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by Timothy Poston,C. T. J. Dodson

Download Tensor Geometry: The Geometric Viewpoint and its Uses (Graduate Texts in Mathematics) fb2
Author: Timothy Poston,C. T. J. Dodson
ISBN: 354052018X
Language: English
Pages: 434 pages
Category: Mathematics
Publisher: Springer; 2nd edition (November 15, 2009)
Rating: 4.6
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FB2 size: 1496 kb | EPUB size: 1632 kb | DJVU size: 1404 kb
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Graduate Texts in Mathematics Bibliographic Information. The Geometric Viewpoint and its Uses.

Graduate Texts in Mathematics. November 1990 Kit Dodson Toronto, Canada Tim Poston Pohang, Korea Contents Introduction. XI O. Fundamental Not(at)ions . 64 Basic geometry and examples, Lorentz geometry 2. Maps. 76 Isometries, orthogonal projections and complements, adjoints 3. Coordinates . Bibliographic Information. Christopher T. J. Dodson.

T. Dodson, Timothy Poston

T. Dodson, Timothy Poston.

The emphasis throughout is on the geometry of the mathematics, which is greatly enhanced by the many illustrations presenting figures of three and more dimensions as closely as book form will allow

T. Series: Graduate Texts in Mathematics, 130. File: PDF, 1. 6 MB. Читать онлайн. Categories: Mathematics\Geometry and Topology. We have been very encouraged by the reactions of students and teachers using our book over the past ten years and so this is a complete retype in TEX, with corrections of known errors and the addition of a supplementary bibliography. Thanks are due to the Springer staff in Heidelberg for their enthusiastic sup port and to the typist, Armin Kollner for the excellence of the final result. Once again, it has been achieved with the authors in yet two other countries.

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The emphasis throughout is on the geometry of the mathematics, which is greatly enhanced by the many illustrations presenting figures of three and more dimensions as closely as book form will allow

This treatment of differential geometry and the mathematics required for general relativity makes the subject accessible, for the first time, to anyone familiar with elementary calculus in one variable and with some knowledge of vector algebra. The emphasis throughout is on the geometry of the mathematics, which is greatly enhanced by the many illustrations presenting figures of three and more dimensions as closely as the book form will allow.

Comments (7)
This is one of those books that is very pleasing to the eye of a mathematical physicist, but lacks a facility of mechanical derivation and simplicity of formalism involved to present the essence of the theory, which here really is general Relativity. A more appropriate title would have involved relativity, perhaps Relativity for Mathematical Physicists. or Relativity for Mathematicians, as it is very clear from the beginning the text is going to be tied in with it. At the end of the book, he gives an exposition of the theory, however the formalism involved seems to be unnecessary and not very motivated as to how much it deviates from standard presentations involving tensor calculus and Levi-Civita. It seems he avoids the traditional tensor calculus in favor of the mathematician's modern formulation of differential geometry to appease that audience. But as a mathematician myself, I do not favor unnecessary abstraction unless it is necessary and simplifies the theory. In this case, it adds extra weight and time to the book. I would prefer to read a the relevant portions of Kreyszig, and then develop the theory of GR. Even Novikov and Dubrovin in Modern Geometry, though less discussion on the physical side of things, mathematically are much more efficient in presenting all kinds of mathematics useful not just to GR, but even inclusive of Yang-Mills and Jacobi Variational theory. I think there is valuable insight in this book, however if you are looking for efficiency in a text, I do not suggest this one.
The authors of this excellent text include a memorable passage in the Introduction that perfectly captures the purpose and primary strength of the book:

"The title of this book is misleading. Any possible title would mislead somebody. 'Tensor Analysis' suggests to a mathematician an ungeometric, manipulative debauch of indices, with tensors ill-defined as 'quantities that transform according to' unspeakable formulae. 'Differential Geometry' would leave many a physicist unaware that the book is about matters with which he is very much concerned. We hope that 'Tensor Geometry' will at least lure both groups to look more closely."

Dodson and Poston's text is a welcome entry in that all-too-small class of books that attempt to bridge the conceptual gulf that separates mathematicians from physicists when they write about differential geometry and general relativity. Modern mathematical treatments of both Riemannian and Lorentzian geometry are typically written primarily in concise and conceptually rich coordinate-free notation; physicists, in sharp contrast, tend to write almost exclusively in a notation that stresses the use of local coordinate systems and index manipulation. A person who is educated in one of these traditions must apply himself with diligence to become proficient in the other; however, this "bilingual" proficiency is surely necessary for the serious students of general relativity, who must study literature written in both styles.

Dodson and Poston's book provides an accessible introduction to the mathematics of general relativity, and it should be particularly useful to both mathematicians and physicists as they develop their abilities to read and write in both coordinate-free and index-based notations. The book is written at a level that should make it accessible to anyone who has studied multi-variable calculus and linear algebra. It is not a complete introduction to either modern differential geometry or general relativity, nor do the authors claim that it is. After all, Spivak devoted five volumes to Riemannian geometry alone and still failed to provide an exhaustive introduction; the subject is enormous in scope.

Mathematicians who find this book helpful in their studies of general relativity might consider looking into the following books, each of which is written in the same mathematical style: (1) Gravitational Curvature by Theodore Frankel (offers a beautiful derivation of the Raychaudhuri Equation); (2) Manifolds, Tensor Analysis and Applications by Abraham, Marsden and Ratiu (Chapters 6, 7 and 8 offer an exceptionally lucid introduction to differential forms, integration on manifolds, the Hodge star operator, the codifferential, and applications of these materials to physics); (3) The Geometry of Kerr Black Holes by Barrett O'Neill (if you want to UNDERSTAND the use of the Weyl curvature tensor in defining the Petrov Type of a spacetime, then read Chapter 5 of this wonderful book); (4) Semi-Riemannian Geometry with Applications to Relativity by Barrett O'Neill (makes an excellent companion text to Dodson and Poston as a mathematically rigorous introduction to GR); (5) General Relativity for Mathematicians by Rainer Sachs and Hung-Hsi Wu (a masterpiece, difficult to find today but worth the effort). For a more far-ranging treatment of geometry with applications beyond GR, Theodore Frankel's The Geometry of Physics is also highly recommended.

After one has used Dodson and Poston and some of these other references as a sort of "Rosetta Stone," then one can become reasonably proficient in deciphering both coordinate-free and coordinate-based literature and translating one into the other. It is sad that the educational process is necessarily so inefficient, but we must be grateful for books like Dodson and Poston's that help us in the endeavor.
Considering the fact that I am a high school student and I had no problem understanding the majority of this text, I would call this book wonderful! It is a book that no student of general relativity or differential geometry can do without. It develops insightful geometric premises early on so that the whole picture of tensors can be absorbed, not just those "definitive" transformation equations. The book is not dry like most math books either. It contains almost witty sections as well as enlightening mathematical ideas. I would recommend this book to anyone studying tensors on their own, or as a supplementary text book for a class.
I agree with previous reviewers, and only wish to add a few comments: 1. This book assumes very little on the part of the reader, which makes it ideal for beginners, as long as they're mature readers. 2. Like many books out there, everything in this book is real and finite dimensional, which is a bit disappointing. 3. It's not as advanced as the writers or reviewers would like to think. For instance, no differential forms, no killing vectors, and although there's a chapter on lie groups it treats only their geometrical aspects and not the algebraic ones. 4. However, it contains two (extensive) chapters on SR and GR which are pure gold, I say! Everything is done from the geometrical point of view, and only AFTER all of the math has been introduced, so the discussion is mature and elegant. In short, this is a good book to read for the geometrical intuition but don't count on it to explain everything about differential geometry. Enjoy!
I found learning GR very frustrating, as I am a mathematician by training and found the woolly "picture the vector" approach of say Graviatation (Misner et al) very loose. it drove me crazy. When I found this I was pleased, and found thye rigourous footing I wanted for the subject. Bundles! After this I could read Gravitation with some comfort.
NOT repeat NOT for anyone but mathematically trained readers.
This book is excellent, but I can't understand that this second edition contains so many "typographical" errors! So, the novice reader will have to consult reference material to make sure he doesn't overlook something, which somehow defeats the purpose of the book.
Besides providing in clear-cut fashion the mathematics essential to research in cosmology, the authors simplify many concepts the physicists make opaque.