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Geometrical Methods of Mathematical Physics [Kindle Edition]

Bernard F. Schutz
4.3 out of 5 stars  See all reviews (3 customer reviews)

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Book Description

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.


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Review

'Although there are a welter of books where similar material can be found, 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 undergraduate should be about to cope with most of it), and the applications should suffice to persuade any physicist or applied mathematician of its importance … Schutz's book is a triumph …' The Times Higher Education Supplement

Book Description

The methods of modern differential geometry have become important in theoretical physics and have applications in relativity, 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 applications to theoretical physics.

Product details

  • Format: Kindle Edition
  • File Size: 5921 KB
  • Print Length: 264 pages
  • Simultaneous Device Usage: Up to 4 simultaneous devices, per publisher limits
  • Publisher: Cambridge University Press; 1 edition (28 Jan. 1980)
  • Sold by: Amazon Media EU S.à r.l.
  • Language: English
  • ASIN: B00D2WQ48A
  • Text-to-Speech: Enabled
  • X-Ray:
  • Word Wise: Not Enabled
  • Average Customer Review: 4.3 out of 5 stars  See all reviews (3 customer reviews)
  • Amazon Bestsellers Rank: #635,728 Paid in Kindle Store (See Top 100 Paid in Kindle Store)
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Most Helpful Customer Reviews
21 of 21 people found the following review helpful
5.0 out of 5 stars excellent 13 Nov. 1999
Format:Paperback
This book provides a unique introduction to differential geometry and its applications. The only prerequisites are a general knowledge of algebra and calculus. Applications in areas such as mechanics, thermodynamics, electromagnetism and especially general relativity are explained in detail. An almost essential book for the advanced undergraduate or beginning graduate student of theoretical or mathematical physics.
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5 of 5 people found the following review helpful
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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12 of 14 people found the following review helpful
4.0 out of 5 stars A comprehensive intro to Differential Geometry 1 Mar. 2001
By A Customer
Format:Paperback
It is a bit informal exposition in comparison with other more mathematical rigorous titles. Definitions of difficult concepts like tensors or manifolds are very accessible and the same to differential forms. It is suitable for any first course in modern geometry applied to physics, above all in relativity theory. As a suggest I think the Riemannian geometry chapter should be increased.
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Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com: 4.0 out of 5 stars  14 reviews
52 of 53 people found the following review helpful
4.0 out of 5 stars A Very Accessible Book ! Buy It ! 6 Nov. 2000
By Andy Gregory - Published on Amazon.com
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.
32 of 33 people found the following review helpful
5.0 out of 5 stars Introduction to Differential Geometry for physicists 2 July 1999
By Bay Area Educator and Tech Worker - Published on Amazon.com
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.
24 of 24 people found the following review helpful
5.0 out of 5 stars Terrific geometry book for physicists 7 May 2006
By Dean Welch - Published on Amazon.com
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.
17 of 17 people found the following review helpful
5.0 out of 5 stars Great for self-study: concise, clear, intuitive; yes, even enjoyable! 1 Sept. 2010
By gengogakusha - Published on Amazon.com
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). Other books I have used repeatedly and highly recommend include Tu's Introduction to Smooth Manifolds (Graduate Texts in Mathematics), Lee's An Introduction to Manifolds (Universitext) on manifolds; Stillwell's Naive Lie Theory (Undergraduate Texts in Mathematics), Tapp's Matrix Groups for Undergraduates (Student Mathematical Library,) and Hall's Lie Groups, Lie Algebras, and Representations: An Elementary Introduction on Lie groups, etc. and Weintraub's Differential Forms: Integration on Manifolds and Stokes's Theorem, Darling's Differential Forms and Connections and at a more advanced level, Morita's lucid and concise Geometry of Differential Forms (Translations of Mathematical Monographs, Vol. 201) on differential forms.

Schutz states that the aim of "this book is to teach mathematics, not physics". In general, I do not think one's math should be learned soley from physics books (having experienced the inadequate job done on mathematics in typical general relativity and quantum mechanics books). This book is no exception despite its exceptional lucidity. The claim that one only needs reasonable familiarity with "vector calculus, calculus of many variables, matrix algebra ... and a little operator theory ..." is overly optimistic. In some narrow sense, it might be true that this is all that is required to follow the basic logic of the mathematical development, but to really understand the text, I believe some background in differential geometry, forms and Lie groups -- preferably acquired from math books written by mathematicians -- is required.

As I said, despite the caveats immediately above, I found the book both illuminating and enjoyable to read. In fact, I return to it quite often to refresh my memory on various topics.
15 of 15 people found the following review helpful
5.0 out of 5 stars A Great Introduction to Diff. Geometry 7 Aug. 2000
By Glen Aultman-Bettridge - Published on Amazon.com
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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