Differential Geometry and Lie Groups for Physicists [Hardcover]

Marián Fecko (Author)

Book Description

ISBN-10: 0521845076 | ISBN-13: 978-0521845076 | Publication Date: October 30, 2006 | Edition: 1

Differential geometry plays an increasingly important role in modern theoretical physics and applied mathematics. This 2006 textbook gives an introduction to geometrical topics useful in theoretical physics and applied mathematics, covering: manifolds, tensor fields, differential forms, connections, symplectic geometry, actions of Lie groups, bundles, spinors, and so on. Written in an informal style, the author places a strong emphasis on developing the understanding of the general theory through more than 1000 simple exercises, with complete solutions or detailed hints. The book will prepare readers for studying modern treatments of Lagrangian and Hamiltonian mechanics, electromagnetism, gauge fields, relativity and gravitation. Differential Geometry and Lie Groups for Physicists is well suited for courses in physics, mathematics and engineering for advanced undergraduate or graduate students, and can also be used for active self-study. The required mathematical background knowledge does not go beyond the level of standard introductory undergraduate mathematics courses.

Editorial Reviews

Review

"The contents of this book cover a lot (if not most) of what a theoretical physicist might wish to know about differential geometry and Lie groups."
Hans-Peter Künzle, Mathematical Reviews

"All basic material that is necessary for a young scientist in the field of geometrical formulation of physical theories is included ... ordered and represented in a very appropriate manner ... with a great respect to the reader. ... I truly believe that reading this book will bring a real pleasure to all physically inclined young mathematicians and mathematically inclined young physicists ... a very good high level textbook ... [I] recommend it to all young scientists being interested in finding correspondence between harmony in the physical world and harmony in geometrical structures. ... well written, very well ordered and the exposition is very clear."
Journal of Geometry and Symmetry in Physics

"the contents of this book covers a lot (if not most) of what a theoretical physicist might wish to know about differential geometry and Lie groups. particularly useful may be that the modern formalism is always related to the classical one with tensor indices still mostly used in the physics literature."
American Mathematical Society

"... the presentation is almost colloquial and this makes reading rather pleasant. The author has made a concerted effort to give intuitive interpretations of complicated ideas such as: the Lie derivative, tensors, the Hodge star operator, Lie group representations, Hamiltonian and Lagrangian mechanics, parallel transport, connections, curvature, gauge theories, spinors and Dirac operators. This will be much appreciated by students (and even researchers, I think). ... an excellent reference for geometers."
Zentralblatt MATH

"Marián Fecko deftly guides you through the material step-by-step, with all the rigor, but without the pain. When going through the chapters, definition by definition, proof by proof and hint by hint, you get an impression of a caring, experienced (and often quirkily funny, but never boring) tutor who really, really wants you to succeed."
Sergei Slobodov, UBC for Physics in Canada

Book Description

Covering subjects including manifolds, tensor fields, spinors, and differential forms, this 2006 textbook introduces geometrical topics useful in modern theoretical physics and mathematics. It develops understanding through over 1000 short exercises, and is suitable for advanced undergraduate or graduate courses in physics, mathematics and engineering.

Product Details

• Hardcover: 714 pages
• Publisher: Cambridge University Press; 1 edition (October 30, 2006)
• Language: English
• ISBN-10: 0521845076
• ISBN-13: 978-0521845076
• Product Dimensions: 10 x 6.9 x 1.5 inches
• Shipping Weight: 3.4 pounds (View shipping rates and policies)
• Average Customer Review: 3.6 out of 5 stars  See all reviews (5 customer reviews)
• Amazon Best Sellers Rank: #2,252,652 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

15 of 19 people found the following review helpful:

2.0 out of 5 stars not for starter or self-learning, March 26, 2007

By 

Z. Tan (Californica, USA) - See all my reviews
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This review is from: Differential Geometry and Lie Groups for Physicists (Hardcover)

The book covers a good range of topics in Differnetial geometry with lots of exercises. One literarily has to do the exercises to develop the concept. Ecah chapter ends with a concise summary of the key equations. The problem is that all the exercises are mixed with the main context. It lacks any exposition or concept development for most of the topics, no definition, no prove, and every page is filled with exercises. This style make it difficult for someone to learn the subjects the first time or to use it as a reference.

Separately, there are too few graphs to assist the reader to visualize the ideas. The prints are also small making it hard to read.

Nakahara's book (Geometry, topology and physics) is a much better choice on the same subject.

9 of 11 people found the following review helpful:

2.0 out of 5 stars Fairly good content, very bad exposition, July 15, 2008

By 

Đorđe Radičević - See all my reviews

This review is from: Differential Geometry and Lie Groups for Physicists (Hardcover)

There's no doubt about it: the material in this book is incredibly interesting and important for an ambitious physics student. The organization of the book is fairly good: informal passages relating the necessary theory alternate with exercises which are all written as "Check that..." or "Prove that...", which allows you to choose which results to prove and which to take as given facts if you -- for any reason -- don't feel like proving them.

However, the book also has some serious shortcomings. The most important one seems to be the horrid style. A book of mathematics for physicists should not be written just like a standard math textbook without proofs -- and this is exactly what this book is like. The definitions that are given are "mathematical" at heart; very rarely can one find an intuitive picture of what is going on immediately after a concept is introduced. On the other hand, the propositions that are not left as excercises are never proven. Granted, they might be intuitively clear, but that doesn't mean that their proofs are obvious. Due to all this, I have always felt a bit confused and certainly not comfortable with new concepts. The author's occasional attempts to "raise morale" by inserting jokes would always backfire because these jokes are so trivial that they seem offensively condecending. Take, for example, the sentence that finishes the introduction of a vector as the equivalence class of tangency of curves:

"And a good old arrow, which cannot be thought of apart from the vector, could be put at P in the direction of this bunch, too (so that it does not feel sick at heart that it had been forgotten because of some dubious novelties)." (p. 25)

So... first of all, this is probably not particularly funny. But more seriously: are we to conclude that the notion of vectors as "directed lines" is important only because otherwise the "good old arrow" (and the reader alike) would feel "sick at heart"? This is an example of a concept so intuitive that a joke like this is generally harmless; however, trouble arises when the same kind of explanation is applied to more abstract concepts (e.g. why not study non-Hausdorff spaces? The explanation given on p. 4 relates to Amazon Basin Indians).

Another important issue is that a large part of this book teaches you the principles of the mathematics behind the physics. This is fine, provided you learn how to operate with these principles; however, the book seldom teaches you how to *work* with the most basic concepts, and that's what the author promises to deliver in the preface.

Unfortunately, there are other issues as well. Introducing new, vital ideas in exercises *only* is one of them. Also, one would desire to know which ideas are crucial or well-worth meditating upon, and this is generally not given in the text. Finally, the excessively informal style prevents this book from being even a good reference.

All in all -- it is possible to learn a lot of new things from this book, but the effort probably isn't worth it.

5 of 7 people found the following review helpful:

5.0 out of 5 stars Differential geometry, July 21, 2007

By 

Michal Tarana (Prague, Czech Republic) - See all my reviews
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This review is from: Differential Geometry and Lie Groups for Physicists (Hardcover)

Marian Fecko's textbook covers well fundamental elements of modern differential geometry and introduction to the Lie groups (not only) from geometrical point of view. Geometrical formulations of the classical mechanics, gauge theory and classical electrodynamics are discussed.

The textbook expects the reader to be familiar with mathematical analysis on the level of the standard course usual in the physics undergraduate study programs. Understanding of the parts dealing with physical applications (classical mechanics and electrodynamics) expects knowledge of fundamental principles of these subjects. Organization of the book allows the reader to concern on particular part, i. e. understanding of later parts doesn't require reading of all previous parts (reading of parts concerning on the classical dynamics does not require reading of parts dealing with electrodynamics). However, relations between different subjects of the theory are explained instructively.

The main advantage of this textbook is that reader "builds" the subject himself by solving the exercises usually appended by hints. It makes all the elements of the theory natural to the reader during study. This way is a little bit more time consuming when compared with other textbooks dealing with this subject. It provides good starting point for study of mathematical aspects of the general relativity and field theories. I recommend this book to everybody who wants to understand fundamental concepts in differential geometry in detail.