow, most of them you can pick up if you read carefully enough.If you aren’t able to read this book carefully, then you probably need to get a more solid foundation in the more basic topics, this book isn’t for you yet (even though it is very introductory, as-is).

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