The Geometry of Physics: An Introduction, Second Edition

Book Description

Theodore Frankel explains those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms essential to a better understanding of classical and modern physics and engineering. Key highlights of his new edition are the inclusion of three new appendices that cover symmetries, quarks, and meson masses; representations and hyperelastic bodies; and orbits and Morse-Bott Theory in compact Lie groups. Geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space. First Edition Hb (1997): 0-521-38334-X First Edition Pb (1999): 0-521-38753-1

Customer Reviews:

Fantastic - for the scientist.......2007-07-18

A very good book: buy it. But only if you are a scientist or student of physics/mathematics. This is not popular-science-common-public level.

a book worth keeping.......2007-05-01

This book can be quite confusing if you start without any background on the idea of manifold or knows nothing about general relativity. However, it does have strong points:

1. The notation is very up-to-date, and is entirely coordinate-independant approach.

2. The author explains in great details of formulation of modern differential geometry, and the details are comparatively lacking in other reference books.

3. The author never hesitate to use graphs and diagrams to illustrate points, and stroke nice balance in between mathematics rigor and physical insight.

Although it appears quite verbose at some point, it is mainly because differential geometry is such a heavy subject. Another book nice to have as companion reading is Goldburg's "Tensor analysis on Manifold", a terse, well-written text book.

Phenomenal .......2006-11-13

I just finished reading this book and I found it phenomenal. The physical ideas are made very clear in a natural mathematical framework.

You should buy this, despite its flaws.......2006-03-03

The other reviews on this page give this book anywhere from 1 to 5 stars, and they are all correct in their own way. The book is inspired, deep and full of physics applications and insights. On the other hand, it skims over mathematical rigor to a large degree and focuses more on defining things, getting a feel for them and moving on to application.

My advice: buy the book for its strengths, and read other books in parallel if you need more rigor. But still, buy it.

Also, things can be confusing on the first two or three reads, but keep at it and you will be glad you did.

The perfect first book in differential geometry.......2005-01-28

Differential geometry can be a very intimidating subject due to its heavy formalism. There are complete books (such as Kobayashi& Nomizu) very good as reference books, and there very few books that show the reader the picture behind the formulas.

This is one such book. It tells you the intuition behind each construction and from this point of view it has many things in common with Arnold's famous book on Math. Methods in Classical Mechanics. But where as Arnold does not pay too much attention to formalism, this book achieves this task as well. It shows the reader how to do those impossible computations as well.

This is definitely the first place to look at if you want to really learn differential geometry. If it seems difficult it is only because the subject is so.

Average customer rating:

Excellent
Essential for grad students in geometry/topology

Spin Geometry. (PMS-38)
H. Blaine Lawson , and Marie-Louise Michelsohn
Manufacturer: Princeton University Press
ProductGroup: Book
Binding: Hardcover

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ASIN: 0691085420

Book Description

This book offers a systematic and comprehensive presentation of the concepts of a spin manifold, spinor fields, Dirac operators, and A-genera, which, over the last two decades, have come to play a significant role in many areas of modern mathematics. Since the deeper applications of these ideas require various general forms of the Atiyah-Singer Index Theorem, the theorems and their proofs, together with all prerequisite material, are examined here in detail. The exposition is richly embroidered with examples and applications to a wide spectrum of problems in differential geometry, topology, and mathematical physics. The authors consistently use Clifford algebras and their representations in this exposition. Clifford multiplication and Dirac operator identities are even used in place of the standard tensor calculus. This unique approach unifies all the standard elliptic operators in geometry and brings fresh insights into curvature calculations. The fundamental relationships of Clifford modules to such topics as the theory of Lie groups, K-theory, KR-theory, and Bott Periodicity also receive careful consideration. A special feature of this book is the development of the theory of Cl-linear elliptic operators and the associated index theorem, which connects certain subtle spin-corbordism invariants to classical questions in geometry and has led to some of the most profound relations known between the curvature and topology of manifolds.

Customer Reviews:

Excellent.......2001-12-22

Who would have known that the equation discovered by P.A.M. Dirac in the 1920's would have the enormous appllications to mathematics that it currently has. This book is an excellent overview of these applications, written by two individuals who are responsible for the development of many of these. Dirac's theory of course had its origins in physics, and physicists, particularly those working in high energy physics, will find this book interesting and helpful.

The authors give a brief introduction and then move on to the representation theory of Clifford algebras and spin groups in chapter 1. The reader can see the origin of Clifford algebras and an introduction to the Pin and Spin groups. Clifford algebras are classified as matrix algebras over the real or complex numbers, and the quaternions. It is the representation theory of Clifford algebras however that has resulted in the impressive results outlined in the book Noting that the tensor product of Clifford algebras is not necessarily a Clifford algebra, the authors introduce a Z(2)-grading on a Clifford algebra, which results in a multiplicative structure in the representations of Clifford algebras. The Lie algebras of the Pin and Spin groups are discussed along with applications to geometry and Lie groups. By far the most interesting discussion though is on K-theory, which allows one to define a ring structure on vector bundles. Distinguishing a base point in the base space, relative K-groups are defined, and shown to be equal for the base space and its i-fold suspension. Bott periodicity results are stated but their proof is delayed until chapter 3. A detailed discussion is given of the Atiyah-Bott-Shapiro isomorphism and KR-theory.

The connection between spin and differential geometry is discussed in chapter 2. The first few sections is a review of standard results in the spin structure of vector bundles, such as Stiefel-Whitney classes and spin cobordism. For Riemannian vector bundles, each fiber has a quadratic form that gives rise to a Clifford algebra on the fiber. The question as to when a vector bundle over the Riemannian base space can be found that has fibers each an irreducible module over this Clifford algebra leads to a consideration of spin manifolds and spin cobordism, when the total space is chosen to be the tangent bundle. The Dirac operator acting on a bundle over this Clifford bundle allows the construction of all the standard elliptic operators such as the signature, Atiyah-Singer, and the Euler characteristic. The authors discuss these constructions in detail along with the notion of of Cl(k)-linear operators.

The Dirac operator can be viewed in Euclidean space as the square root of a Laplace operator, but over general manifolds it is the Laplacian with a correction term dependent on the curvature and Clifford multiplication. The Bochner vanishing theorems are discussed in great detail, along with the results on the existence of exotic spheres.

An entire chapter is spent on index theorems, wherein the authors present the results in terms of the approach used by Atiyah and Singer, instead of the heat kernel methods of Gilkey and Patodi. Physicists might prefer the later approach, due to its connections with applications, but the abstract K-theory approach undertaken by the authors is elegant and their presentation is excellent. The role of physics in index theorems is a fascinating one though, especially the use of supersymmetry to simplify the proofs of some of the results. The authors do not discuss this approach, but point out, interestingly, that it does not work when one is dealing with torsion elements in K-theory. These cannot be detected using cohomology nor can the modulo-two invariants appearing in the index theorems be computed from local densities.

The last chapter is a long one and discusses applications in differential topology and geometry, emphasizing index thoerems and Riemannian manifolds of positive scalar curvature. The authors outline just when the indexes are integers (the integrality theorems) and use spin geometry to discuss the immersion problem for manifolds and the vector field problem. Exotic n-spheres again make their appearance, wherein it is shown that some of these have very few symmetries and are very asymmetric objects. A short introduction to elliptic genera is given. Interestingly, C*-algebras are briefly mentioned as tools to decide whether for every compact spin manifold with positive scalar curvature all higher A-genera must be zero. Spin-c manifolds are not treated, the authors instead concentrating their attention to Kahlerian geometry. In this context the Clifford algebra multiplication has a beautiful relationship with the complex structure. A brief discussion is given of the pure spinors of Cartan and twistor spaces. The theory of holonomy and calibrations, the later due to one of the authors, is discussed in great detail. The discussion begins in the consideration of when universal covering spaces are not Riemannian manifolds and their holonomy groups have been classified. The idea of a calibration arises from the consideration of submanifolds that are homologically volume-minimizing. These become calibrations when the integrals of p-forms on them are the volumes, and these p-forms have vanishing differentials on oriented tangent p-planes on the manifold. The authors give an interesting discussion of the relation between spinors and calibrations.

Essential for grad students in geometry/topology.......1998-12-23

As a graduate student in mathematics I survived on this encyclopedic work. Anyone interested in differential geometry or differential topology will eventually need something in this book.

Prerequisites are graduate-level algebra and analysis, and some topology and differential geometry. He introduces the subject of pseudodifferential operators and Sobolev spaces, but it's easy to get lost in that part unless you first read Shubin's book "Pseudodifferential operators and Spectral theory". Also, the quick shuffling of Lie group information can be disheartening if you're not used to it. Harvey's book "Spinors and Calibrations" is a more elementary book if this is the case.

Book Description

This book provides the first lucid and accessible account to the modern study of the geometry of four-manifolds. It has become required reading for postgraduates and research workers whose research touches on this topic. Pre-requisites are a firm grounding in differential topology, and geometry as may be gained from the first year of a graduate course. The subject matter of this book is the most significant breakthrough in mathematics of the last fifty years, and Professor Donaldson won a Fields medal for his work in the area. The authors start from the standpoint that the fundamental group and intersection form of a four-manifold provides information about its homology and characteristic classes, but little of its differential topology. It turns out that the classification up to diffeomorphism of four-manifolds is very different from the classification of unimodular forms and that the study of this question leads naturally to the new Donaldson invariants of four-manifolds. A central theme of this book is that the appropriate geometrical tools for investigating these questions come from mathematical physics: the Yang-Mills theory and anti-self dual connections over four-manifolds. One of the many consquences of this theory is that 'exotic' smooth manifolds exist which are homeomorphic but not diffeomorphic to (4, and that large classes of forms cannot be realized as intersection forms whereas distinct manifolds may share the same form. These result have had far-reaching consequences in algebraic geometry, topology, and mathematical physics, and will continue to be a mainspring of mathematical research for years to come.

Customer Reviews:

An excellent summary of Donaldson theory.......2000-06-16

This book brings together the brilliant work Donaldson did at Oxford during the early 1980s. The unique properties of 4-manifolds are clearly and concisely written out with concentration on explaining field theories like Yang-Mills and gauge theory with a truly firm mathematical foundation, presented in a book for the first time. A great companion for any researcher in the field of geometry and topology, or even loop quantum gravity!

Protein Geometry, Classification, Topology and Symmetry: A Computational Analysis of Structure (Series in Biophysics)

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Protein Geometry, Classification, Topology and Symmetry: A Computational Analysis of Structure (Series in Biophysics)
William R. Taylor , and Andras Aszodi
Manufacturer: Taylor & Francis
ProductGroup: Book
Binding: Hardcover

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ASIN: 0750309857

Book Description

Using a geometric perspective, Protein Geometry, Classification, Topology, and Symmetry reviews and analyzes the structural principals of proteins with the goal of revealing the underlying regularities in their construction. It also reviews computer methods for structure analysis and the automatic comparison and classification of these structures with an analysis of the statistical significance of comparing different shapes. Following an analysis of the current state of protein classification, the authors explore more abstract geometric and topological representations, including the occurrence of knotted topologies. The book concludes with a consideration of the origin of higher-level symmetries in protein structure. The authors focus on simple geometric methods that are deterministic rather than probabilistic and on the more abstract simplifications of protein structure that allow a better understanding of the overall fold of the structure. Most of the methods described in this book have corresponding computer programs. These can be found (as C source code) at the ftp site of the Division of Mathematical Biology at the National Institute for Medical Research. This collection of ideas contains pedagogical material that make it ideal for post-graduate courses as well as new ideas and results essential for researchers investigating protein structures.

Geometry, Topology and Physics, Second Edition (Graduate Student Series in Physics)

Average customer rating:

Geometry Topology and Physics: A condesed view
An excellent book
A great reference book.
Great book.
Flat spheres and more

Geometry, Topology and Physics, Second Edition (Graduate Student Series in Physics)
M. Nakahara
Manufacturer: Taylor & Francis
ProductGroup: Book
Binding: Paperback

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ASIN: 0750306068

Book Description

Differential geometry and topology have become essential tools for many theoretical physicists. In particular, they are indispensable in theoretical studies of condensed matter physics, gravity, and particle physics. Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields. The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the development of the subject. The book features a considerably expanded first chapter, reviewing aspects of path integral quantization and gauge theories. Chapter 2 introduces the mathematical concepts of maps, vector spaces, and topology. The following chapters focus on more elaborate concepts in geometry and topology and discuss the application of these concepts to liquid crystals, superfluid helium, general relativity, and bosonic string theory. Later chapters unify geometry and topology, exploring fiber bundles, characteristic classes, and index theorems. New to this second edition is the proof of the index theorem in terms of supersymmetric quantum mechanics. The final two chapters are devoted to the most fascinating applications of geometry and topology in contemporary physics, namely the study of anomalies in gauge field theories and the analysis of Polakov's bosonic string theory from the geometrical point of view. Geometry, Topology and Physics, Second Edition is an ideal introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics.

Customer Reviews:

Geometry Topology and Physics: A condesed view.......2007-05-13

This book provide a complete and useful review of geometrical instuments of mathematical physics from the beginnig to the most advanced topics of interest. It can be used by students at the beginnig of thei studies in this topics, and it's found to be a useful gallery for higher level students (or scholar).

An excellent book.......2006-09-19

This is the best book of its type, that is, a book that contains almost all if not all the advance mathematics a theoretical physicist should know. I have studied chapters 2-9 and it has the perfect balance between rigorous presentation of topics and practical uses with examples. The level is for advance graduate students. The range of topics covered is wide including Topology topics like Homotopy, Homology, Cohomology theory and others like Manifolds, Riemannian Geometry, Complex Manifolds, Fibre Bundles and Characteristics Classes. I believe this book gives you a solid base in the modern mathematics that are being used among the physicists and mathematicians that you certainly may need to know and from where you will be in a position to further extent (if you wish) into more technical advanced mathematical books on specific topics, also it is self contained and brings lots of exercises that help learn the concepts presented, my advice, get it is a superb book!

A great reference book........2006-06-19

No doubt, the interplay of topology and physics has stimulated phenomenal research and breakthroughs in mathematics and physics alike.

Unfortunately, there is so much mathematics to master that the average graduate physics student is left bewildered.....until now.

The text is an excellent reference book. I emphasize reference. The book presupposes an acquaintance with basic undergraduate mathematics including linear algebra and vector analysis.

The author covers a wide range of topics from tensor analysis on manifolds to topology, fundamental groups, complex manifolds, differential geometry, fibre bundles etc.

The exposition in necessarily brief but the main theorems and IDEAS of each topic are presented with specific applications to physics. For example the use of differential geometry in general relativity and the use of principal bundles in gauge theories, etc.

Unfortunately, there are very few exercises necessitating the use of supplementary texts. However, to the author's credit appropriate supplementary texts are provided. The author goes to great lengths to show which texts inspired the chapters and follows the same line of presentation.

Perhaps the greatest attribute of the text is to take disparate branches of mathematics and coallate them under one text with applications to physics. In doing so one gains a better grasp of how the fields of mathematics interact in the domain of physics.

Great book........2005-03-26

This is a very useful book for understanding modern physics. You absolutely need such a book to really understand general relativity, string theory etc. For instance, Wald's book on general relativity will make much more sense once you go through Nakahara's book. It is very complete, clearly written, comprehensive and easy to read. I would also recommend Morita's "Geometry of differential forms' and Dubrovin,Novikov and Fomeko's 3 volume monograph, if you can find it. All in all, Nakahara's book is one of the best buys, precious book.

Flat spheres and more.......2004-01-21

Highly stimulating and extremely hard to read, written for mathematicians in physics. However, the chapter on Riemannian Geometry can be worked through, up to a point, without any knowledge of exterior differential forms, and is notable if for only one fact alone: a simple calculation is provided that explains explicitly that spheres in four and eight dimensions (3-spheres and 7-spheres) are flat with torsion! I don't know another reference that a physicist without special background in math can consult to understand this highly nonintuitive fact.

Geometry, Relativity and the Fourth Dimension

Average customer rating:

With few exceptions, it is a readable, stepwise explanation of how the universe is structured
The best book ever in its field
explain dimensions very well
See what's outside the box
excellent book, fascinating author, start your exploration! :)

Geometry, Relativity and the Fourth Dimension
Rudolf Rucker
Manufacturer: Dover Publications
ProductGroup: Book
Binding: Paperback

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ASIN: 0486234002

Book Description

Exposition of 4th dimension, concepts of relativity as Flatland characters continue adventures. Popular, easily followed yet accurate, profound. Topics include curved space time as a higher dimension, special relativity, and shape of space-time. Accessible to lay readers but also of interest to specialists. Includes 141 illustrations.

Customer Reviews:

With few exceptions, it is a readable, stepwise explanation of how the universe is structured.......2007-06-29

To understand relativity, it is necessary to understand geometry, specifically how a straight line can be curved. For nearly everyone, any attempt to understand four-dimensional space begins with understanding how a three-dimensional creature would appear to a two-dimensional one. One of the earliest and still the greatest of all introductions to going up a dimension is "Flatland" by Edwin A. Abbott. Quite naturally and sensibly, Rucker starts with Abbott's rendition of the properties of Flatland.
Rucker then moves on to the idea of curved space, where the shortest distance between two points is a "straight line", which is curved by the properties of the space. The space that we occupy is curved by the presence of matter, as Einstein claimed in his relativity theories. Furthermore, movement causes shrinkage in the direction of the movement and the slowing of time, which causes time to become just another dimension of space. As counterintuitive as this may appear, Einstein's relativity theory has been verified over and over again to a large number of significant figures.
One of the best things about this book is that Rucker has included problems at the end of each chapter. These problems reinforce the concepts of the chapter; it is unfortunate that no solutions were included.
In this book, Rucker steps the reader through all of the background material necessary to understand relativity and four-dimensional space. With few exceptions, the accounts are understandable to anyone with an understanding of college algebra.

The best book ever in its field.......2007-04-19

This book has presented the most difficult topics of our world with the easiest words. After reading this book many of my questions that I had in my mind for a long time were answered. It's worth thousands more than its price.
Congratulation to Mr. Rudolf Rucker for his great book.

explain dimensions very well.......2007-03-31

it is published years before but it is almost new for today and it explain dimensions and shape of space well and clearly .thanx to amazon for sending me timely.

See what's outside the box.......2007-03-30

Over two millenia ago, Euclid wrote his masterpiece Elements and stated in his fifth postulate that only one perpendicular line could pass through any one point adjacent to another line.

One hundred fifty years ago, it was proven that yet another geometry could be described by asserting that more than one parallel line could pass through such a point.

Building on these ideas, Rucker briefly yet thoroughly surveys the relevant mathematics outside the box of Euclidian geometry.

It's a fascinating place too because it involves considerations of hyperspace, four dimensional travels and ultimately Einstein's theory of relativity.

Copiously filled with illustrations to help drive home his points, Rucker has produced a book that meaningful helps one visualize and better understand the fourth dimension.

This book is an excellent read along with Choas, Coincidences and All that Math Jazz, The Fourth Dimension Simply Explained, Einstein's own Relativity and Hyperspace by Michio Kaku which discusses all these ideas as well as contemporary string theory (which purports to pull it all together).

excellent book, fascinating author, start your exploration! :).......2005-12-31

Mr. Rucker is a 'genius educator' in my opinion :) he can open your mind and get you started - no matter what direction you wanna take! :) whether it be philosophy, math, physics, or even spirtual things - Mr. Rucker can get you going! to me, he is one of the great men of these modern times :) ah, i remember! pay particular attention to visualizing hyper-dimensional objects .. it can be done! good luck and may god bless all of you! :)

Average customer rating:

A below average text book
Sometimes annoying but instructive
A bad book for 7 th graders like me
Opinión general
Good book for applications of fractal geometry, but....

Fractals Everywhere
Michael F. Barnsley
Manufacturer: Morgan Kaufmann
ProductGroup: Book
Binding: Paperback

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ASIN: 0120790696

Book Description

This volume is the second edition of the highly successful Fractals Everywhere. The Focus of this text is how fractal geometry can be used to model real objects in the physical world.

This edition of Fractals Everywhere is the most up-to-date fractal textbook available today.

Fractals Everywhere may be supplemented by Michael F. Barnsley's Desktop Fractal Design System (version 2.0) with IBM for Macintosh software. The Desktop Fractal Design System 2.0 is a tool for designing Iterated Function Systems codes and fractal images, and makes an excellent supplement to a course on fractal geometry

* A new chapter on recurrent iterated function systems, including vector recurrent iterated function systems.
* Problems and tools emphasizing fractal applciations.
* An all-new answer key to problems in the text, with solutions and hints.

Customer Reviews:

A below average text book.......2006-11-13

Barnsley's book has a shortcoming common to many math text books -- it's poorly written. Barnsely's writing style is superfluous and rambling. What I learned from this book was in spite of Barnsley's writing, not because of it.

Furthermore, the book's illustrations are substandard. There are over five different fonts used in illustrations (including hand written text). This leads to confusion when you're unsure if the text in Barnsley's illustrations is referring to Greek letters or the conventional alphabet. Another shortcoming is that Barnsley intermingles end of chapter exercises with new concepts. You may have problems 1.1 to 1.5 reviewing what you've already learned, and then problem 1.6 introduces completely new material. This is a problem throughout the book, as important concepts are introduced in exercises or otherwise illogical locations.

On the positive side, solutions to most exercises are presented at the end of the book. Overall the book was useful, but learning the material was unnecessarily difficult due to the book's shortcomings.

Sometimes annoying but instructive.......2002-08-27

Although instructive, this book is sometimes annoying to read. The author seems to be playing his cards very close to the vest and not telling us everything.

For instance, there is little or no instruction on how to implement the IFS attractors presented as a panacea for data compression. This seems to be proprietary to his company. It also seems that hands-on manipulation is crucial to the images' production, contrary to the author's claims.

If you can understand the mathematics you may find the book useful, as I did when writing my book Fractals in MUsic.

A bad book for 7 th graders like me.......2001-11-28

this is a bad and very confusing book for a young student in, say... 7th grade, like me. The language is incomprehensible and there are no visual aids.

Opinión general.......2001-11-19

HUmmm!! parece interesante este librito. Pero la verdad busco uno donde encuentre aplicaciones a la ingeniería.
Estos libros de teoría suelen ponerse aburridos al no tener sufuciente información sobre aplicaciones.
De todos modos apenas lo tenga en las manos y lo mire doy una opinión más seria de este.

Good book for applications of fractal geometry, but...........2001-06-23

This is a good book on applications of fractals; the chapters on modeling natural objects with iterated function systems (IFS) and fractal interpolation are especially useful. Many standard topics are included, for example, fractal dimension, Julia and Mandelbrot sets, chaos and the shift dynamical system. Some of the illustrations are captivating.

However, the book is not well organized, and the writing is extremely wordy to the point of being irritating. Some paragraphs read as if they belonged to a "Dummies" handbook. Also, I have to agree with one reviewer that the treatment of fractal dimension is poor. For one thing, it does not fully develop the intuition behind the concept-- much less the math. This same remark holds for the chapter on chaotic dynamics.

Book Description

This elegant book by distinguished mathematician John Milnor, provides a clear and succinct introduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. The author presents proofs of Sard's theorem and the Hopf theorem.

Customer Reviews:

Exactly would it should be.......2005-12-30

I would suggest to use this book as a companion to more serious books on topology. Weighing in at a mere 51 pages, this book accomplishes what it needs to: a brief, succinct introduction to topology mostly based on the work of Brouwer. There is a nice mixture of topics, ranging from Sard's theorem to Poincare-Hopf theorem. The proofs and ideas are not fully rigorous or developed, but that would be quite a bit to expect from such a short exposition.

best math book ever written.......2005-02-11

Despite the lovely subject matter covered in this book, it more importanty gives one a taste of Mathematics as an intellectual discipline. It in outline shows how a mathematical theory - in this case Differential Topology - is constructed and consquently what mathematicians actually do and think about.
Anyone who would like to appreciate Mathematics as a field of study rather than just learn some math should open this book.

Better still, the prerequisite is only multivariate calculus!I have long thought this book should be the third year of calculus rather than differential equations or complex analysis.

Additionally, for the novice it is the only entry I know of into the mysteries of high dimensional geometry, that amazing almost unbelieveable accomplishment of the human mind.

There is a Star Trek episode in which a blind woman wears a dress of sensors which enable her to know more about her environment than a person can know from seeing. She knows exact distances and dimensions, can detect minute movements, can process the complete spectrum of light. In some sense she sees better. Modern topology and geometry are like that sensor dress for seeing higher dimensions. While we can not visualize the sphere in 5 dimensions, we know more about it from these mathematical theories than a five dimensionally sighted being ever could.

Today, mathematics is often considered to be just a practical tool - like a spread sheet - or a toaster oven. We forget its power to widen our imagination, to frame the unimaginable. This book reminds us of this and shows why Mathematics is the Queen of Sciences.

Compact and useful.......2005-01-29

This book packs a lot of interesting material into a small volume. E.g., I picked up another book recently that started talking about cobordisms right off the bat; despite my having a couple of shelves full of well-known Dover, Springer, Cambridge UP etc. books on topology, differential geometry, mathematical physics, etc., Milnor's tiny book was the only one I found that could help me understand what cobordisms are right away. The book also uses many illustrations to help understanding.

I demote this to 4 stars only because Princeton UP's price is a bit high; many years ago I was lucky enough to find a used copy of the old U. Virginia edition, and paid much less.

Yet another popular (YAP) Math text.......2004-12-01

In all practicality, for general math students this book is nice for the library, but by no means is it essential. In fact, it's not worth the $25 in most cases. There are so many outstanding Math texts / topics out there, it is doubtful and political that Milnor's Differential Viewpoint deserves its popularity.

I found this book helpful as supplementary reading for Calculus on Manifolds, so I am a minority student. (Majority student = Linear Algebra Done Right.) Spivak's book motivated the need to look carefully at the first few sections of Milnor's book. The definitions in Milnor coincide perfectly with Spivak.

I left Milnor's book with a good intuition about the inverse function theorem, manifolds, and the rank theorem. I also gave a small study to Sard's Theorem, but I had no need to venture into what apparently was the meat of the book...

It is arguable that the Inverse Function Theorem, Manifolds, and the Rank Theorem alone warrant buying this book, despite it only represents the first 2 sections of it, and is far from the total purpose of the book. Nonetheless, that's all I wanted to gain at the time I was reading it.

On the other hand, if you really are a Differential Topology student (small minority), you are the one who wouldn't need the review because you would know what you are trying to get from the book.

On the other hand, students who buy this book who don't know why will probably do nothing more than collect dust with it.

Take full advantage of the clear, encompassing exposition:.......2003-05-18

Do the exercises. Many were Ph.D. dissertation-level problems in the 1960s; today, they're aptly described as "elementary"- because Milnor MADE them elementary.

This book forms part of the toolkit you will need to fully explore the more modern work in dynamics, complexity, and applications (e.g., economics, physics).

Book Description

This unique book places emphasis on the understanding of the material presented by adopting a reflective approach towards the scientific method used. Knowledge of algebra, geometry and trigonometry is required however, the authors introduce more advanced mathematical methods in the context of the physical problems which are used for analysis. Modern physics topics, including quantum mechanics and relativity are introduced early and are integrated with more "classical" material from which they have evolved.

Customer Reviews:

Not a wasted word and that's good.......2005-05-07

This is one of those rare physics books where every word has the physical information of ten pages of american textbooks. I must confess, I DETEST american introductory physics textbooks such as the ones written by Giancoli, Serway or Freeman something. All those books have three things in common.

1. They're big and fat and spend about 50 pages per chapter.

2. They are written from a purely commercial viewpoint instead of trying to write a decent textbook. When was the last time you saw one of these titles that didn't come with a student solutions manual, study guide and a ton of other crap meant to make spend your annual bookbudget on a single title.

3. They all give answers to only odd numbered exercises which run into more than hundred. After spending all that money on a book the least they could do is give me the ability to check the answer after spending all that time and energy trying to solve it.

And this is where understanding phsyics comes to the rescue. It's thinner than all introductory physics textbooks you ever saw. The exercises never number more than 20 and each and every one of them is meant to teach you something useful instead of rehashing skills you learned in high school, which brings me my next point.
This textbook actually assumes you had physics in high school. It picks up right where you left at graduation and builds on it. In less then 740 pages it teaches you stuff most 1200 page textbooks never come by. Stuff you thought you would only get in your second or third year is explained in a way you can understand.

Best of all, this book is meant to fill the heads of students instead of the wallets of the authors. There are no student solutions manuals, study guides, or other useless nonsense meant only to line the pockets of the writers. All the answers to exercises are given.

Like one reviewer said this book only uses black blue and gray, which is a good thing. You're paying for all those pretty pictures, and in the end they don't advance your understanding anyway. Besides all those colours make the pages busy and difficult for me to concentrate.

To sum up, buy this book if you want to learn real physics instead of relearning what you learned in high school.

Not very pedagogic.......1999-12-02

I am an introductory student in physics and our techer told us to buy this book since he thought it was a very good book.

I don't agree.

The book contains a lot of interesting material, but has few examples which can tell you how to use the new knowledge. The examples in the book is either very easy or hard to understand beacuse they don't tell you how they came up with the equation they use so solve the problem.

Don't buy this book. Buy University Physics instead. Much better

relatively speaking.......1999-11-29

I am currently taking a course in which this text is our primary learning manual. Relative to physics textbooks in general, it went into great detail concerning fairly advanced topics. However, it did not do a good enough job of laying down the fundamental principles of physics, which makes it more difficult to comprehend the more involved topics. The book also lacks a glossary and a comprehensive index. The examples are either too complex or too simplistic, and I have been forced to use several other books as auxiliary materials. If you are a physics instructor, I implore you, please DO NOT USE this text in an introductory course, unless the primary goal of your class is to drive away students from the study of physics, ranting madly about.

Excellent book.......1999-11-26

A great book, I just love it!

An excellent book.......1999-11-24

This is a super book. I found it particularly useful in refreshing my memory!

Average customer rating:

This is where it started
Excellent introduction to fiber bundles
Still attractive.

The Topology of Fibre Bundles. (PMS-14)
Norman Steenrod
Manufacturer: Princeton University Press
ProductGroup: Book
Binding: Paperback

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ASIN: 0691005486

Book Description

Fibre bundles, now an integral part of differential geometry, are also of great importance in modern physics--such as in gauge theory. This book, a succinct introduction to the subject by renown mathematician Norman Steenrod, was the first to present the subject systematically.

It begins with a general introduction to bundles, including such topics as differentiable manifolds and covering spaces. The author then provides brief surveys of advanced topics, such as homotopy theory and cohomology theory, before using them to study further properties of fibre bundles. The result is a classic and timeless work of great utility that will appeal to serious mathematicians and theoretical physicists alike.

Customer Reviews:

This is where it started.......2002-06-26

For those individuals who want an in-depth, insightful, and solid understanding of fiber bundles this book must be read. In spite of its date of publication, it still is of considerable value in this regard. Modern treatments of fiber bundles are very formal and the underlying motivation gets swept away in the thirst for rigor. Fiber bundles are now ubiquitous in differential topology, algebraic topology, differential geometry, and algebraic geometry, and have also found a place in theoretical physics, thanks to the success of gauge field theories. Therefore a mastery of fiber bundles is essential for entering any of these fields. But fiber bundles are fascinating objects in and of themselves, and studying them for their own sake needs no apology.

The author does use some antiquated notation, but that is not really a hindrance to the study of the book. The reader will no doubt have some background in differential geometry and topology before attempting this book, so the appropriate translation to more modern notation should be straightforward. Once started, and with a little thought adjustment to the idiosyncracies of the author's writing style, the reader will find a plethora of neat examples and insights into the subject. In particular, part 3 on the cohomology theory of bundles is exceptionally valuable in that it gives the reader a detailed overview of the origin of what are not called Stiefel-Whitney classes. The theory of characteristic classes has of course advanced and matured extensively since this book first appeared, but all of the modern treatments are lacking in that they do not give the reader an appreciation of the fundamentals of the subject. Indeed, the construction of the obstruction to the construction of a cross-section to a bundle is the starting point for many of the ideas in obstruction theory that one finds in differential topology. And yes, the procedures the author uses can be "cleaned-up" and made more concise, but the price one pays in such an endeavor is the loss of an appreciation of the concepts behind the scene.

Since the book is a monograph, there are no exercises, and this is probably the only minus to the book. Also, some knowledge of the German language would be useful to a reader who has it, since the author makes references to papers written in German and much of the terminology in the book shows its roots in the German language. One good example of this is the Reidemeister theory of cohomology groups based on a bundle of coefficients, called Uberdeckung by Reidemeister.

There is no question as to why this book remains in print, and it will no doubt continue to be well into the 21st century. IT is a good example of the idea that something new may not be something better. After finishing it, the reader will be amply prepared to enter into the continually-evolving theory of fiber bundles and their applications, all of which are interesting and important.

Excellent introduction to fiber bundles.......2002-02-25

This book supplies a lot of intuition and background that more modern texts seem to assume of the reader. Steenrod's writing is meticulous and extremely clear. My opinion is that one can learn just as much out of this seemingly outdated text and probably even more than from the modern texts.

... True, more slick machinery has been developed since Steenrod's time, but those big machines are hardly transparent. Steenrod assumes very little of the reader; he even has a quick course in homotopy groups, although he assumes the reader knows the basics of homology/cohomology. Perhaps most importantly, since many of the ideas in the book were new at the time, he doesn't assume that the reader is already comfortable with those ideas. All together this makes a very accessible book indeed.

Still attractive........1999-11-22

A nostalgic but still attractive book on (homotoy theory of) fiber bundles. This book is not very accessible as it predates the development of modern machinery of algebraic topology, but is worth reading.

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