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Trade in Geometry, Topology and Physics, Second Edition (Graduate Student Series in Physics) for an Amazon Gift Card of up to Â£11.00, which you can then spend on millions of items across the site. Trade-in values may vary (terms apply). Learn more

"a very impressive book.." -- Australian and New Zealand Physicists" The clarity of the presentation is enhanced by explicit calculations and diagrams; the proof of a theorem is given only when it is instructive and not very technical. There is also a large number of exercises and problems, and last but not least, an index superb layout" -- Zentralblatt fur Mathematick un ihre Grenzgebiete "I believe that the book will not only boost modernization of the traditional courses of theoretical physics but will prompt the specialist in topology and differential geometry to have a closer look at the applications. So I welcome this second edition." --Christopher Gilmour

I must say that this edition contains some severe errors. In several places the mathematics has been wrongly transferred over from the old edition in key definitions such as those of the wedge product (those should be tensor products on the RHS) and topological spaces (that's a J in (ii), not a T) and elsewhere in the book. I'm not sure how many there are in total so I write this as a caution that if you aren't sure what you're doing it might be wise to check out an older edition. I don't know how it's happened but it has.

Other than that, class book and extremely useful for a starting-out theoretical physicist.

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I think this is a nice book for people who start work in the area of mathematical physics. The content of the first chapter gives a broad introduction to quantum mechanics paying special attention to path integral quantization. The rest of the book deals mainly with topology (homology groups, homotopy classes, topological invariants like Chern numbers, etc.) and its application in quantum field theory, condensed-matter theory or general relativity. Throughout the chapters one can find a lot of examples and exercises which help to understand the theory. Therefore, I highly recommend this book as an introductory lecture.

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Nakahara's book is one of the best introductions to geometry and topology that I have read. I constantly use the book as the starting place for just about any topic in geometry and topolgy.

After reading the book you will not be able to jump straight into research work, but it does bridge the gap between more advanced texts and papers.

Everybody should have a copy.

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To complete this book, there should be a section on general curvilinear coordinate transformations, the ultimate foundation of tensor calculus.This is a defficiency this book shares with many differential geometry texts.But maybe this can be forgiven at graduate level, for which this book is a decent pedagogical text- if a little terse at times. The book begins with a survey of those areas of physics to which diff geom are applied , then develops some topology, and goes on to a comprehensive discussion of the theory of finite dimensional manifolds-including a chapter on complex manifolds.You will learn basic exterior calculus, lie derivatives and covariant derivatives , and so on.A first choice for those who have had a little preparation at undergraduate level.

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Most Helpful Customer Reviews on Amazon.com ^{(beta)}

Amazon.com:
18 reviews

93 of 97 people found the following review helpful

Best in its genre17 Sept. 2002

By
"jjf-sp"
- Published on Amazon.com

I suppose I should preface this by saying that I read this book *after* reading similar books, so my ability to understand this book is probably better than others, but that said, I think that my comparative evaluation is free from this bias... There seem to be a few books on the market that are very similar to this one: Nash & Sen, Frankel, etc. This one is at the top of its class, in my opinion, for a couple reasons: (1) It's written like a math text that covers physics-related material, not a book about mathematics for physicists. I prefer this; you may not. As a consequence, this book is more rigorous than its alternatives, it relies less on physical examples, and it cuts out a lot of lengthy explanation that you may not need. Of course, there are drawbacks to all of these "features" -- you need to decide what you need and what's best for you. (2) It's most comprehensive, with Frankel coming in second, and Nash & Sen least comprehensive (though they have quite a bit on Fibre bundles and related topics). Nakahara has a chapter on complex manifolds, which is absent from the other two. Nakahara also concludes with a nice intro to string theory, which is absent from the other two as well (though nothing you couldn't find in Polchinski or the like). Actually -- I modify this slightly. Frankel covers less subjects than Nakahara, but with more depth (though also more wordiness -- I quit Frankel about 2/3 through because it wasn't succinct enough and I got tired of it). Depending on your tastes, I would recommend this book before the other two. It presupposes that you have an understanding of algebra (groups, rings, fields, etc.) but it has an introduction to the necessary components of topology within. Frankel has presupposes both algebra and topology; Nash & Sen presupposes only algebra.

71 of 74 people found the following review helpful

A wonderful exposition on the mathematics of modern physics22 July 2001

By
hsurreal
- Published on Amazon.com

Format: Paperback

Nakahara is one of my favorite books. It gives the reader the necessary knowledge in differential geometry and topology to understand theoretical physics from a modern viewpoint. Each chapter in Nakahara would normally take a full semester mathematics course to teach, but the necesseties for a physicist are distilled with just the right amount of rigor so that the reader is neither bored from excessive proof nor skeptical from simple plausibility arguments. The first few chapters (homotopy, homology) are rather dry, but the text picks up after that. The manifold chapter is really good, particularly the Lie groups section which gives a geometric viewpoint of the objects which get very little attention in a typical particle physics course. Unfortunately, nothing is said on representation theory, but that can be found in Georgi's book. The cohomology chapter is wonderfully quick and to the point. I found myself having to tell myself to slow down because of the excitement I had in reading it. The Riemannian geometry chapter reads wonderfully and serves as a great reference for all those general relativity formulae you always forget. The end of that chapter has an exquisite little bit on spinors in curved spacetime. The complex geometry chapter is also wonderful. I find myself going back to it all the time when reading Polchinski's string text. The chapters on fiber bundles seem a bit on the overly mathy side, but then again, all the pain is in the definitions which becomes well worth it in the end. I haven't read the last few chapters (spending all of my time in Polchinski!) but I definitely will when I have some spare time. The notation in Nakahara is also really self explanatory and standard. It is written with the physicist in mind who doesn't mind a bit of sloppiness or ambiguity in his notation. With regards to Frankel, Nakahara is much more modular than Frankel. Each chapter of Nakahara is pretty much self contained whereas Frankel kinda needs to be read straight through. I find it very difficult to just look up a random thing in Frankel and learn about it on the spot, whereas this seems to work in Nakahara just fine. Frankel is a bit more respectful of proper mathematics which also makes it a harder text to read for physicists. Nakahara is a great text. When I visited Caltech I noticed it on the bookshelf of every theorist that I talked to. Anyone who wants to understand how it is that geometry is so important in modern theoretical physics would do himself a favor in buying this book.

29 of 31 people found the following review helpful

Great book.25 Mar. 2005

By
Gargantua
- Published on Amazon.com

Format: Paperback

This is a very useful book for understanding modern physics. You absolutely need such a book to really understand general relativity, string theory etc. For instance, Wald's book on general relativity will make much more sense once you go through Nakahara's book. It is very complete, clearly written, comprehensive and easy to read. I would also recommend Morita's "Geometry of differential forms' and Dubrovin,Novikov and Fomeko's 3 volume monograph, if you can find it. All in all, Nakahara's book is one of the best buys, precious book.

27 of 29 people found the following review helpful

Excellent Survey of Mathematical Tools18 May 2000

By
A Customer
- Published on Amazon.com

Format: Paperback

This text acts as a well-rounded introduction and overview of much of the mathematics underlying modern physics. While far from rigorous, the physics student will come away with a good understanding of how to use a wide variety of mathematical tools. This book is a necessity for every theoretical physicist. When used in a course (probably advanced undergrad or beginning grad), it should definitely be supplemented with more thorough texts, such as Geometry of Physics by Frankel. After such a course, one should be fully prepared for texts such as Spin Geometry by Michelson & Lawson, and String Theory by Polchinski. As for the mathematics presented in the book, go to one of the many excellent intro books to algebraic topology (Fulton, Munkres, Massey, Bott & Tu) and fibre bundles (Steenrod, Husemoller) for proper treatments of the subjects.

34 of 40 people found the following review helpful

Too many errors to be useful for study24 July 2008

By
njdj
- Published on Amazon.com

Format: Paperback

Reading all the glowing reviews of this book, I wonder whether the reviewers actually tried to use the book to understand the material, or just checked the table of contents. There are so many misprints, throughout, that one wonders if the book was proofread at all. Some of the mistakes will be obvious to every physicist - for example, one of the Maxwell equations on page 56 is wrong - others are subtle, and will confuse the reader. The careful reader, who wants to really understand the material and tries to fill in the details of some of the derivations, will waste a lot of time trying to derive results that have misprints from intermediate steps which have different misprints! Some chapters are worse than others, but the average density of misprints seems to be more than one per page. The book might be useful as a list of topics and a "road map" to the literature prior to 2003, but that hardly justifies the cost (or the paper) of a whole book.