This book is intended to provide a working knowledge of those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms that are essential for a deeper understanding of both classical and modern physics and engineering. Included are discussions of analytical and fluid dynamics, electromagnetism, thermodynamics, the deformation tensors of elasticity, soap films, special and general relativity, the Dirac operator and spinors, and gauge fields, including Yang-Mills, the Aharonov-Bohm effect, Berry phase, and instanton winding numbers. Before discussing abstract notions of differential geometry, geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space; consequently, the book should also be of interest to mathematics students. This book will be useful to graduate and advanced undergraduate students of physics, engineering and mathematics. It can be used as a course text or for self study.
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' … extremely helpful for students in physics and engineering … recommended to a wide audience …' European Mathematical Society
'The layout, the typography and the illustrations of this advanced textbook on modern mathematical methods are all very impressive and so are the topics covered in the text.' Zentralblatt für Mathematik und ihre Grenzgebiete
'The layout, the typography and the illustrations of this advanced textbook on modern mathematical methods are all very impressive and so are the topics covered in the text.' Zentralblatt für Mathematik und ihre Grenzgebiete
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This book is intended to provide a working knowledge of those parts of geometry that are essential for a deeper understanding of both classical and modern physics and engineering. This book will be useful to graduate and advanced undergraduate students of physics, engineering and mathematics.
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Die hilfreichsten Kundenrezensionen
13 von 13 Kunden fanden die folgende Rezension hilfreich
Format:Taschenbuch
This book introduces the methods of modern differential geometry and its uses in theoretical physics. The only prerequisites are a good working knowledge of multivariable calculus and linear algebra. The book is very much written for a physics audience(i.e. the book is actually READABLE unlike so many graduate texts in mathematics, and there is an emphasis in actually learning how to CALCULATE things, rather than just staring weary eyed at mathematicians beloved polished proofs that only they can understand) There is an emphasis on physical understanding of the mathematical structures and not too many proofs. Proving things is not a bad thing, but Dr. Frankel seems to know when its most appropriate to do this, and doesn't get too bogged down in the proofs. There is a lot of material in this book (22 chapters) The book is broken into 3 main sections. The first section is on "Manifolds, Tensors, Exterior Forms" Differential forms are not that familiar to physicsts and this is a great place to learn about them. There is very nice section on how to relate Forms to vector Analysis in 3 space that physicists love dearly (see page 94). The second section is on "Geometry and Topology"-mainly Riemannian Geometry and Some Algebraic Topology like DeRham Cohomology, and the third is "Lie Groups, bundles, and Chern Forms". In this third section there is a Chapter on the Dirac equation, and its relation to Spin geometry. The only thing that the book is lacking is that there is no complex algebraic geometry (for aspiring string theorists). It would be nice if some day Dr. Frankel could write a book on this subject, since at this time none exist. I think that even mathematicians could learn a thing or two from this book. Most of differential geometry originated in Physics, not the reverse.
9 von 10 Kunden fanden die folgende Rezension hilfreich
Von Aaron Warren
Format:Taschenbuch
This book is ideal for a full year advanced undergrad or beginning grad course designed as an intro to theoretical and mathematical physics. Frankel writes clearly and is very careful not to slip anything past the reader - every statement is fully justified by text.
The biggest problem with this book is that in trying to justify everything, Frankel often forgets the big picture while focusing on details. The focusing on details is great, but he isn't explicit in formulating where these details will be used and how they are incorporated into the global structure of the theory.
This means that anyone teaching from this book should not be shy about slashing out extraneous material and coordinating developments in this book with other texts, such as Goeckler & Shuker, or Nakahara. These books act as good surveys of the mathematics behind the physics and better illuminate how all the details tie together to form a coherent theory.
5.0 von 5 Sternen
A modern and complete introduction to differential geometry for physicists
31. März 2012
Von M.M.
Format:Taschenbuch|Von Amazon bestätigter Kauf
I owe the author of this book my first real understanding of contemporary differential geometry, on a basic level. The book of T. Frankel covers all the mainstream topics a theoretical physicist will be interested in: manifolds, Riemannian geometry, a bit of algebraic topology, Lie groups and classical Yang-Mills theory.
I started my study of differential geometry with the classical coordinate-based approach as layed out in the great book of Lovelock and Rund. This gave me all the skills necessary for manipulating tensor fields, covariant derivatives, Riemannian tensor and the like, but I had no coherent understanding of the mathematical theory behind theses objects. The latter was fully delivered by Frankels book.
Frankels book has a presentation style which uses all the spectrum between chatty motivation up to the rigorous mathematical theory corset. If someone finds the book lacking direction, I would recommend to use a second book in parallel which develops the topics quicker or in the more terse mathematical style. I did so with success and consulted the book of Hou and Hou on differential geometry, which has an opposite style of presentation and covers many more topics. But the quality of Frankel for self study is unsurpassed. In Frankel I liked the introduction to Cartans notation of tensor-valued forms a lot, as it is not found everywhere. The third edition devotes a separate chapter for this notation. I would consider Frankels book to be even better if a few peripheral topics were omitted as the reader sometimes must decide where to focus most and some previous familiarity with the material will be helpful.
In summary: Frankels book on differential geometry is modern, very complete, accessible and written with pedagogical carefulness.
I started my study of differential geometry with the classical coordinate-based approach as layed out in the great book of Lovelock and Rund. This gave me all the skills necessary for manipulating tensor fields, covariant derivatives, Riemannian tensor and the like, but I had no coherent understanding of the mathematical theory behind theses objects. The latter was fully delivered by Frankels book.
Frankels book has a presentation style which uses all the spectrum between chatty motivation up to the rigorous mathematical theory corset. If someone finds the book lacking direction, I would recommend to use a second book in parallel which develops the topics quicker or in the more terse mathematical style. I did so with success and consulted the book of Hou and Hou on differential geometry, which has an opposite style of presentation and covers many more topics. But the quality of Frankel for self study is unsurpassed. In Frankel I liked the introduction to Cartans notation of tensor-valued forms a lot, as it is not found everywhere. The third edition devotes a separate chapter for this notation. I would consider Frankels book to be even better if a few peripheral topics were omitted as the reader sometimes must decide where to focus most and some previous familiarity with the material will be helpful.
In summary: Frankels book on differential geometry is modern, very complete, accessible and written with pedagogical carefulness.
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5.0 von 5 Sternen
Great collection of almost everything
I've been reading this book more or less for fun for the last 9 months, mostly while taking the bus to the university. Lesen Sie weiter...
Veröffentlicht am 16. September 2010 von David
4.0 von 5 Sternen
Excellent book covering a variety of topics
I used this book primarily to study Exterior Calculus, so I can not comment on some of the more esoteric topics discussed on the later chapters. Lesen Sie weiter...
Veröffentlicht am 5. März 2010 von Orestis Vantzos
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