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Geometrical Methods of Mathematical Physics [Paperback]

Bernard F. Schutz
4.0 out of 5 stars  See all reviews (13 customer reviews)

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Book Description

January 28, 1980 0521298873 978-0521298872 First Published
In recent years the methods of modern differential geometry have become of considerable importance in theoretical physics and have found application in relativity and cosmology, high-energy physics and field theory, thermodynamics, fluid dynamics and mechanics. This textbook provides an introduction to these methods - in particular Lie derivatives, Lie groups and differential forms - and covers their extensive applications to theoretical physics. The reader is assumed to have some familiarity with advanced calculus, linear algebra and a little elementary operator theory. The advanced physics undergraduate should therefore find the presentation quite accessible. This account will prove valuable for those with backgrounds in physics and applied mathematics who desire an introduction to the subject. Having studied the book, the reader will be able to comprehend research papers that use this mathematics and follow more advanced pure-mathematical expositions.

Frequently Bought Together

Geometrical Methods of Mathematical Physics + Tensors, Differential Forms, and Variational Principles (Dover Books on Mathematics) + Differential Geometry (Dover Books on Mathematics)
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Editorial Reviews

Review

"...excellent. It would require a great deal of delving in the literature to produce equivalent treatments....a very useful introduction...." J. M. Stewart, Journal of Fluid Mechanics

"...Schutz has such a mastery of tthe material that it soon becomes clear that one is in authoritative hands....this book is the most lucid I have come across at this level of exposition. It is eminently suitable for a graduate course (indeed, the more academically able undergraduates should be able to cope with most of it), and the applications should suffice to persuade any physicist or applied mathematician of its importance." Ray d'Inverno, Times Higher Education Supplement

Book Description

For physicists and applied mathematicians working in the fields of relativity and cosmology, high-energy physics and field theory, thermodynamics, fluid dynamics and mechanics. This book provides an introduction to the concepts and techniques of modern differential theory, particularly Lie groups, Lie forms and differential forms.

Product Details

  • Paperback: 264 pages
  • Publisher: Cambridge University Press; First Published edition (January 28, 1980)
  • Language: English
  • ISBN-10: 0521298873
  • ISBN-13: 978-0521298872
  • Product Dimensions: 0.7 x 5.9 x 9 inches
  • Shipping Weight: 14.9 ounces (View shipping rates and policies)
  • Average Customer Review: 4.0 out of 5 stars  See all reviews (13 customer reviews)
  • Amazon Best Sellers Rank: #42,603 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews
49 of 50 people found the following review helpful
4.0 out of 5 stars A Very Accessible Book ! Buy It ! November 6, 2000
Format:Paperback
This is a very enjoyable and clearly written book. From a physics point of view the approach is rather abstract, so although differential geometry is developed from 'scratch', it is probably better to have studied a more elementary text on the theory of 2-surfaces in 3-space first (eg Faber's book Differential Geometry and Relativity Theory ). The first chapter sets the mathematical background expected of the reader. The rudiments of analysis, topology, calculus of many variables and basic linear algebra is reviewed.The ensuing chapters cover differential geometry from a 'modern' viewpoint but the style is quite relaxed and the links to 'co-ordinate approach' are well explained. The exercises concentrate on the abstract approach. Throughout the book the underlying structure of manifolds is concentrated upon. No extra 'structure' eg connections and 'distance' concepts are added until the final chapter on Riemannian spaces. For example the metric tensor throughout the body of the book is merely used as a map between a tangent space and its dual space. It is only used as a 'distance' operator in the final chapter.For the purposes of independent study this is a sound book, there are hints and partial solutions for many of the exercises, which is always a welcome feature for those studying entirely on their own.
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30 of 31 people found the following review helpful
Format:Paperback
A heuristic and intuitive intro. to manifolds, fiber bundles, connections etc. Some applications are briefly touched upon. This is a good book to study for those that feel they didn't learn enough geometry from their GR class. Note: no complex algebraic geometry here, so this book would be considered too elementary for those looking for a mathematics book for strings.
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21 of 21 people found the following review helpful
5.0 out of 5 stars Terrific geometry book for physicists May 6, 2006
Format:Paperback
Advanced mathematics, such as differential geometry and topology, plays an important role in many areas of physics. This excellent book covers one of these topics, differential geometry. This is a topic essential for understanding general relativity and gauge theory. There are several good books aimed at physicists that cover differential geometry. While some of these have a broader scope than this book, nevertheless this book is my favorite one for differential geometry.

The topics covered include those necessary for reading advanced treatments of general relativity (such as Wald or Misner/Thorne/Wheeler). These include manifolds, fiber bundles, tangent/cotangent bundles, forms, Lie derivatives, Killing vectors and Lie groups.

Following this basic material a chapter covering some applications to physics, one example is electromagnetism. Up to this point the consideration of manifolds had been fairly general. In the final chapter the implications of adding a connection, and then a metric, are considered.

Why do I think this book is so good? It's not the breadth of material covered, this book is very focused on a limited range of material. It's the quality of the presentation for what it does cover. The development follows a logical order, the writing is exceptionally clear and the diagrams are very useful since Schutz explains them so well.
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14 of 14 people found the following review helpful
5.0 out of 5 stars A Great Introduction to Diff. Geometry August 7, 2000
Format:Paperback
This book presents the basic concepts of differential geometry in a clear, concise manner using modern notation. Schutz's writing style is very readable and there is a considerable breadth of coverage. In areas where one might wish for greater depth, Schutz provides excellent references. My only regret is that the physical applications chapters weren't longer. An excellent starter book and a good quick reference if you continue in differential geometry, GR or field theory.
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13 of 13 people found the following review helpful
Format:Paperback|Verified Purchase
The exposition in this book is concise, intuitive and, for the most part, quite lucid. It's really tops for getting the larger view, relating key mathematical concepts to applications in physics. As an autodidact, I've found it particularly productive to use this book in conjunction with a more detailed treatment of some particular topic, e.g. tensors, representation theory for groups and the relationship between that and Lie groups and algebras. I've also found it enlightening to supplement this book with the typically more detailed and superb expositions on some topic in Frankel's The Geometry of Physics: An Introduction, Second Edition and in Wasserman's apparently lesser known but phenomenal Tensors and Manifolds: With Applications to Physics. With some foundation/supplementation, the book can profitably be used to solidify and extend one's intuitive understanding of these mathematical topics and come to understand how they are of use in physics. One can also use the book to identify weakness in one's understanding and to determine what else one needs to study to make further progress. In addition, Schutz provides solutions or hints to the exercises. It's a comparatively quick read and overall, quite enjoyable. Highly recommended for self-study but see the caveats below.

Despite my high praise, I think that this book is best used as a supplement to more thorough treatments of the math covered (mainly, differential manifolds, forms, Lie derivatives, Lie groups).
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Most Recent Customer Reviews
4.0 out of 5 stars Very good introduction
Very good introduction, but I felt that waiting until chapter 5 to introduce any examples was a little weird of a decision. Read more
Published 9 months ago by Nate
4.0 out of 5 stars Importand
It is just what I wanted, this book was new, cheaper and very good material for someone who wants to learn more about geometrics methods in physics.
Published 13 months ago by edward q.
5.0 out of 5 stars The best introductory book in modern differential geometry for...
I have used this book as a student (junior level and up) and I recommend it to undergraduate students making their way to general relativity and gauge theories. Read more
Published 18 months ago by Konstantinos N. Anagnostopoulos
3.0 out of 5 stars too much stuff, too little time to explain
This book is only readable AFTER you have read Schutz "Introduction to general relativity", the latter is a much better book. Read more
Published on January 13, 2009 by Hexogen
3.0 out of 5 stars Integrability conditions discussed
Written in a attractive and even seductive way, relying more on Lie algebraic language than is typical, this book is probably as stimulating an intro. Read more
Published on January 21, 2004 by Professor Joseph L. McCauley
3.0 out of 5 stars Not as good as "a first course in general relativity"
I had read first the "first course in general relativity"and was exited,so i fygured out that this book from the same author would reach the same standards,but it... Read more
Published on October 22, 2001 by "dimitrish@kasparovchess.com"
5.0 out of 5 stars A Great Introduction to Diff. Geometry
This book presents the basic concepts of differential geometry in a clear, concise manner using modern notation. Read more
Published on August 7, 2000 by Glen Aultman-Bettridge
1.0 out of 5 stars a little time and no love
I am very angry with paying this much money for a book which obviously dosen't live up to it's name. Read more
Published on December 29, 1999 by Slakhami Bjorkan
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