Geometrical Methods of Mathematical Physics [Paperback]

Bernard F. Schutz (Author)

Book Description

ISBN-10: 0521298873 | ISBN-13: 978-0521298872 | Publication Date: January 28, 1980 | Edition: 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.

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.

See all Editorial Reviews

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: 8.9 x 6 x 0.7 inches
• Shipping Weight: 14.9 ounces (View shipping rates and policies)
• Average Customer Review: 3.9 out of 5 stars  See all reviews (10 customer reviews)
• Amazon Best Sellers Rank: #595,064 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

38 of 39 people found the following review helpful:

4.0 out of 5 stars A Very Accessible Book ! Buy It !, November 6, 2000

By 

Andy Gregory (Cleveland England) - See all my reviews

This review is from: Geometrical Methods of Mathematical Physics (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.

25 of 26 people found the following review helpful:

5.0 out of 5 stars Introduction to Differential Geometry for physicists, July 2, 1999

By 

A graduate student of Physics (Santa Barbara, CA) - See all my reviews

This review is from: Geometrical Methods of Mathematical Physics (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.

14 of 14 people found the following review helpful:

5.0 out of 5 stars Terrific geometry book for physicists, May 6, 2006

By 

Dean Welch - See all my reviews
(REAL NAME)   

This review is from: Geometrical Methods of Mathematical Physics (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.