Book Description
Presenting an introduction to the mathematics of modern physics for advanced undergraduate and graduate students, this textbook introduces the reader to modern mathematical thinking within a physics context. Topics covered include tensor algebra, differential geometry, topology, Lie groups and Lie algebras, distribution theory, fundamental analysis and Hilbert spaces. The book also includes exercises and proofed examples to test the students' understanding of the various concepts, as well as to extend the text's themes.
Customer Reviews:
A fast introduction to mathematics in physics.......2006-01-02
The book does not assume prior knowledge of the topics covered. However, the reader will find use of prior knowledge in algebra, in particular group theory, and topology. Compared to texts, such as Arfken Weber, Mathematical Methods for Physics, A Course in Modern Mathematical Physics is different, and emphasis is on proof and theory. The text is reasonably rigorous and build around stating theorems, giving the proofs and lemmas with occasional examples. The style is not the strictest, although making the text more reader friendly, it is easy to get confused with which assumptions have been made, and the direction of the proof. Sometimes only the "if" part is proven.
Students familiar with algebra will notice that the emphasis is on group theory, interestingly the concept of ideals is left mostly untouched. For more on representation theory a good reference is Groups Representations and Physics by H.F. Jones where solutions to some of the exercises can be found, and examples of the use of the fundamental orthogonality theorem applied to characters of represenations.
The first 6 chapters are relatively straight forward, but in chapter 7 Tensors the text becomes much more advanced and difficult. Chapter 10 on topology offers some lighter material but the reader should be careful, these consepts are to re-appear in the discussion of differential geometry, differentiable forms, integration on manifolds and curvature. These are not the most simple subjects and it is clear that they deserve entire courses of their own.
The book has insight and makes many good remarks. However, chapter 15 on Differential Geometry is perhaps too brief considering the importance of understanding this material, which is applied in the chapters thereinafter. The book is suitable for second to third year student in theoretical physics.
Jumping over the Gap.......2005-12-30
Most physicists avoid mathematical formalism, the book attacks this by exposing mathematical structures, the best approach I've ever experience. After reading the first chapter of this books I can assure is a must for everyone lacking mathematical formation undergraduate or graduate.
It surely jumps over this technical gap experienced by most physics opening the gate for advanced books an mathematical thinking with physic intuition.
Unfortunately is very expensive, i hope i could have it some day.
A serious, wide spectrum introduction to modern mathematical physics.......2005-10-10
This book covers almost every subject one needs to begin a serious graduate study in mathematical and/or theoretical physics. The language is clear, objective and the concepts are presented in a well organized and logical order. This book can be regarded as a solid preparation for further reading such as the works of Reed/Simon, Bratteli/Robinson or Nakahara.
Not a review, only a little more information.......2004-12-11
Since I don't yet have this book, I cannot review it; however, I have found the contents of this book on the publisher's web site in case it would help anyone decide to purchase it or not.
Contents
Preface
1. Sets and structures
2. Groups
3. Vector spaces
4. Linear operators and matrices
5. Inner product spaces
6. Algebras
7. Tensors
8. Exterior algebra
9. Special relativity
10. Topology
11. Measure theory and integration
12. Distributions
13. Hilbert space
14. Quantum theory
15. Differential geometry
16. Differentiable forms
17. Integration on manifolds
18. Connections and curvature
19. Lie groups and lie algebras
I will return at a later date to properly review it in case I need to change the rating I gave it.
Average customer rating:
- Excellent if it si still as good as the edition from 20 years ago
- A totally ineffective method of teaching Calculus
- Brilliant method
- excellent for basic calculus ....
- The worst math book ever
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Calculus and Analytic Geometry
Sherman K. Stein , and
Anthony Barcellos
Manufacturer: McGraw-Hill Science/Engineering/Math
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ASIN: 0070611750 |
Book Description
A revision of McGraw-Hill's leading calculus text for the 3-semester sequence taken primarily by math, engineering, and science majors. The revision is substantial and has been influenced by students, instructors in physics, engineering, and mathematics, and participants in the national debate on the future of calculus. Revision focused on these key areas: Upgrading graphics and design, expanding range of problem sets, increasing motivation, strengthening multi-variable chapters, and building a stronger support package.
Customer Reviews:
Excellent if it si still as good as the edition from 20 years ago.......2005-09-20
I can only review an edition of this book dated at least 20 years ago. I bought it when I was in high school (in 10th grade actually) and used it by myself to learn calculus.
When I got to the actual class, I knew basicaly everything and I mean everything that the teacher taught us in 12th grade. I didn't even need anything for my first 2 university level courses in calculus, just took the notes in the class and that was enought to get A+ in both, differential and integral calculus (course 1) and Vetor calculus.
The explanation of derivatives was great and my teache just enhanced my knowledge there. The book was specially great when teaching integration. There was a chapter devoted to that. I skept the section about using tables for integration and only learned a few basic formulas. The book taught the methods and still now, after 20 years, I can integrate pretty much anyhing without any difficulty at all.
If the current edition is as good as it was 20 years ago, then this book is definitely a winner.
A totally ineffective method of teaching Calculus.......2005-02-25
I am a student at CSUSB and I have had to use this book for 3 quarters of Calculus. At the school all the professors say the book is horrible and can't wait until the department changes. In 2005 they finally changed the book to Caculus by Larson, Hostetler, Edwards. It is ironic in desperation I asked a fellow student how he was handing the problems of this book. He said that he borrowed a book from his friend who said the book was excellent. It turned out that it was the book written by Larson, Hostetler, and Edwards. I then obtained a copy for myself and found the book to be excellent.
Stein's and Barcellos's book has very poor explanations in the chapters and very few examples to explain to you the concepts. Whereas Larson's book has excellent explanation of concepts and follows it up with good examples that make the concepts easy to understand. At the present we are studying the disk and washer methods of finding volume. Stein covers these topics in approximately 3-4 pages of very poor explantions. The Larson book had 2 chapters on the subject and 9 pages just on the 2 methods. I currently have an A in the last quarter of Calculus and I attribute that to using the book by Larson.
As you can see I can not say enough bad about the book by Stein and Barcellos. Good luck and I hope you make the right choice, but don't buy this book.
Brilliant method.......2004-05-30
This book is literally the best basic calculus text you can possibly get. Anyone wanting to start learning calculus NOW should get this. No real previous mathematical knowledge is necessary. There are several appendices on algebra and series etc. The book discusses trigonometry, so you can learn the book practically without knowing a thing. The "feeling" of the book is inexplicable. Reading this book really gave me an true understanding of basic calculus. Excellent for people like me who need proofs (especially visual ones) have a solid grasp on concepts. If your only goal in learning calculus is to do well on examinations, this book is definately not for you. You should get "Calculus for Dummies" or something like that. The great thing about the text is that it appeals to almost everyone. If there is a certain chapter you don't care for or doesn't matter to you, for example on methods of graphing, you can just skip it, and it will not do any harm. Highly intelligently organized. If you want some help in you physics class on basic vector algebra, just turn to chaper 18 and just read! This book is full of applications, which is great. It also has several historical notes. The colors make the book very engaging to read. Unless your colorblind, this will help engage your interest. Very adequate spacings on the paper as to keep you clearheaded and focused. The drawings rival those of Picasso. They show calculus to be a LIVELY subject. The examples (inside the chapters) are very helpfull. Stein offers several suggestions on how to solve certain problems. Its a shame; this book does not attract the amount of attention it deserves. I did get stuck a couple times; but that is inevitable. Definately get this one: its a gem to have. I can understand how some people would hate this book; its not very concise. That should not be a hindrance. If you feel there is no need to read on about a subject, skip some pages. At the end of chapters it all comes together with a summary of the most important concepts. The book prepares you for study of calculus-based sciences such as physics, and for more advanced mathematical topics as well. I worship this book.
excellent for basic calculus ...........2004-05-25
This book is the best place to start to learn calculus. It starts from very basic principles and also contains some more advanced stuff like Stokes Theorem etc. Some readers may find that the book contains too many basic trivial explanations, but I see this as a strength. When you start learning calculus, I think it is a good idea to explain even the trivial, to make sure that you have a good understanding of really everything. I am sure that most readers will benifit from this, even those already having some more advanced math knowledge. Yes : even this latter group will appreciate the benifits of this book, they can always skip some explanations but will benifit from the very clear exposition of more advanced concepts like Stokes Theorem etc ... Myself for instance, I like the more rigorous and abstract math like "real mathematical analysis", but when I need to refresh some calculus and geometry techniques, this book is really the best to sharpen my intuition and understanding of calculus.
Another excellent feature of the book : this book should serve as an example for the layout of math books : it contains a lot of spacing (handy to make personnal annotations), contains a lot of examples, and contains a lot of excellent pictures illustrating a concept... Also some nice anecdotes are added to keep the reader interested. I wish all math books were like this.
If math is not your strongest skill and you need to learn some higher calculus this book will be your excellent companion helping you to gain the insight and intuition you need. If you are busy with more advanded and abstract math, this book also has something to offer to you : this book serves as a fallback point for sharpening your mathematical intuition and refreshing some concepts that you might have forgotten.
Small drawbacks are : -Some more advanced concepts (like Stokes Theorem) are very well explained, but others are explained without proof (convergence of series...) or with simplified proofs (for instance limited to two dimensional cases, though excellent again to gain mathematical intuition). Maybe this is acceptable for a calculus course, but may disappoint the reader who is looking for rigour.
Conclusion : perfect book to gain insight in calculus, it suits well on the shelf of everybody busy who needs calculus...
The worst math book ever.......2004-02-08
If you have to buy it, buy it. Otherwise, avoid it like the plague. I was forced to buy it because this ass was a professor at UC Davis and made all Davis math students buy it. Even the other professors hated it, but we were stuck with it becuase of his tenure.
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.
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.
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.
Average customer rating:
- Mathematical Handbook of Formulas and Tables
- Handbook of formulas and Tables
- great reference
- one of the best
- Very useful in a pinch
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Schaum's Mathematical Handbook of Formulas and Tables
Murray R Spiegel
Manufacturer: McGraw-Hill
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ASIN: 0070382034 |
Book Description
Students and research workers in mathematics, physics, engineering and other sciences will find this compilation of more than 2000 mathematical formulas and tables invaluable. They will see quickly why half a million copies were sold of the first edition! All the information included is practical -- rarely used results are excluded. Topics range from elementary to advanced-from algebra, trigonometry and calculus to vector analysis, Bessel functions, Legendre polynomials and elliptic integrals. Great care has been taken to present all results concisely and clearly. Excellent to keep as a handy reference!
Customer Reviews:
Mathematical Handbook of Formulas and Tables.......2007-09-25
A very useful book that gathers all the mathmatical formals 'as the title states. As an Engineering Student it is very helpful to have everything in one text instead of getting your old books and digging through them to find them.
Handbook of formulas and Tables.......2007-01-04
It is a good quick reference to getting formulas for math problems.
great reference.......2007-01-04
tables are concise with out missing any important integrals. the table is my constant companion for undergrad physics and mathematics.
one of the best.......2006-11-10
one of the best books i've ever got....
it has every thing i need
Very useful in a pinch.......2006-11-10
As a tabular summary of many useful mathematical relations, the book is very job-specific; however, it contains most of the functions and functional relations that a scientist or engineer might need. The layout is clean and very well organized. It's a useful reference, but does not actually derive anything, so if one is looking for derivations, then try looking at applied mathematics textbooks.
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?
Average customer rating:
- Differential Geometry review
- Good as a basic textbook and a source of solve problems
- Differential Geometry - A Schaum's Outline Series
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Schaum's Outline of Differential Geometry (Schaum's)
Martin M. Lipschutz
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Differential Geometry
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.
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.
Average customer rating:
- Not the best
- Great book for amatures
- Volume 1: A nice study of de Rham Cohomology
- The Great American Differential Geometry Book
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A Comprehensive Introduction to Differential Geometry, Volume 1, 3rd Edition
Michael Spivak
Manufacturer: Publish or Perish, Inc
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Binding: Hardcover
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Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus
ASIN: 0914098705 |
Customer Reviews:
Not the best.......2007-09-28
Spivak's text gets a lot of good reviews, and it is a fine text. In fact, it's one of the best I've ever seen. Read a few other books on the subject, and you'll agree that this is a massive improvement on them. So why only 3 stars? Because there's a much better text on the subject: John Lee's "An Introduction to Smooth Manifolds". This book outshines Spivak's in so many ways. Sure, Spivak is great at motivating major developments in the theory (for instance, he really helps you understand why we need to define a tangent space and why it is the way it is), but he fails pretty bad when it comes to developing some actual theory.
Reading Spivak's text is like taking a stroll, a fresh break from the usual mathematics textbook style. But you also hit a bunch of brick walls on this stroll. It'll be a great discussion, and then you'll come to a theorem. You'll have no idea what its for (some of the time) and you'll struggle to work through its proof (most of the time). Furthermore, the organization is... well, there is no organization! As a result, Spivak can seem to droll on. Lee isn't as good at giving the overall big picture as well as Spivak, but he does everything else exceptionally. Leave Spivak for bed time reading, but do your real studying out of Lee.
Great book for amatures.......2006-02-23
If you want a book that is rich with examples then this is it. The proofs are, for the most part, clear and concise, thus a person who is learning the material without the aid of an instructor can follow the logic. However, the author could have spent some more time developing topological ideas (thought he does have an appendix section that does a fair job of it) within the flow of the first chapter. I personally find appendices to be too distracting and tend to slow down the flow of the material in a particular chapter. Other than that, this is a great book if you want to learn differential geometry and the theory of smooth manifolds.
Volume 1: A nice study of de Rham Cohomology.......2005-09-06
This book is the first volume of the 3rd edition in a five volume series on differential geometry. The emphasis on this first volume is the study of differential forms and de Rham Cohomology Theory. Spivak also considers two 'bonus' topics: integral manifolds & foliations and Lie groups.
You'll need some prerequisites to get started. For the differential topology material (including Sard's Theorem and Whitney's 2n+1 Embedding Theorem), I recommend Hirsch's "Differential Topology". For results on determinants and symmetric groups, I use Hungerford's "Algebra", now in its 12th printing. For the general topology material (Hausdorff spaces, Urysohn metrization, etc.), I recommend Munkres "Topology", 2nd edition.
Spivak begins this volume with a review of topological manifolds in Chapter 1. The author provides the basic definitions and gives lots of examples of surfaces and other manifolds. The discussion of manifolds and surfaces continues in the Chapter 1 Exercises. (The author routinely used the exercise set to continue the thread of discussion.) Quick mention of the surface classification theorem is made, although for the proof of this, you'll need to look in Hirsch or Munkres. The reader gets to have fun gluing topological handles onto and cutting disks out of the 2-sphere.
Chapter 2 reviews some of the basic concepts from differential topology, including the fundamental Whitney Embedding Theorem and Sard Critical Point Theorem. Basic properties of smooth maps are also studied.
Chapter 3 studies the general vector bundle and specializes to the tangent bundle of a smooth manifold. The author is keen on the idea that the reader 'grok' (i.e. understand intuitively) the tangent bundle and the associated induced maps and commutative diagrams. The notion of orientability is also introduced.
Multilinear forms and their tensor product are studied in Chapter 4. This is a key building block in the construction of de Rham cohomology. The author gets side tracked a bit with a discussion of differences in classical/modern notion.
Chapter 5 is a very nice chapter on vector fields. Instead of just appealing to results from differential equations (as is usually done) to build integral curves and the flow of a vector field, Spivak establishes these needed results from differential equations using a very accessible integral equations/fixed point argument. Once the flow of a vector field is show to exist (locally), Lie derivatives and Lie brackets are then studied.
Following the integral curves & vector fields material in the previous chapter, the author detours a bit and studies the problem of integral manifolds of dimensions other than 1 along with applications to foliations in Chapter 6. Spivak establishes a basic version of the Frobenius Integrability Theorem and uses examples to motivate the result before diving into the proof.
The basics of de Rham cohomology are established in Chapter 7 and Chapter 8. Alternating and skew-symmetric forms are discussed, although is may be easiest to establish some of the needed results on the symmetric group of permutations after reviewing Hungerford's algebra text. Differential forms and their wedge product are defined, and Frobenius' Theorem can now be restated in terms of differential forms. Two versions of Stokes Theorem are established and this result is applied to integrating forms on manifolds and studying properties of the degree of a proper map of between manifolds. The formal definition of the de Rham cohomology groups is given and some basic calculations are carried out.
The author does something curious with one of the main results of de Rham cohomology, namely the homotopy-invariance property. He starts this with a discussion section in Chapter 7 (not a called out theorem) in which contractible manifolds are show to have zero cohomology in all dimension by an explicit calculation showing all closed k-forms are exact. The results that the author establishes in Chapter 7 for this `one-off' calculation are precisely what are needed to show the more general result that homotopic maps induce equivalent homomorphisms of de Rham cohomology later in Chapter 8.
Chapter 9 is a very nice chapter covering several foundational topics of Riemannian geometry; include the Riemannian metric, geodesics, the exponential map, geodesic completeness and tubular neighborhoods.
Chapter 10 is a short chapter on Lie groups and is something of a detour from the main thread. The author uses the material as a source of application of the material from the first nine chapters.
Returning to de Rham cohomology in Chapter 11, more foundational results from algebraic topology are studied, including exact sequences, Poincare Duality, the Thom class and the index of a vector field.
The book contains many wonderful geometric diagrams which help motivate the material. In most cases, the author is very careful to highlight theorems, propositions and lemmas. Occasionally key results will be 'buried' in a series of discussion paragraphs, which makes referring to these results later on somewhat difficult. The author never, ever calls out or highlights any of his definitions. This can be somewhat frustrating, especially when trying to track down one of these definitions. Fortunately the index to the book is reasonably good.
The Great American Differential Geometry Book.......2003-07-10
Michael Spivak begins these five volumes stating his modest aim to write the "Great American Differential Geometry book." He surely has. Instead of listing the numerous subjects Spivak treats clearly and beautifully in these volumes, I'd like to call out the delightful travelogue style in which they are written, using history, anecdotes, and opinion to explain, illuminate, and, when possible, motivate the gleaming modern edifice. Spivak's opinions are sprinkled lightly here and there like easter eggs. How could you not love a math book that uses the subtitle "The Debauch of Indices," or dismisses Eric Temple Bell's history as "supercilious remarks of questionable taste"? Also, don't miss the annotated bibliography in volume 5. The fact that legions of professionals refer to these books in their original *typewritten* format [1st & 2d editions] is a further testament to their quality. The third edition is typeset using TeX and, though beautiful, still manages to retain a little of the quirky typewritten appearance. One quibble: I was disappointed to see that this edition did not use Richard Bassein's bizarre artwork [think 70s psychedelic] for the covers; I admit that this stuff weirded me out originally, but have grown to love it -- where else could I see fuzzy trolls in crowns made from Enneper's minimal surface?
Let Spivak take you "All the Way With Gauss-Bonnet."
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