Three-Dimensional Geometry and Topology

Book Description

This book develops some of the extraordinary richness, beauty, and power of geometry in two and three dimensions, and the strong connection of geometry with topology. Hyperbolic geometry is the star. A strong effort has been made to convey not just denatured formal reasoning (definitions, theorems, and proofs), but a living feeling for the subject. There are many figures, examples, and exercises of varying difficulty.

This book was the origin of a grand scheme developed by Thurston that is now coming to fruition. In the 1920s and 1930s the mathematics of two-dimensional spaces was formalized. It was Thurston's goal to do the same for three-dimensional spaces. To do this, he had to establish the strong connection of geometry to topology--the study of qualitative questions about geometrical structures. The author created a new set of concepts, and the expression "Thurston-type geometry" has become a commonplace.

Three-Dimensional Geometry and Topology had its origins in the form of notes for a graduate course the author taught at Princeton University between 1978 and 1980. Thurston shared his notes, duplicating and sending them to whoever requested them. Eventually, the mailing list grew to more than one thousand names. The book is the culmination of two decades of research and has become the most important and influential text in the field. Its content also provided the methods needed to solve one of mathematics' oldest unsolved problems--the Poincaré Conjecture.

Thurston received the Fields Medal, the mathematical equivalent of the Nobel Prize, in 1982 for the depth and originality of his contributions to mathematics. In 1979 he was awarded the Alan T. Waterman Award, which recognizes an outstanding young researcher in any field of science or engineering supported by the National Science Foundation.

Customer Reviews:

A refreshing style of writing.......2001-06-21

Stanislaw Ulam once compared learning mathematics to learning a language, in that some people learn mathematics by "grammar" while other learn it by ear. Thurston's book is a bit like learning by ear.

fun and geometric-intuition-minded.......1998-12-23

A must for anyone entering the field of three-dimensional topology and geometry. Most of it is about hyperbolic geometry, which is the biggest area of research in 3-d geometry and topology nowdays.

Most of it is readable to undergraduates. Its target audience, though, is beginning graduate students in mathematics. If not already familiar with hyperbolic geometry, you might want to get an introduction to the subject first. Once with this background, though, you will discover there is another level of understanding of hyperbolic space you never realized was possible. One imagines Thurston able to skateboard around hyperbolic space with the kind of geometric understanding he conveys here.

Customer Reviews:

Good book........2006-08-18

Most popular textbook on Differential geometry. Feynman once said the most popular one may not be the best, which is not completely true here. If you want another view of differential geometry, Su Buqing's Lectures on Differential Geometry is a great little book.

Best DG book out there.......2005-04-21

This book is rather expensive, but when compared to the other books available, it is not a waste of your money. It has plenty of exercises, many of them with answer or hints in the back of the book, and its exposition is broad, very clear and concise.

It is hard to tell being a math student, but I think anyone with a solid knowledge in multi-variable calculus (Apostol's book would be perfect) or, better yet, who has taken multi-variable analysis course would find this book accessible. One of the advantages of this book is that it is self-contained, so even though it uses, for example, the inverse function theorem (which is something unavoidable for a DG book), it has an appendix on differentiability and continuity which covers this.

The exercises range from easy to very hard, but because of the exposition and of the way the exercises are stated (the tougher ones are many times itemized so that they drive you to the answer) it is rare to find a problem that the reader will not be able to solve upon a little thinking.

The greatest advantage of this book is its clear and well-written exposition. It has very few errors, mostly typographical. It covers a lot of topics and its notation is extremely simple and suggestive, which, believe me, is of great help in a DG book. In short, if you want or have to learn differential geometry, save your time and get this book. As another reader very intelligently put it, there is a reason why this is a classic.

There is a reason why it is a classic........2005-04-03

Before talking about the book itself, let me tell you that I am a mathematician, and when I took a differential geometry course and used do Carmo's book, I already knew I wanted to be a mathematician. So, is this a book for mathematicians? Well, yes, but not exclusively. It is certainly written from a mathematician's point-of-view, and it assumes some maturity on the part of the reader. For instance, the exercises contain very little in the way of drill, and are used to enhance the theory (as pointed out by another reviewer). It seems to me that the author believes that mature readers can provide their own `drill' exercises. So, you won't find many exercises asking for you to find pricipal curvatures for this or that surface, and that other as well; exercises in this book have a theoretical flavor to them. This, of course, makes for some hard exercises, and I do remember spending a lot of time over them, often working together with other students taking the same course. The upside is that we learned the material, and thoroughly. Also, the author provided plenty, plenty of examples. The figures are very well drawn and really allow you to see what is going on - even though these days, with powerful computer packages like Maple, Mathematica, Matlab, and others, any student can provide his/her own pictures. But just because now we can use computers, I wouldn't say the text shows signs of age. It is jus as clear now in its exposition of topics and concepts as it was many years ago. So, even though there are many good alternatives in the market, if I had to teach a course now on this subject, or even better, if I were a student now taking this subject, I would certainly have this book at the top of my list of possible textbooks.

engineers should get another book.......2005-03-16

If you are a mathematician this book probably suits your purposes, BUT if you are an engineer (interested in shell structures for instance) I would recommend "Elementary Differential Geometry" by Pressley.
This book focuses on many sub-topics that are not of interest to an engineer and many of the exercises are abstract and of very little practical value. Also "Elementary Differential Geometry" focuses more on real 3-D shapes and their properties, and thus it is more readable.

Very Difficult.......2004-01-22

Although some claim it is classic, don't expect it to be readable. The book's definitions cna be quite confusing, and it is often difficult to understand many of the definitions or problems without a great deal of effort. If you are using this book for a class, I would reccomend getting a more readable text for reference.

Average customer rating:

Great for self-study
If we make the assumption that "good book" means a book
Best Book Evar!!11!!11!
A must-have text for any grad student!
Great book

Introduction to Smooth Manifolds
John M. Lee
Manufacturer: Springer
ProductGroup: Book
Binding: Paperback

General | Science | Subjects | Books

General | Mathematics | Science | Subjects | Books

General | Medicine | Subjects | Books

All Amazon Upgrade | Amazon Upgrade | Stores | Books

Medicine | Amazon Upgrade | Stores | Books

Professional & Technical | Amazon Upgrade | Stores | Books

Science | Amazon Upgrade | Stores | Books

All Titles | Qualifying Textbooks - Fall 2007 | Stores | Books

Medicine | Qualifying Textbooks - Fall 2007 | Stores | Books

Professional | Qualifying Textbooks - Fall 2007 | Stores | Books

Science | Qualifying Textbooks - Fall 2007 | Stores | Books
Similar Items:

Accessories:

ASIN: 0387954481

Book Description

This book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research--- smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more. The approach is as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, while making full use of the powerful tools that modern mathematics has to offer. Along the way, the book introduces students to some of the most important examples of geometric structures that manifolds can carry, such as Riemannian metrics, symplectic structures, and foliations. The book is aimed at students who already have a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis. John M. Lee is Professor of Mathematics at the University of Washington in Seattle, where he regularly teaches graduate courses on the topology and geometry of manifolds. He was the recipient of the American Mathematical Society's Centennial Research Fellowship and he is the author of two previous Springer books, Introduction to Topological Manifolds (2000) and Riemannian Manifolds: An Introduction to Curvature (1997).

Customer Reviews:

Great for self-study.......2007-09-28

I should say first that I was already familiar with manifold theory before picking up this book. I had already wrestled with some of the definitions, theorems, and whatnot, so I can't necesarily say I was a complete beginner before reading this book. Also, I'm not sure if I can say how great this book would be if you have no idea what a manifold (or tangent space, etc.) is. However, that stuff aside, this is an amazing text. I'm studying this book on my own, and it's great. The concepts are woven throughout the text instead of being lumped into chapters devoted to them (though some people might prefer the latter). Also, they're used to reinforce and build on each other.

As an example, Spivak doesn't treat Lie groups until the second to last chapter. Lee introduces them in the second chapter, uses them as examples throughout the text, builds up the theory of Lie groups as the book goes on, uses Lie groups (and their actions on other manifolds) in developing certain other areas (it really streamlines the development) and ends with a nice big chapter on them. Of course, this is just one example.

Lee developes manifold theory so that it would appeal to a physicist, geometer, algebraist, topologist, etc. Everything gets talked about! This means, however, that he can't treat any one subject in too much detail. For instance, he leaves curvature and other parts of Riemannian geometry to his other Riemannian Geometry text, but it's definitely worth the trade off. This book trashes Spivak. Buy it today!

If we make the assumption that "good book" means a book.......2007-07-24

that lends itself to self-studying then this is not a good book, but excellent. All complaints reported in other reviews are actually answered in the preface: the book is about the mathematical machinery ordinated under the title smooth manifold theory. It is not a book on riemannian geometry that's why there is no extensive treatment of metrics or any treatment of connections. Each topic comes up whenever the prerequisite tools are built and enough motivation can be given, that's why it is a pleasure to read this book. If you like encyclopedic expositions there are plenty of them out there. It is obvious that the author belongs to that group of people who like to excel in whatever they do. All books written by J.M. Lee not only teach you the subject of their titles but also how to write a book if it happens to reach that point in your mathematical career. They are in some sense both books and meta-books on mathematics :)
This review is not intended to comment on other reviews, but let us be honest and agree on the fact that an author never faces the danger of being too clear: as to the length and the pace of the book, I wish this book were only one volume of a series from the same author starting with topology and culminating with the interplay of differential geometry and pdes. There is a drawback however, reasonably not anticipated. Most math books are not written to be actually read (aphoristic but true). This book makes an exception and thus the usual binding proves insufficient quickly. A hardcover version would be convenient. Suggestion for "clever" math students: learn the stuff from Lee and then pretend you are reading Lang's "introduction"...

Best Book Evar!!11!!11!.......2007-03-30

I really like this book. Physically, it looks much like Lang's algebra book, but I assure you that it contains none of the snide remarks. Though, it does have a picture of the author in a berra which is odd. I'm sure I mis-spelled that, but it's the french hat that people like to use to make fun of artist types.

In any case, this book is long and contains a lot of problems for you to do. Unfortunately I do not do them, but that is a different story. I'm nowhere near finishing all the stuff this book has to tell me, but whenever I need to find something I don't know this book tends to have it. The index is great. It might be the best of any book I've used. The greatness of this book is a little surprising juxtaposed with Lee's book on Riemannian geometry which is not exceptional.

Since this book is so large, and it says it's a graduate math book right on the cover, I like to take it out with me when I go out on the town. I find it's a great ice breaker with the ladies. I only wish it was the nice burnt orange of the newer springer books.

All in all, this is a great book, and really puts Spivak to shame.

A must-have text for any grad student!.......2007-02-11

We're using Gullemin and Pollack's text for our differential topology course. I found it rather difficult to learn from it. A friend of mine strongly recommended this book by Lee (actually, he recommended the whole series.) The definitions are concrete, and the proofs are rigorous. Lee provides some great motivations for the ideas presented in this text. Ultimately, I find that it's a well written topology book and should be on any mathematicians bookshelf.

Great book.......2005-10-27

It's very readable. He has a good descriptive, conversational style. It's also very thorough. For example after he gives his definitions of the tangent space he copmares and it to the competitors and shows equivalence. There is plenty of work in coordinates but things are defined in the proper coordinate invariant ways. Nice coverage of vector bundles and a whole chaptor on the cotangent bundle which is nice.

Lots of Lie groups... he introduces symplectic manifolds and talks about Hamiltonian mechanics on the cotangent bundle. What I'm saying is all and all he talks about a lot of wicked good stuff.

One warning: The word transversality appears I believe once in the whole book and that's in an exercise. Intersection theory does not seem to be covered at all. That's not a complaint. That stuff is in lots of good books that don't go anywhere near a lot of the things that are in Lee's book. I'm just saying if you are thinking of using this as a reference for a course that has transversality on the syllabus you will need a second book. Let's say Hirsch's differential topology for the classic, or Guillemin and Pollack's book by the same name for something that doesn't have function spaces as it's second chapter.

So yeah. Good book. Thanks Dr. Lee.

Riemannian Manifolds: AN INTRO TO CURVATURE (Graduate Texts in Mathematics)

Average customer rating:

As always
Nice graduate text.
A nice modern treatment.
Excellent reading, even for a layman!

Riemannian Manifolds: AN INTRO TO CURVATURE (Graduate Texts in Mathematics)
JOHN M. LEE
Manufacturer: Springer
ProductGroup: Book
Binding: Paperback

General | Science | Subjects | Books

General | Mathematics | Science | Subjects | Books

All Deals | Blowout Books | Stores | Books

Science | Blowout Books | Stores | Books

All Amazon Upgrade | Amazon Upgrade | Stores | Books

Professional & Technical | Amazon Upgrade | Stores | Books

Science | Amazon Upgrade | Stores | Books

All Titles | Qualifying Textbooks - Fall 2007 | Stores | Books

Professional | Qualifying Textbooks - Fall 2007 | Stores | Books

Science | Qualifying Textbooks - Fall 2007 | Stores | Books
Similar Items:

ASIN: 0387983228

Book Description

This text is designed for a one-quarter or one-semester graduate couse in Riemannian geometry. It focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds. The book begins with a careful treatment of the machinery of metrics, connections, and geodesics, and then introduces the Riemann curvature tensor, before moving on the submanifold theory, in order to give the curvature tensor a concrete quantitative interpretation. The remainder of the text is devoted to proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's Theorem, and a special case of the Cartan-Ambrose- Hicks Theorem. This unique volume will especially appeal to students by presenting a selective introduction to the main ides of the subject in an easily accessible way. The material is ideal for a single course, but broad enough to provide students with a firm foundation from which to pursue research or develop applications in Riemannian geometry and other fields that use its tools. Of special interest are the "exercises" and "problems" dispersed throughout the text. The exercises are carefully chosen and timed so as to give the reader opportunities to review material that hasjust been introduced, to practice working with the definitions, and to develop skills that are used later in the book. The problems that conclude the chapters are generally more difficult. They not only introduce new mateiral not covered in the body of the text, but they also provide the students with indispensable practice in using the

Customer Reviews:

As always.......2007-09-03

prof. Lee sets the norm of mathematical exposition. I would give it 5 stars if it were several hundred pages thicker. There is so much to say about riemannian manifolds and it would be a pleasure to see them under the light prf. Lee sheds on such difficult concepts.

Nice graduate text........2007-03-30

I used this book to teach about half a year of a graduate Riemannian manifolds course. It is a very good introductory text. I wish it has a bit more background on curves and surfaces, but otherwise it was excellent. It doesn't get into a lot of more advanced topics, but the treatment of Jacobi fields and so forth is really good.

A nice modern treatment........2005-10-27

I just got this fella, and I'm really just through the first four chaptors but so far I'm very pleased. He really tries to tie the definitions and theorems to something you can think about. He gives three "model spaces", the n-sphere, R^n, and hyperbolic space and keeps coming beck to them as he does new things. I like that after he defines connections he shows some in R^n. You know, things like that. Anyway, I'm not a specialist but this seems to me as good an introduction to Reimannian curvature as you could ask for. At least as good in my opinion as Del Carmo's book.

So thanks again Dr. Lee. You keep writing them and we'll keep reading them.

Excellent reading, even for a layman!.......2005-10-20

I never had much use for formal education and quit school back in the 10th grade. I work on the line at a fish cannery and do an honest day's work for an honest day's wage. I don't understand people who make a living sitting around all day just thinking or writing things. What's getting made? How do you just think about things and expect people to pay you for it?

Normally I kick back with a cold brew and whatever sports is playing on the tube. Last book I read was in school. I was too busy with football, basketball and girls to waste time with studying. So you might think, what in the world would make me pick up "Riemannian Manifolds" and start reading a graduate text in mathematics? I don't know, something about the title just grabbed me.

You know what? It's a pretty good book. I'm not saying I understood everything Mr. Lee was talking about. I mean, I sorta remember stuff like algebra and geometry and triangles and proofs and things like that, and all that math stuff helped me get through the first four chapters. But when I got to chapter 5, talking about Riemannian geodesics, I got kinda lost. I took a piece of string, used it to connect two cities on a globe, and then I understood. After that, the book picked up pace and finished really strong with comparisons of manifolds on both positive and negative curvatures. I'm thinking I'll read "The Laplacian on a Riemannian Manifold" next. Who ever thought all this math stuff could be so interesting?

Schaum's Outline of Differential Geometry (Schaum's)

Average customer rating:

Differential Geometry review
Good as a basic textbook and a source of solve problems
Differential Geometry - A Schaum's Outline Series

Schaum's Outline of Differential Geometry (Schaum's)
Martin M. Lipschutz
Manufacturer: McGraw-Hill
ProductGroup: Book
Binding: Paperback

General | Science | Subjects | Books

General | Mathematics | Science | Subjects | Books

Study Guides | Reference | Subjects | Books

All Titles | Qualifying Textbooks - Fall 2007 | Stores | Books

Professional | Qualifying Textbooks - Fall 2007 | Stores | Books

Reference | Qualifying Textbooks - Fall 2007 | Stores | Books

Science | Qualifying Textbooks - Fall 2007 | Stores | Books
Similar Items:

ASIN: 0070379858

Book Description

Students will find all the information covered in the standard textbooks--and more--explained clearly and concisely in this powerful study tool. Unusually detailed, it elucidates all the most difficult-to-grasp concepts that class studies and texts sometimes gloss over. The hundreds of problems with fully explained solutions illuminate important points and teach students sound problem-solving skills. Ideal, also, for independent study.

Customer Reviews:

Differential Geometry review.......2007-01-04

I have found this to be an excellent addition to my library.

Good as a basic textbook and a source of solve problems.......2006-06-05

This book is intended to assist upper level undergraduate and graduate students in their understanding of differential geometry, which is the study of geometry using calculus. Usually students study differential geometry in reference to its use in relativity. I personally have a rather oddball application for the subject - modeling of curved geometry for computer graphics applications. The fundamental concepts are presented for curves and surfaces in three-dimensional Euclidean space to add to the intuitive nature of the material.
The book presumes very little in the way of background and thus starts out with the basic theory of vectors and vector calculus of a single variable in the first two chapters. The following three chapters discuss the concept and theory of curves in three dimensions including selected topics in the theory of contact.
Great care is given to the definition of a surface so that the reader has a firm foundation in preparation for further study in modern differential geometry. Thus, there is some background material in analysis and in point set topology in Euclidean spaces presented in chapters 6 and 7. The definition of a surface is detailed in chapter eight. Chapters 9 and 10 are devoted to the theory of the non-intrinsic geometry of a surface. This includes an introduction to tensor methods and selected topics in the global geometry of surfaces. The last chapter of the outline presents the basic theory of the intrinsic geometry of surfaces in three-dimensional Euclidean space.
Exercises are primarily in the form of proofs, and there are plenty of worked examples. Since the examples are kept to no more than three dimensions, the outline contains plenty of good instructive diagrams that illustrate key concepts. This Schaum's outline has quite a bit of instruction in it past the bare required minimum, but you might still want a good primary textbook. My personal favorite is Pressley's "Elementary Differential Geometry". Overall I find this to be a very good outline and source of solved problems on the subject and I highly recommend it.

Differential Geometry - A Schaum's Outline Series.......2000-06-25

As with all of the Schaum's Outline Series, this book is particularly useful if the readers intent is to gain a working knowledge of the subject. The subject of Differential Geometry is no exception. Dr. Lipschultz has done an excellent job of communicating the essential aspects of differential geometry to the reader. The book assumes a fairly low level of mathematical ability having calculus as the primary prerequisite. From this humble beginning, Dr. Lipschultz takes the reader through the necessary discussions of vector functions, curvature, fundamental forms, and tensor analysis. Given the theoretical nature of the subject, Dr. Lipschultz has included most of the theorems and associated proofs necessary for a general understanding of the subject. However, this book is not a substitute for a serious study of differential geometry. In addition most of the problems are limited to two dimensional surfaces and this reader would have enjoyed a more adventurous investigation of higher dimensional spaces. Like all Schaum's series, the text is chock full of problems and their solution. I recommend this book for anyone interested in quickly gaining a working knowledge of the subject.

Geometry of Differential Forms (Translations of Mathematical Monographs, Vol. 201) (Translations of Mathematical Monographs)

Average customer rating:

Self contained introduction to techniques of classifying manifolds.
A very good book.

Geometry of Differential Forms (Translations of Mathematical Monographs, Vol. 201) (Translations of Mathematical Monographs)
Shigeyuki Morita
Manufacturer: American Mathematical Society
ProductGroup: Book
Binding: Paperback

General | Science | Subjects | Books

General | Mathematics | Science | Subjects | Books

Mathematical Analysis | Mathematics | Science | Subjects | Books

All Titles | Qualifying Textbooks - Fall 2007 | Stores | Books

Professional | Qualifying Textbooks - Fall 2007 | Stores | Books

Science | Qualifying Textbooks - Fall 2007 | Stores | Books
Similar Items:

ASIN: 0821810456

Book Description

Since the times of Gauss, Riemann, and Poincaré, one of the principal goals of the study of manifolds has been to relate local analytic properties of a manifold with its global topological properties. Among the high points on this route are the Gauss-Bonnet formula, the de Rham complex, and the Hodge theorem; these results show, in particular, that the central tool in reaching the main goal of global analysis is the theory of differential forms.

The book by Morita is a comprehensive introduction to differential forms. It begins with a quick introduction to the notion of differentiable manifolds and then develops basic properties of differential forms as well as fundamental results concerning them, such as the de Rham and Frobenius theorems. The second half of the book is devoted to more advanced material, including Laplacians and harmonic forms on manifolds, the concepts of vector bundles and fiber bundles, and the theory of characteristic classes. Among the less traditional topics treated is a detailed description of the Chern-Weil theory.

The book can serve as a textbook for undergraduate students and for graduate students in geometry.

Customer Reviews:

Self contained introduction to techniques of classifying manifolds........2007-01-10

This text is phenomenally easy to read and well organized. The author starts you on a journey by first explaining the importance and power of classifying manifolds namely by certain invariants preserved by certain mappings ( diffeomorphisms ).

For example, like Euler, we could count the number of holes in the surface and using this combinatorial method we are led to homology theory.

Or like Gauss, we could use a differentiation and integration to come up with the idea of curvature as an intrinsic feature of the surface.

Modern approaches use differential forms to represent homology and cohomoly groups.

The author also deals with fibre bundles demonstrating their importance in analyzing manifolds specifically how the number of fibre bundles possible with given Lie groups as structure groups over the manifold can be answered by characteristic classes such as the Chern and Pontrjagin classes. The use of differential forms is indispensible.

Perhaps the most satisfying aspect of this book is that it clarifies the notions of connection, connection form, curvature, curvature form for manifolds and fibre bundles.

There are plenty of exercises to boot.

A very good book........2005-03-28

This is probably the most clearly written self-contained book on the basics of differential geometry. The author does a great job explaining the ideas behind purely mathematical 'dry' constructions. On the other hand, everything is defined correctly and precisely. A very readable and useful book with the perfect combination of formal math. and intuition. I would recommend it to students in theoretical physics area, together with the Nakahara's fantastic book.

Schaum's Outline of Tensor Calculus (Schaum's)

Average customer rating:

Tensor Calculus
Easy reading for a complex subject
An evergreen book
Helpful in Learning Relativity
Agreed, a must have...

Schaum's Outline of Tensor Calculus (Schaum's)
David C. Kay
Manufacturer: McGraw-Hill
ProductGroup: Book
Binding: Paperback

General | Science | Subjects | Books

General | Mathematics | Science | Subjects | Books

Mathematical Analysis | Mathematics | Science | Subjects | Books

Study Guides | Reference | Subjects | Books

All Amazon Upgrade | Amazon Upgrade | Stores | Books

Nonfiction | Amazon Upgrade | Stores | Books

Professional & Technical | Amazon Upgrade | Stores | Books

Reference | Amazon Upgrade | Stores | Books

Science | Amazon Upgrade | Stores | Books

Schaum's Outlines | Amazon Upgrade | Stores | Books

All Titles | Qualifying Textbooks - Fall 2007 | Stores | Books

Entertainment | Qualifying Textbooks - Fall 2007 | Stores | Books

Professional | Qualifying Textbooks - Fall 2007 | Stores | Books

Reference | Qualifying Textbooks - Fall 2007 | Stores | Books

Science | Qualifying Textbooks - Fall 2007 | Stores | Books
Similar Items:

ASIN: 0070334846

Book Description

This lucid introduction for undergraduates and graduates proves fundamental for pactitioners of theoretical physics and certain areas of engineering, like aerodynamics and fluid mechanics, and exteremely valuable for mathematicians. This study guide teaches all the basics and efective problem-solving skills too.

Customer Reviews:

Tensor Calculus.......2007-01-04

I have found it to be an excellent refernce for an otherwise difficult subject.

Easy reading for a complex subject.......2006-06-26

I've many books about tensors and this one is the best one to start learning such a difficult subject. It does not omit the things that are assumed you must know. It explains everything, even the simplest things in a easy way. However, you should know vector analysis and multidimensional calculus in order to understand the complex things in the last chapters.

An evergreen book.......2005-10-25

It's a pity Amazon doesn't declare the age of books that sells (date of edition).
This book - in spite of its oldness - is very useful to everyone that needs to know something about tensors.
Lionello Cantoni

Helpful in Learning Relativity.......2005-09-24

I have been studying a textbook in general relativity. A lot of the material that they gloss over is detailed in an understandable way in this volume.

Agreed, a must have..........2005-06-22

...for students, educators and practicing scientists and engineers. Great reference and learning/teaching tool.

Average customer rating:

Tubes (Progress in Mathematics)
Alfred Gray
Manufacturer: Birkhäuser Basel
ProductGroup: Book
Binding: Hardcover

General | Science | Subjects | Books

General | Mathematics | Science | Subjects | Books

All Amazon Upgrade | Amazon Upgrade | Stores | Books

Professional & Technical | Amazon Upgrade | Stores | Books

Science | Amazon Upgrade | Stores | Books

All Titles | Qualifying Textbooks - Fall 2007 | Stores | Books

Professional | Qualifying Textbooks - Fall 2007 | Stores | Books

Science | Qualifying Textbooks - Fall 2007 | Stores | Books
ASIN: 3764369078

Book Description

The main subject of the book is the full understanding of Weyl's formula for the volume of a tube, its roots and its implications. Another discussed approach to the study of volumes of tubes is the computation of the power series of the volume of a tube as a function of its radius. The chapter on mean values, besides its intrinsic interest, shows an interesting fact: methods which are useful for volumes are also useful for problems related with the Laplacian. Historical notes and Mathematica drawings have been added to this revised second edition.

Average customer rating:

Introductory level text with empasis on intuition examples and exercise.
Cartan's formulation of differential geometry taken up here.
Good low dimensional calculation
Solid and Modern Introduction
Very useful, but lacking some abstraction

Elementary Differential Geometry
Barrett O'Neill
Manufacturer: Academic Press
ProductGroup: Book
Binding: Hardcover

General | Science | Subjects | Books

General | Mathematics | Science | Subjects | Books

Mathematical Analysis | Mathematics | Science | Subjects | Books

All Amazon Upgrade | Amazon Upgrade | Stores | Books

Professional & Technical | Amazon Upgrade | Stores | Books

Science | Amazon Upgrade | Stores | Books

All Titles | Qualifying Textbooks - Fall 2007 | Stores | Books

Professional | Qualifying Textbooks - Fall 2007 | Stores | Books

Science | Qualifying Textbooks - Fall 2007 | Stores | Books
Similar Items:

ASIN: 0120887355

Book Description

Written primarily for students who have completed the standard first courses in calculus and linear algebra, ELEMENTARY DIFFERENTIAL GEOMETRY, REVISED SECOND EDITION, provides an introduction to the geometry of curves and surfaces.

The Second Edition maintained the accessibility of the first, while providing an introduction to the use of computers and expanding discussion on certain topics. Further emphasis was placed on topological properties, properties of geodesics, singularities of vector fields, and the theorems of Bonnet and Hadamard.

This revision of the Second Edition provides a thorough update of commands for the symbolic computation programs Mathematica or Maple, as well as additional computer exercises. As with the Second Edition, this material supplements the content but no computer skill is necessary to take full advantage of this comprehensive text.

*Fortieth anniversary of publication! Over 36,000 copies sold worldwide
*Accessible, practical yet rigorous approach to a complex topic--also suitable for self-study
*Extensive update of appendices on Mathematica and Maple software packages
*Thorough streamlining of second edition's numbering system
*Fuller information on solutions to odd-numbered problems
*Additional exercises and hints guide students in using the latest computer modeling tools

Customer Reviews:

Introductory level text with empasis on intuition examples and exercise........2005-07-11

If you are looking for abstraction with little in the way of intuition I suggest Conlan " differential manifolds"

If you are an applied mathematician or physicist this book is for you.

I have always beleived that to truly grasp mathematics one must be provided with a reason for WHY things are the way they are and WHAT IDEAS the expression must express. This is best done with examples and exercises.

I digress.

The book restricts is exposition to two and three dimensions. Some of the topics can readily be bootstrapped to higher dimensions.

The book starts with basic ideas of curve, directional derivative and tangent vector in Euclidean space with a sprinkling of differential forms to wet the appetite.

It then moves into the notion of frame fields along curves resulting in the Frenet formulas which express how the frame fields change along the curve. These are expressed in terms of the frame field themselves giving ideas of curvature and torsion.

The book then abstracts these concepts to show how we can talk about change of frame fields along arbritrary directions not just along the curve. The tools used to do this are the covariant derivative and connection forms which can then be used to develop connection equations ( abstracted analogue of frenet formulas ) and then the cartan structural equations.

The book talks about isometries and defines euclidean geometry as those properties preserved by isometries. It then abstracts once again to surfaces in R3 using patches and appropriate conditions on the overlap without introducing manifolds although these are briefly mentioned later.

We then see how calculus in euclidean space can be adapted to surfaces using these patches. The corresponding concepts of function, differentiability and tangent vectors on these objects is introduced. Forms on these surfaces are introduced and their application to integration theory on these surfaces is developed showing how forms on the surface are " pulled back" to euclidean space using the idea of differential of a map and integrated there. The integration gives the volume ( area ) of that surface. Stokes theorem is introduced.

We now move into the idea of shape operators on the surface and show how these describe how the normal vector on the surface move in various directions giving ideas of mean and gaussian curvature . We see a very nice interplay of algebraic analysis leading to a geometric analysis.

The book then deals with studying geometrical properties on surfaces using the Cartan methods described earlier.

We then see how to define intrinsic geometry of any surface. Namely those properties of the surface that are preserved by isometries. From the definition of isometry we see that these rely on on the concepts of tangent vector and inner products. Shape operators and mean curvature are not intrinsic.

We now study the geometry of surfaces specifically the intrinsic geometry without reference to an imbedding space ( R3). An abstract "surface" is endowed with an inner product. A different inner product gives a different geometry. We talk about gaussian curvature and covariant derivative which are intrinsic.

Geodesics are introduced as is the gauss bonnet theorem which relates a geometric property to a topological one.

The book concludes with a chapter on global properties ( 2 d surfaces ) especially how gaussian curvature influences geodesics and how the two completely determine the geometry of the surface.

Cartan's formulation of differential geometry taken up here........2003-11-30

My first encounter with this book was during the academic year of 2000-2001, when it was used as the main text for an upper division course on differential geometry. The class --taught by a distinguished scholar-- was only meant to be a brief excursion into the realm of continuous math, beyond analysis and topology. After finishing the term however, I decided to change direction and as time went on, I drifted more and more towards geometry as the field of further concentration. The original second edition (from 1997) contained numerous typos, but luckily, the revised 2006 issue takes care of these and also streamlines the section numbering formats which had made the referencing and following through with the material a bit cumbersome. As some of the other reviewers have mentioned, the emphasis here is on the low (= 2 and 3) dimensional geometry, formulated in the language of differnetial forms (Cartan's early 20th century approach).

Within the eight chapters of the book (seven in the 1966 edition), the reader is first introduced to some preliminaries such as tangent vectors, directional derivatives, and differential forms. In chapter two, the author presents the Frenet frame formulas, covariant derivatives, connection forms, and Cartan's structural equations, which are generalizations of the Frenet frame formulas for surfaces. In chapters three and four, there is a healthy dose of Euclidean geometry and calculus on surfaces. In chapter five, discussion is on the study of the shape operators and normal and Gaussian curvatures, where also some useful computational examples have been presented. Geometry of surfaces is the subject of chapter six, where the crucial Gauss' egregium theorem is proved, and in chapter seven students are introduced to the basics of the Riemannian geometry, culminating in the famous Gauss-Bonnet theorem. In chapter eight (which is highly topological) complete surfaces, covering spaces, Jacobi fields, and the subject of classification of surfaces are explored. The appendices include help on using popular computer algebra systems (with updates in the latest revised edition), and another appendix providing solutions to many of the odd-numbered exercises in the book.

Please note that the author leaves out a discussion of several essential tools, for example, the Schwarz-Christoffel symbols, tensors, and Lie derivatives. The exposition does not fully explore some other important topics such as the first and second fundamental forms, and parallel translation, which only show up in the exercises. Then again, perhaps to keep the level of exposition elemantary and the size limited, Dr. O'Neill has preferred to skip some topics. One remedy is to back his text up with Manfredo Do Carmo's 1976 classic, which is mathematically more rigorous, and covers more of the above-mentioned topics. Afterwards, one can certainly continue the study of the essentials by reading other advanced material such as William Boothby's "An Introduction to Differentiable Manifolds and Riemannian Geometry". There is also a somewhat obscure title by Richard W. Sharpe with the title "Differential Geometry: Cartan's Generalization of Klein's Erlangen Program", from the Springer-Verlag GTM series that's worth checking into. Finally, other elemantary-level sources to keep in mind for a beginning student are the recent texts by Andrew Pressley (2001) and Wolfgang Kuhnel (2002) both available on amazon.com's catalog.

[Review updated in May 2006]

Good low dimensional calculation.......2003-07-29

It's easy to read with enough examples. Suitable for self study after your advanced calculus (inverse function thm/implicit function thm should be covered here) and linear algebra classes. Tons of exercises will help you familiarize yourself with the calculation in low dimension. (Do I love the exercises on minimal surfaces and surfaces of revolution in chapter 5 and 6!) Most of them are workable. This is the strength of the book. Since the author limits the material to low dimensions, some definitions are a bit misleading, such as the definition of exterior derivative of 1-form in chapter 4, where another term to be added happens to be zero. I think there is a big gap in style and level of difficulty between this book and author's "Semi-Riemannian Geometry: With Applications To Relativity".

After this book, probably you want to read Hicks' "Notes on differential geometry", if you can find a copy in some lib. Darling's "Differential Forms and Connections" is also highly recommended. It is modern but not much topological stuff.
Company it with Warner's "Foundations of differentiable manifolds and Lie groups" for topology also much higher algebra.

Solid and Modern Introduction.......2000-10-18

I worked through the first edition of this book some years back. After finishing this book I was ready for more abstract treatments of Riemannian Geometry. For example, having seen covariant derivatives on 2-surfaces embedded in R^3 motivates the abstract definition of connections on manifolds.

Chapter 1 is a decent introduction to pullbacks and pushforwards of differntial forms and tangent vectors respectively. In fact, all the subsequent geometry is based on pullbacks and pushforwards.This itself motivates the more abstract definition of a differentiable manifold with its coordinate charts. True,tangent vectors are not described in the most abstract fashion (e.g. as derivations on the algebra of functions) but this is not appropriate for a first course.

Chapter 2 describes the language of frame field and connection forms and derives the Frenet-Serret equations in terms of moving frames and structure equations. We associate this with the methods of Elie Cartan, who used moving frames in a systematic manner.

Chapter 3 deals with isometries; frankly speaking I never understood the raison d'etre for such a long chapter on such a topic.

Chapter 4 discusses coordinate patches. Again, this is thoroughly modern, and you won't find this in Struik or Kreyszig. The idea of piecing together coordinate patches to get geometric or topological information is a twentieth-century conception.

Chapter 5 introduces the Shape Operator, which is subsequently used in Chapter 6 to derive the equations of surface theory. This is really moving frames again, in another guise.

Chapter 7 finally tries to put this in a more abstract setting by defining abstract surfaces with an intrinsically defined covariant derivative.Holonomy and the Gauss-Bonnet theorem are discussed.

After reading this book, one would be equipped to handle do Carmo's book on Riemannian geometry, or O'Neill's book on Semi-Riemanninan geometry, or the more recent book by Lee, again on Riemannian geometry.

Very useful, but lacking some abstraction.......2000-03-15

I like this book very much because it helps me frequently when I need to remember some definitions or formulas, but I think it could be improved if some topics were treated in a more abstract way, as all the material on differential forms, for example.

Some Nonlinear Problems in Riemannian Geometry (Springer Monographs in Mathematics)

Average customer rating:

Some Nonlinear Problems in Riemannian Geometry (Springer Monographs in Mathematics)
Thierry Aubin
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

General | Science | Subjects | Books

General | Mathematics | Science | Subjects | Books

Mathematical Analysis | Mathematics | Science | Subjects | Books

All Amazon Upgrade | Amazon Upgrade | Stores | Books

Professional & Technical | Amazon Upgrade | Stores | Books

Science | Amazon Upgrade | Stores | Books

All Titles | Qualifying Textbooks - Fall 2007 | Stores | Books

Professional | Qualifying Textbooks - Fall 2007 | Stores | Books

Science | Qualifying Textbooks - Fall 2007 | Stores | Books
ASIN: 3540607528

Book Description

During the last few years, the field of nonlinear problems has undergone great development. This book consisting of the updated Grundlehren volume 252 by the author and of a newly written part, deals with some important geometric problems that are of interest to many mathematicians and scientists but have only recently been partially solved. Each problem is explained, up-to-date results are given and proofs are presented. Thus, the reader is given access, for each specific problem, to its present status of solution as well as to the most up-to-date methods for approaching it. The main objective of the book is to explain some methods and new techniques, and to apply them. It deals with such important subjects as variational methods, the continuity method, parabolic equations on fiber.

Books:

Books Index

Books Home

Recommended Books