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1. Principles of Mathematical Analysis (International Series in Pure & Applied Mathematics)
by WalterRudin
list price: $132.81
our price: $132.81
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Asin: 007054235X
Catlog: Book (1976-01-01)
Publisher: McGraw-Hill Science/Engineering/Math
Sales Rank: 33187
Average Customer Review: 4.22 out of 5 stars
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Reviews (69)

5-0 out of 5 stars These Five Stars Need an Explanation
I write this review from the perspective of a mathematician who first encountered this book as an undergraduate in the 1970s and who has most recently had the enjoyable experience of teaching from it during the 1999-2000 academic year. "Baby Rudin" is like no other "elementary" text I have ever encountered. I agree with the other reviewers who criticize the book for its lack of pictures, its lack of historical motivation, its lack of "soul." Yet, in the hands of a professor who is prepared to present the pictures, the motivation, and the "soul" that the text itself lacks, this book can form the basis of a deep, rich introduction to the glorious world of real analysis.

Every time I return to this book I discover new and wonderful things in it. For example, in his treatment of the limits of elementary sequences (that are "normally" treated using the log and the exponential function), Rudin uses the binomial theorem with a deftness and facility that contemporary students rarely encounter. Although Rudin's text presents minimal historical background, it is at the same time more faithful to the historical development of the subject than any other text I can think of.

That the book is small and easy to carry around is no disadvantage. Who says that a calculus book has to be the size of the Manhattan phone directory to be valuable?

5-0 out of 5 stars Possibly the best (math) textbook I've seen yet
This book served as my introduction to analysis and to higher math itself. It uses strict "French-style" definition-theorem-proof format, and you might find yourself spending hours on one page. Some so-called "maturity" might be needed in order to submit to what look like unrelated strands of thought in order to reach desired results. But any bright student will be able to handle the material. Let me emphasize this: NO prior introduction to mathematics of ANY sort is NEEDED for reading this book. Nothing besides a logical, patient mind is a prerequisite. Although tastes may vary, I have a revulsion towards math books which do not use this format, but go for a looser, more conversational style. I believe they are much less clear and more difficult for studying introductory material. I suggest that you use this book as your primary introduction to analysis, and look into other "looser" ones whenever you are having trouble. This book is also excellent reference. If you are trying to learn analysis, and ESPECIALLY IF THIS IS YOUR FIRST COURSE IN HIGHER MATH (!), for God's sake don't get a looser book. You will have made a mistake.

5-0 out of 5 stars A Classic "Baby" Analysis Book
This book is a standard undergrad. introduction to Analysis. It provides a nice foundation, making you work at reading proofs and solving problems while getting familiar with the basic concepts -- limsups and infs, basics of continuity, compactness, etc. You would perhaps be better served if this using this book is not your first experience with really doing mathematics, e.g. formal proofs, etc. -- though not Spivak's Calculus on Manifolds or one of J.P. Serre's Arithmetic books, this book is more concise than many. Important theorems such as the Stone Weierstrass are proven in a very clean brief way (this may not lead to the most useful of proof styles -- you may find yourself expending precious time on cleaning up proofs -- "does leaving this step in make me look stupid?" -- and perhaps cutting so much that proofs may look "infelicitous."). I also do not remember this book being strong on Lebesgue theory and don't remember discussion of Littlewood's principles, Radon Nikodym, etc. These, the real substance of Real Analysis, are best seen in Royden or Rudin's Real and Complex book.Moreover, some professors prefer the sigma algebra approach to measures -- the wonderful S. Kakutani, for example, who briefly guest taught the class in which I used this book insisted on reteaching measures using sigma algebras.

5-0 out of 5 stars If you are serious about doing math...
then I suggest you use this book for your introduction to analysis. I divide up my critique into the following sections:

The author of this book expects you to be comfortable with mappings, set theory, linear algebra, etc. I would recommend that you use either Munkres' book on topology, or (if you can't afford that) the Dover book, Introduction to Topology by Bert Mendelson (you should read all of Ch. 3 BEFORE starting Rudin if you want to pick up on which things could be even more general than they are in Rudin - refer to earlier chapters if you don't recognize something). I suggest also looking at continuity in one of the topology books I mentioned. Also, look up the following things and at least know what they are before getting past Ch. 4, so you have some supplemental language to use: Banach space, boundary, basis for a topology, functional.

Like I said, this book is for serious people, and it requires strong focus for you to pick up on all the subtle arguments made through his examples. I do not agree with some people who say this book is bad for an introduction, in fact I think it is the best because Rudin REFUSES to be tied down to single variable concepts which could be explained just as easily in the context of more general spaces. If you are one of those kids who think's you're great at math because you do well in competitions, steer clear; your place is playing with series, inequlities, and magic tricks. If you are a get-your-hands-dirty kind of mathematician, then you should never let this book leave your side.

I think that it may be a different style than most people are used to, but once you get past that I think I would call the readability nearly perfect. He strips away most general useless commentary (for example, in Gallians poor algebra book, "In high school, students study polynomials with integer coefficients, rational coefficients, and perhaps even complex coefficients"). In Rudin, you get no nonsense -- only math.

The real trick to getting in his swing of things is to MAKE SURE YOU COMPLETE HIS PROOFS. They are extremely slick and often are polished in such a way that it's like his little secret. If you can't do one on your own, just ask the prof in office hours or put it aside for later. The proofs are not presented in this way as to imply that you should just accept them, he wants you to dig in and justify the intermediate steps for yourself, so do it and you'll be good by Ch. 3, I promise.

Many exercises in this book are often found as theorems in other books. What's so unique about this book is that very few problems are solved by simple definition pushing, especially as you go further into the book. That's why I call this the get-your-hands-dirty book, because you'll be forced to, and believe me you'll recognize changes in the way you think if you do this diligently. So, do as many exercises as you can, esp in Ch. 2 and Ch. 4, they will help you the most in this book. What's great about the problems is that they challange you to make REAL connections between ideas and create your own equivalent ways of thinking about the subject. I often have to conjecture and prove several lemmas to avoid wimping out and using "clearly" in my proofs.

If you really really love math and know in your heart that you need to get better to be happy in life, you should cover Ch.1-Ch.6 before Juior year of college and finish it before grad school. I also suggest using this book as a stepping stone to more advanced books -- see Halmos' Measure Theory and know it before grad school.

Finally, DO NOT BE AFRAID! You really have to commit to this book before getting into it, do not be afraid. My best advice to any mathematician is to know your weaknesses, BUT to respond promptly to them.

4-0 out of 5 stars If you take your mathematics seriously,otherwise runlikehell
I previously reviewed this book and gave it 5 stars. But as time passes, my opinion of this book changes shade. 5 stars is too much for this book - I have other books that are much more useful to me than Rudin's.

This book is hyped up a lot by intimidating professors (and competitive students), but does not deliver the goods. Many people feel that Rudin is concise and effective. But to me, Rudin is terse and weak.

It is not hard to discover why his book is in fact so ineffective. The reason is that he is trying to cover too much ground in too few pages.

The core of this goal, is probably a sick conspiracy: to achieve the impossibe --- to be the most bought math book in history (required text for every math curriculum), yet at the same time cover all the difficult topics that 99% of Math majors will never master without graduate studies.

This all reaches a peak in his neglectful treatment of multivariate functions. It would be a shame if a student really had to learn Multivariate analysis from this book. (However, Rudin is good to keep handy if you are doing problems from Spivak's book.)

The end result, is that this book is extremely demanding for even the eager student, who is seeing it for the first time. Nobody I know, in result, has benefited much from this book.

One final criticism. For those, like myself, who haven't worked all the problems in this book, Rudin is a pretty terrible reference. I once had the misfortune of trying to reference his proof of L'Hospitals. In conclusion, I found it easier to reprove L'Hospital myself than to read his cryptic use of the real axioms.

Now with so many criticisms, I must explain why I have given 4 stars.

There comes a point in time, for any respectable math student, that he must develop the ability to solve difficult, abstract problems with little explanation of how and why.

In this regards, Rudin's book could be an extremely valuable resource. He has left a trail (THE PROBLEMS!!) which goes through many crucial ideas in Mathematics. Few books, at the undergraduate level, have such a vast amount of problems - aimed at the budding math student. In this respect, Rudin should get no less than 5 stars.

But I stand at 4. Regretfully, Mathematics departments everywhere have forced the Rudin pedagogy on everyone. I believe the student should make this choice (i.e. which books to study in detail).

And since it was forced on me, I have a voice in this matter: This book should not be on the undegraduate curriculum. And in fact, I don't like his style, I don't like this book, and I'll do problems elsewhere, thanks.


p.s. If you happening to be struggling through the book at this time, here is some advice: Keep your freshman Cal book handy. Don't become a victim, and don't go through this course not knowing how to prove the limit laws, the definition of a derivative, Mean value theorem, derivative laws the proof of the fundamental theorem of calculus, and theorems involving integrals of continuous functions, convergence divergence tests, power series representations, partial derivatives. Note that all of these topics are indeed in a freshman cal course. (Well, this is what popped into my head, not a formal and complete list..)

It is here where calculus actually can become very useful. For example you can define the logarithm, exponential function - and this leads to a definition of a real exponent without using inf / sup 's as Rudin does in a Chapter 1 problem. ... Read more

2. A First Course in the Finite Element Method
by Daryl L. Logan
list price: $128.95
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Asin: 0534385176
Catlog: Book (2001-04-05)
Publisher: Thomson-Engineering
Sales Rank: 299669
Average Customer Review: 5 out of 5 stars
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Book Description

This third edition provides a simple, basic approach to the finite element method that can be understood by both undergraduate and graduate students. It does not have the usual prerequisites (such as structural analysis) required by most available texts in this area. The book is written primarily as a basic learning tool for the undergraduate student in civil and mechanical engineering whose main interest is in stress analysis and heat transfer. The text is geared toward those who want to apply the finite element method as a tool to solve practical physical problems. ... Read more

Reviews (1)

5-0 out of 5 stars Excellent first course in FEM
This book may be the best text book I have come across in years.
If you have the basis of engineering (or physics or maths) you barely need an instructor.
The explainations are clear, and I haven't found any mistakes yet (if there are mistakes they will be few).
Lots of nice problems with answers provided.

This book is excellent for whom desires to learn on which basis FEMs work (undergraduate and graduate). ... Read more

3. Real Analysis (3rd Edition)
by Halsey Royden
list price: $103.00
our price: $103.00
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Asin: 0024041513
Catlog: Book (1988-02-02)
Publisher: Prentice Hall
Sales Rank: 236211
Average Customer Review: 3.69 out of 5 stars
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Book Description

Real Analysis is a shorter version of the author's Advanced Calculus text, and contains just the first nine chapters from the longer text. It provides a rigorous treatment of the fundamental concepts of mathematical analysis for functions of a single variable in a clear, direct way. The author wants students to leave the course with an appreciation of the subject's coherence and significance, and an understanding of the ideas that underlie mathematical analysis. ... Read more

Reviews (16)

3-0 out of 5 stars Excellent Reference, Poor Introduction
This book is an excellent reference for analysis. The proofs are modern and generally excellent, and the treatment of measure and integration is very advanced. However, I would not recommend this book as an introduction; not very much motivation or intuition is provided for the material, and the double-coverage of abstract and real cases is awkward. Recommended substitute: Walter Rudin, Real and Complex Analysis, and Kolmogorov, Introductory Real Analysis.

4-0 out of 5 stars This is a pretty average Real Analysis book
The book presents a nice introduction to Lebesgue integration theory. There are a lot of typos so it isn't a good book to learn on your own from because his notation is not rigors. The book doesn't make a good reference either, being that a lot of the important results are left as exercises. So I would have to say this book isn't great, but I would recommend it over a lot of the other analysis books out there. I personally found that Halmos "Measure Theory," was pretty good.

5-0 out of 5 stars this book is just plain good.
I began as a graduate student in applied maths less than a year ago; all of the students that I spoke with prior to that said that real analysis with rudin's book was their worse & hardest class..
So when I walked into MTH 5111 Real Variables I thought oh *&^% what am I in for?? but then I picked up the Royden book and I understood the way he was presenting the materail.. the book is very stright to the point + leaves channelgning problems to the HW sets but the autor clearly outlines. I have learned more from this book and course than any other...

4-0 out of 5 stars Not bad for self-study, excellent for reference
I used Royden (2nd edition) as a graduate student over 30 years ago, and have been away from real analysis pretty much ever since (not because of the book(!), but because of being in computers). I've taken a renewed interest in the subject (I'm a pretty random person) and have been surprised at how the material has come back to me, I think because of the readability of the text. It's true, Royden challenges the reader at every turn, but if one has acquired the level of mathematical maturity commensurate with strong interest in analysis, the challenges are appropriate, in my opinion

4-0 out of 5 stars I'm surprised too.
All the negative oppinions say that It repeats allmost identical
contents twice. I cannot understand why so many people feel so uneasy about that. Acutally, It does not repeat the "same" thing. In part one, measure theory on the real line is presented and, after you get pretty good understanding and image in the "real world", the abstraction (or equivalently, axiomazation)of measure In abstract space is given. I think It's the best way of explaining something. I know that many people in this field love the "rudin Style" - Books which contains Definitions and Theorems only.
Oh I envy them. I wish I had the ability to understand something from the essense without rumbling the world I can touch for some time. but It's pretty hard thing to do for me, and I'm sure that most others also are. If you agree with the negative oppinions than you can start from part three. you can get all the contents of Part One. and hopefully, You would be able to Understand all the materials. But I don't think that you will get an clearer image than those who have startde from part one.Even those who complain the system of this book could do that cause they already have read the part one and studied the "same thing" Twice. If they have not read them and started from the part three, they would have complained that It was so abstract. ... Read more

4. Introduction to Analysis
by EdwardD. Gaughan
list price: $130.95
our price: $130.95
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Asin: 0534351778
Catlog: Book (1997-12-10)
Publisher: Brooks Cole
Sales Rank: 229724
Average Customer Review: 4.0 out of 5 stars
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Book Description

This book takes a rigorous approach to, and therefore creates a deeper understanding of, the usual topics handled in one-dimensional calculus--limits, continuity, differentiation, integration, and infinite series. The text was designed to bridge the gap between intuitive calculus courses normally offered at the undergraduate level and the sophisticated analysis courses offered at the senior or graduate level. The author wrote the book with two goals in mind: the development of a rigorous foundation for the basic topics of analysis, and the less tangible acquisition of an accurate intuitive feeling for analysis. ... Read more

Reviews (6)

5-0 out of 5 stars Excellent Introduction
I believe the author has given students a wonderful text for the beginning of analysis.I disagree with the student who stated that the proofs are unclear.When I took this class last fall, all of the proofs were not clear in the beginning but with a little clarifying by my professor, they quickly became understandable.The problems at the end of the sections are designed to help students fully understand the chapter and while some of the problems require a lot of extra work, they are not impossible.It seems to me if you gave this book a bad review, it was because you were unwilling to put forth the time and effort to fully understand the material.This book was extremely helpful and guided the class in a clear and concise manner.

1-0 out of 5 stars Extremely unclear
As a math major, I am of course interested in the subject of Real Analysis. But I cannot agree with the other reviews in that the book is very UNACCESSIBLE to undergraduates! I have poured through the first 3 chapters until I almost have them memorized and the concepts are NOT CLEAR. Instead of explaining what is going on, the author rambles on about the "beauty" of his subject and how "we" as students should "carefully" digest the ideas. The examples and proofs are NOT clear and are NOTHING like the HW problems! In fact the proofs are so incomplete, that proving them in the fashion that the author does, does not convince my professor??? This book is a BIG waste of over $100 and a class that I might have enjoyed if the author could see outside of his ego and stop trying to show off to undergraduates who do not know the subject yet!

5-0 out of 5 stars Great book by Great Guru
In my college days ,i found Edward D gaughan's 'Introduction to Analysis',amidst pile of other books.I read the preface countless times.With his words of encouragement and admonishment ,writer makes you more comfortable ,in your long journey ofMathematics which is real and abstract at once!! I liked this book;After a decade still i fondly remember this book !!I recommend this book whole heartedly, to any one who is sincere in his approach and wants to become first rate mathematician.

5-0 out of 5 stars Excellent
At first glance, the book appears too thin.But once you've delved a few pages into it, you immediately recognize that the author has simply mastered the art of brevity - while remainining eminently readable.The book covers all the material needed for an undergraduate analysis course - and covers it extremely well.The proofs are easy to follow and extremely instructive, so that the book can also be used in a "Bridge to Advanced Mathematics" or "How to do Proofs" that some schools offer to their math majors to ease the transition from problem-based lower-division courses to proof-based upper-division courses.

4-0 out of 5 stars A Good, Rigorous Introduction to Analysis
I used this book for an undergraduate course in Analysis and was pleased with it. It is well-organized with poignant examples. The physical construction of the book, however, proved not to be as good as the contentand it is quite worn. ... Read more

5. The Concepts and Practice of Mathematical Finance (Mathematics, Finance and Risk)
by Mark S. Joshi
list price: $50.00
our price: $39.50
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Asin: 0521823552
Catlog: Book (2003-12-24)
Publisher: Cambridge University Press
Sales Rank: 56094
Average Customer Review: 4.5 out of 5 stars
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Book Description

This introductory text provides a clear understanding of the intuition behind derivatives pricing, how models are implemented, and how they are used and adapted in practice. M. Joshi covers the strengths and weaknesses of such models as stochastic volatility, jump diffusion, and variance gamma, as well as the Black-Scholes. Examples and exercises, with answers, as well as computer projects, challenge the mind and encourage learning how to become a good quantitative analyst. ... Read more

Reviews (4)

3-0 out of 5 stars Good book on the basics
This book comes in between Wilmott and the more technical books.
And its by no means complete, if you want a more comprehensive treatment you may want to buy wilmott.
And if you need something more technical you should
get the book by Oskendal and/or Nielsen.

If you want to get an inexpensive book then go for this.

5-0 out of 5 stars An outstanding book in a crowded field
In recent years bookshelves (and readers) have groaned under the weight of new First Courses in Mathematical Finance. There is, of course, a huge overlap in content and it is no easy task to write a book which is both better than its predecessors and genuinely novel. In both tasks Mark Joshi has succeeded admirably: this book deserves to become the leader in its field.

Finding the right level of mathematical sophistication is a difficult balancing act in which it is impossible to please all readers. Here, the author has had a clear vision that the principal audience is the practising or potential quantitative analyst (or quant) and writes accordingly; it is impossible to do better than taking an approach of this sort. Such a quant must have a certain minimum level of mathematical background (a good degree in a numerate discipline). By definition, this has to be assumed for a decent understanding of the material, but the author always has an eye on what a quant really needs to know. Integrated into this mathematical work is a good deal of information about how markets, banks and other corporations operate in practice, not found in more academically-oriented books.

The first half of the book includes the core material found in any decent first course on the subject including basic stochastic calculus, pricing of European options through discounted expectation under a risk-neutral measure, the Black-Scholes differential equation and so forth. Where this book really stands out, however, is the exceptional clarity with which the key concepts are separated. Not only are three different ways for deriving the Black-Scholes formula presented (through PDEs, expectation, and the limit of discrete tree-models) ; much more significantly, the different roles played by hedging, replication and equivalent martingale measures in enforcing a price are made crystal clear. In whatever way you already think about this material, you will almost certainly come away with something new from reading this treatment. In my case, for example, I gained a much greater understanding of why "risk-neutral" pricing is so called.

The second half of the book, roughly speaking, covers a selection of more sophisticated material. The major areas covered include interest-rate derivatives and models; and more complicated models for stock price evolution (such as stochastic-volatility, jump-diffusion and variance-gamma) that have been proposed to correct inadequacies in the Black-Scholes model such as its failure to explain market smiles. Once the core ideas have been so thoroughly explained in the first half, a great deal of interesting and diverse material can be covered rapidly yet with a great deal of clarity and coherence, relating the new models to core ideas such as uniqueness of prices and hedging issues.

Those with quantitative finance experience are still likely to find a good deal that is new and worthwhile in this book. And if you a thinking about becoming a quant, I cannot think of a better book to read first.

5-0 out of 5 stars Most comprehensive
This is the most comprehensive and up to date textbook on quantitative finance that I have seen so far. Joshi is an excellent mathematician and an excellent quant. He knows finance like the back of his hand, and explains it very well.

5-0 out of 5 stars A must read for anyone interested in mathematical finance
The modern paradigm within mathematical finance is the use of martingale
methods for the pricing of options; an understanding of it is
critcal not only to quants who use these mathematical tools on a day
to day basis, but also to risk professionals in general when understanding the
risks inherent in a new product. At present, however,
there are very few accessible texts that discuss this at a level that
is suitable for the (sizeable) interested audience; texts either do not
have adequate coverage of the martingale methodology, concentrating on the
older less insightful pde methods, or concentrate (too much in the
reviewers opinion) on mathematical rigour and
require a substantial understanding
of probability theory before one is able to understand and appreciate
the finance.

Mark Joshi's book fills this niche admirably: it is mathematically rigorous
where it needs to be, but more importantly "physically" insightful --- the
author takes considerable pain in assisting the reader in developing
an intuition both for the models used and the products that are
priced. However, the mathematics is all there; more importantly
for the finance professional there are details on how to implement the
various models described. Again in marked contrast to other texts available
the book includes a number of relevant exercises (with solutions) and
computer projects --- features which this reviewer welcomes.
The book is also to be applauded on the fact that
it does not end after a discussion of the Black Scholes stock case ! Instead
the second half of the book discusses, admittedly assuming a slightly higher
level of mathematical sophistication (but never beyond, what one would
expect of a good physical sciences/mathematics graduate), multiasset options,
the LIBOR market model, stochastic volatility and jump diffusion models.
This again is a key strength of the text, rendering these subjects far
more accessible to a wider audience.

In short this is a book which anyone who is interested in mathematical
finance should have on their book shelf. ... Read more

6. Probability and Computing : Randomized Algorithms and Probabilistic Analysis
by Michael Mitzenmacher, Eli Upfal
list price: $55.00
our price: $55.00
(price subject to change: see help)
Asin: 0521835402
Catlog: Book (2005-01-31)
Publisher: Cambridge University Press
Sales Rank: 639049
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Book Description

Assuming only an elementary background in discrete mathematics, this textbook is an excellent introduction to the probabilistic techniques and paradigms used in the development of probabilistic algorithms and analyses. It includes random sampling, expectations, Markov's and Chevyshev's inequalities, Chernoff bounds, balls and bins models, the probabilistic method, Markov chains, MCMC, martingales, entropy, and other topics. The book is designed to accompany a one- or two-semester course for graduate students in computer science and applied mathematics. ... Read more

7. The Art and Craft of Problem Solving
by PaulZeitz
list price: $65.95
our price: $65.95
(price subject to change: see help)
Asin: 0471135712
Catlog: Book (1999-02-09)
Publisher: Wiley
Sales Rank: 38955
Average Customer Review: 4.5 out of 5 stars
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Book Description

This text on mathematical problem solving provides a comprehensive outline of "problemsolving-ology," concentrating on strategy and tactics.It discusses a number of standard mathematical subjects such as combinatorics and calculus from a problem solver's perspective. ... Read more

Reviews (14)

5-0 out of 5 stars General Problem Solving Strategies.
Perfect match for all math problem solvers.
Wonderful Book with around 660 problems.
Level National Math competition, IMO, Putnam.
If I have to pick the best two problem solving books so far publish in the English Language
Problem-Solving Strategies (Problem Books in Mathematics) by Arthur Engel and this Book by Paul Zeitz are the clear winners.

This particular book has very clear explanations of the main problem solving strategies illustrated with carefully sample problems. Reading this book brings to my memory the works of Polya. One of the only things I think the book is lacking is on strategies to solve Geometry problems in particular or to use the same strategies in the book to solve more Geometrically flavor problems. Nevertheless is a Joy to read.
Please Paul keep writing this beautiful problem solving books.

5-0 out of 5 stars One of the best
This book is indeed one of the best problem-solving textbook so far. As a frequent lecturer of Taiwan IMO team, I have many many MO books. Most of the books available are well-written by professionals and excellent mathematicians. However, since IMO does really prevail in recent years, these authors could not be the participants themselves (^^). Furthermore, usually these books (except those are merely problems collections) contains a good proportion of "harder" and beautiful problems, and the easier and basic training problems are relatively few. It often get the beginners frustrate.

Now this maybe is the first book written by a member of former MO team, and now a training lecturer. (The author himself won the USAMO and IMO in 1974, and helped train several USA IMO teams, including the 1994 "perfect score team"). So here is the precious experience! Besides, the ratio between the harder problems and the easier problems is really good. In my opinion this is an excellent textbook for ambitious beginners (both teachers and students), for self-studys and problem-solving fans. Highly recommended.

5-0 out of 5 stars Essential for budding (and experienced) problem-solvers
I join the ranks of previous reviewers here who honestly feel that having read this book in high school would have almost certainly changed my life. I, too, did very well in high school math competitions, but the maturity I am gleaning from this gem may have vaulted me into a different league.

It contains hundreds of problems from various levels of competition, from AIME problems all the way through some of the toughest Putnam problems (which, if you know anything about the Putnam, are about as hard as competition problems come). But the biggest help are the vital insights and exciting ways of looking at these problems. Don't take my word for it-- many past IMO contestants have suggested this book too.

You don't have to be a math competition buff to gain from this book, however. If you're simply interested in mathematical puzzles and problems, and looking to expand your repertoire, this book will help you. Anyone with a good dose of intelligence and motivation will benefit.

For an additional problem book, check out Mathematical Olympiad Challenges by Andreescu and Gelca. For purely Putnam treatment, there are several volumes written by Kedlaya. And if you're a CS student, looking for honing those CS math skills to be razor sharp, you should definitely look into Concrete Mathematics by Graham, Knuth, and Patashnik.

Happy solving.

5-0 out of 5 stars The Book I wish I had in High School
When I was in high school, I placed second in the Alabama State Mathematics Contest and won many others. However, I might could have been competitive with the IMO style problems had I had this book and would be much better off today had I seen this book earlier.

This book is for the exceptionally brilliant and the mentally tough. It is absolutely necessary to approach this book in a different way from a standard math textbook. You MUST attempt the examples BEFORE looking at the example solutions, NO MATTER HOW DIFFICULT OR FRUSTRATING. You may be bamboozled by the problems, but even trying to understand the problems before looking at the solutions and thinking about how a solution might proceed will pay huge dividends in the long run.

For example, in the first chapter Zeitz presents an example asking the reader to prove that the product of four consecutive integers cannot be a perfect square. The solution involves some clever algebraic trickery not visible to the inexperienced, but persistence and getting your hands dirty is key.

If you persist in spite of the considerable difficulty, you will find that you get better very, very quickly. You will also notice that it isn't just contest problems it helps you solve. I have found that I have solved my homework sets in the Berkeley graduate engineering program much more easily since working these problems. You will start to see creative and clever solutions where they exist in everything problem oriented.


5-0 out of 5 stars Brilliant
As a high school student that is essentially bored with the regular, ho-hum classes that my school offers, this book is perfect. It gives a problem-solving foundation for math enthusiasts desiring to compete nationally in contests like the AMC, AIME, and USAMO. The problems are excellent and cover a wide range of difficulty (past ASHMEs, USAMOs, and, finally, IMOs); and the solutions are well-written, logical, and intelligible. In short, if you are looking to "get better" at problem solving, this is the book for you.

Note: I also bought Problem-Solving Strategies by Arthur Engle. Those, perhaps more advanced, problem-solvers that want even more of a challenge should purchase this book as well (as both books give very challenging problems, but Engel's is undoubtedly more advanced). ... Read more

8. Schaum's Outline of Calculus
by Elliott Mendelson, Frank Ayres
list price: $16.95
our price: $11.53
(price subject to change: see help)
Asin: 0070419736
Catlog: Book (1999-06-28)
Publisher: McGraw-Hill
Sales Rank: 7932
Average Customer Review: 3.88 out of 5 stars
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Book Description

Students can gain a thorough understanding of differential and integral calculus with this powerful study tool. They'll also find the related analytic geometry much easier. The clear review of algebra and geometry in this edition will make calculus easier for students who wish to strengthen their knowledge in these areas. Updated to meet the emphasis in current courses, this new edition of a popular guide­­--more than 104,000 copies were bought of the prior edition--­­includes problems and examples using graphing calculators.

... Read more

Reviews (8)

2-0 out of 5 stars Good for review but errors abound
A good overview of calculus, though--admittedly--I am using it to review the subject rather than learn it for the first time. Very practical, and offers enough examples to get the hang of it in most cases.

However, the egregious number of serious errors in the book (in a 4th edition?!) can often be frustrating if not misleading. Some errors are misstatements of theorems or errors in the worked problems! Others include mislabeled graphs, incorrect PROBLEMS (yes!), incorrect answers etc. Believe, me, I've spent hours checking my work, assuming I had made the mistake (but have verified using mathematica, graphing calculators etc.) For someone working nearly every problem, this leads to a lot of confusion and a huge waste of time. I estimate that I have found 20-30 major errors already, and I've only finished the chapters covering calculus of a single variable. :(

If they had errata published, it might be a little better, but haven't been able to find any.

Unfortunately, haven't tried other review texts...probably better just to get a real calculus book. I've forgotten the one I used in high school and subsequently sold. :(

2-0 out of 5 stars I wish it had more....
This book was a bummer, man. I LOVED the Schaum's Outline for PreCalculus and it was awesome. This book however, left me wanting more. It needs more SOLVED exercises. In each section you get about 3-5 solved problems and then 10-15 problems with answers without solutions. I bought this book hoping it would be a supplement to my text book, but it just didn't have enough step-by-step solutions which is what I need when I'm learning new material.

5-0 out of 5 stars Schaum's Calculus
I've worked with several versions of the Schaum's Calculus
over the years. This work has excellent coverage of derivatives,
integrals, curvilinear motion, polar coordinates, indeterminate
forms, indefinite integrals, centroids, arc length, tests for
divergence/convergence, partial derivatives, volumes, triple
integrals and a host of exotic areas. There are many multi-
dimensional diagrams to aid in your understanding of this
fairly complex subject. I did well in Intermediate Calculus

garnering an "A". In addition, the Fundamentals of Engineering
Licensure Exam covered quite a bit of basic and intermediate
calculus. This is an excellent supplementary work to complement
the course textbook and class notes.

5-0 out of 5 stars An excellent companion to Calculus
I found this book to be a very good supplement to anyone taking a calculus course. The main highlights (and some, but few) lowlights are as follows:

The Good:
1. LOTS and LOTS of topics covered ranging from limit concepts to l'Hopital's rule to integral tests to multiple integrals, this book covers A LOT (and even a brief intro to differential equations.)

2. Enough practice problems to ensure that the reader will comprehend the material (as is the case with most Schaum Outline books).

3. Lots of graphs for visual learners.

4. A fraction of the price of most calculus books.

The Bad:
1. The only bad thing I could possibly think of in this book is that it explains vector concepts and differentiation and integration of vector functions and gradient, divergence, and curl, but leaves out Green's and Stokes' theorems (must be covered in the vector analysis Schaum book).

For more detail, check out the list of chapter topics on the back cover of the book (it's a pretty thick paragraph)

5-0 out of 5 stars Indispensible if you want the "A"...
My instructor had a nervous breakdown about 1 month into an integral calculus class. He spent the rest of the semester discussing his personal problems during class, instead of teaching. He stopped giving tests and cancelled his office hours. We had a midterm, which I failed (with a 28/100), along with the rest of the class. My entire grade hinged on the final exam. I bought this book and spent the last half of the year using this book to teach me integral calculus. Two weeks before the final, the instructuor told the class that he was throwing out the midterm, and that our grade for the class would be based solely on our performance on the final exam.

I got a 96/100 on the final, and an "A" for the course. This book saved me. (This sounds ridiculous, I know...but it is absolutely true.) ... Read more

9. Introduction to Real Analysis, 3rd Edition
by Robert G.Bartle, Donald R.Sherbert
list price: $113.95
our price: $113.95
(price subject to change: see help)
Asin: 0471321486
Catlog: Book (1999-09-21)
Publisher: Wiley
Sales Rank: 219853
Average Customer Review: 4 out of 5 stars
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Book Description

In recent years, mathematics has become valuable in many areas, including economics and management science as well as the physical sciences, engineering and computer science. Therefore, this book provides the fundamental concepts and techniques of real analysis for readers in all of these areas. It helps one develop the ability to think deductively, analyze mathematical situations and extend ideas to a new context. Like the first two editions, this edition maintains the same spirit and user-friendly approach with some streamlined arguments, a few new examples, rearranged topics, and a new chapter on the Generalized Riemann Integral. ... Read more

Reviews (11)

4-0 out of 5 stars Good guidance to the world of analysis
This book is very helpful to those student who want a advanced calculas process and need a basement to the study of real analysis. This book has many example which are very helpful to the student and we can have a chance to think about the process to the solution. Best textbook of what i have read this year!!

5-0 out of 5 stars An excellent real analysis text !!!
A good introduction to real analysis. Proofs are detailed. This book is definitely for anyone who loves real analysis.

3-0 out of 5 stars A concise and to-the-point Real Analysis book
~I am a non-Math undergraduate but Finance graduate . I found this book quite concise and useful . Mathematical concept are developed in a systematic way and are not too difficult to catch. Graduate Student in Finance may find this book useful to help them develop essential real analysis skills before they move on to study other probability / integration / measure / mathematical finance theory .

However , hints for exercise is not enough , this may create a problem for some beginning~~ undergraduate maths student.~

5-0 out of 5 stars Masterpiece
This is a coherent piece of work. Presentation is crisp. Techniques of proof are rigourous. You need a touch of mathematical mutarity and an informal background in calculus to appreciate the treasures hidden in the pages of this book.

3-0 out of 5 stars It's Not That Good
This text is used at the University at Buffalo, SUNY in the US for the first undergraduate/graduate course (MTH 431/531) in so-called real variable theory.

This text while making some improvements over the years, such as providing more 'examples' in an attempt to help the student understand the theory, it really reflects the major problem in the field of mathematics today.

This problem is the discipline's fixation on abstraction and technique which alienates some less capable and prepared students (and I might add, people in general).

To make my point, the authors, as has been a common complaint, are not really aware of the lack of pedagogy incorporated in the text. This is a major problem with most mathematical and other technical textbooks.

In many of the examples and proofs, the authors leave out important information, expecting that the already stressed and overloaded graduate student will figure out on their own. Many of the examples are not instructive at all, but very frustrating because they are too complicated. There is in many places of the text too much information left out, and in other places points/claims made with no explanation. This is true of most mathematical textbooks and renders them worthless in my opinion for learning.

This textbook is not suitable, in my opinion, for use in a big university where there is poor instruction along with a major lack of faculty/student support for beginning graduate students. It would be better if there was some tutelage along with the texts overkill of brevity.

In conclusion, this book illustrates a major problem in mathematics education in this country (USA) today, along with what appears to me as the problem of professional snobbery of disciplines. ... Read more

10. The Visual Display of Quantitative Information
by Edward R. Tufte
list price: $40.00
our price: $25.20
(price subject to change: see help)
Asin: 0961392142
Catlog: Book (2001-05-01)
Publisher: Graphics Press
Sales Rank: 1137
Average Customer Review: 4.55 out of 5 stars
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Reviews (53)

4-0 out of 5 stars Sets the stage for all information architects
This book will teach you some basics on how to most effectively present quantitative information using various sorts of graphs and charts. Afterwards you will know how and why you should get rid of chart junk (gridlines, tick marks, ornaments, etc.) or alternatively using some of the examples on bad design presented, you will see how to manipulate your audience using the "Lie Factor". Actually the advice given in this book could easily fit within just one piece of paper, but then: This book is simply beautiful. It is state of the art for printed books, you almost feel a passion for it. Mr. Tufte takes his own medicine: No words in this book are superfluous. Illustrations and examples are carefully selected and reprinted with the utmost care. It takes no more than some hours to read the book, but afterwards you can use more than just a few hours to study the examples of timeless graphic displays. The only reason why this book is short of five stars is the following: Mr. Tufte uses quite some space providing statistics about charts found in different publications (chart junk percentages, lie factor. Personally I find this information fairly irrelevant and would have preferred more examples of chart remakes. However this book is definately still a MUST have!

5-0 out of 5 stars I'd give it 6 stars if they'd let me....
Instead I give this book regularly to my students & wish to goodness that more of my colleagues had copies. From the opening pages -where Tufte gives us 4 data sets that are statistically indistinguishable but graphicly at different points of the compass- through the beautifully rendered examples of classical and modern examples of meaningful graphics & "chartjunk" Tufte serves as a wry, witty, and informative guide to the perils & joys of informing or confusing an audience with charts and graphs.
Although in some ways a polemic against the misuse of graphical techniques, Tufte never loses his sense of humor & gives us plenty of really GOOD examples as well as a harsh deconstruction of some truly horrendous images. While this, the first in what has become a series, predates the muddy dawn of computer graphical "presentations" the basic principles outlined in its pages are every bit as applicable to the PowerPoint generation as they were to transparencies & posters. Buy it. read it Use it.

5-0 out of 5 stars You'll Never Make a Chart the Same Way Again
Edward Tufte is a prophet of the Information Age come to warn us that we must repent or be consigned to oblivion.

One of the great advances which has made the Information Age possible has been the development of easy-to-use graphing software to swiftly create charts which used to take skilled draftsmen days to produce.

Unfortunately, the commoditization and automation of this once-dear skill set has resulted in the proliferation of lies, damned lies, and lousy statistics.

Tufte, a Princeton professor and polymath with passionate interest in statistics, information design, and public policy, offers up a thorough diagnosis of what ails our data-rich, information poor society:

- Poor graphical integrity, where the visual proportions are out of synch with the data's proportions

- Chartjunk, unnecessary clutter which reduces the proportion of data-ink in a graphic

- Poor labeling, which robs data of context

- Low-density presentations, where complex and nuanced data are "dumbed down" for the sake of a fleeting aesthetic

Fear not---Dr. Tufte also provides the reader with a course of treatment (called "Graphical Excellence") thoroughly illuminated with real-world examples drawn throughout history.

This is one of those rare works which feeds both your right and left brain. It is a closely-argued work on behalf of clean and clear communications. It is also a wonderful art book depicting the evolution of an often-misunderstood art form.

Whether you're an engineer, a statistician, a businessman, or a teacher, this beautifully-designed book will help you become a more effective communicator.

5-0 out of 5 stars Superb Introduction to Quantitative Information Display
Prof. Tufte uses an excellent assortment of charts and graphics to illustrate his points. I found this book to be a quick read; and one I could return to for years to come, as the principles he describes are quite applicable to web site design. I would recommend this book, in fact, I was impressed enough to sign up for the design seminar.

1-0 out of 5 stars Very Short on Substance; Has Essentially Only A Single Point
Other reviewers have mentioned a few negatives. To me, these mostly boil down to short-on-substance problems. The author is a bit pompous -- which wouldn't matter that much if he had a lot to say. Alas, he does not. In essence, the author makes one -- and only one! -- point with the whole book: eliminate "chart junk" (e.g. 3-D effect bars, etc). He is manically obsessive-compulsive about this point so that he takes it to extremes -- get this: computing "data ink" to "junk ink" ratios he even eliminates the axis line (to increase the ratio). While he's at it, just put tics and only where data are (thus giving marginal distributions of x and y) -- cute idea and it does increase "info"-to-junk ink to the max, but these ideas are nearly absurd extremes. If you really want to learn new techniques and real-value PRINCIPLES get William Cleveland's "Elements of Graphing Data" (original or revised). Don't be put off by publication date -- Cleveland's book is a superbly enjoyable read with eminently useful ideas. I've used principles from Clevelend's book to great effect. I've been graphing for decades, but with Cleveland's book I recently made a very large jump in the quality of my graphical communication. Skip the low-on-substance, one-note Tufte and go for the full-of-substance, emminently useful Cleveland. ... Read more

11. An Introduction to Numerical Analysis
by KendallAtkinson
list price: $109.95
our price: $109.95
(price subject to change: see help)
Asin: 0471624896
Catlog: Book (1989-01)
Publisher: Wiley
Sales Rank: 270674
Average Customer Review: 3.25 out of 5 stars
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Book Description

This Second Edition of a standard numerical analysis text retains organization of the original edition, but all sections have been revised, some extensively, and bibliographies have been updated. New topics covered include optimization, trigonometric interpolation and the fast Fourier transform, numerical differentiation, the method of lines, boundary value problems, the conjugate gradient method, and the least squares solutions of systems of linear equations. Contains many problems, some with solutions. ... Read more

Reviews (4)

5-0 out of 5 stars Excellent Book - not for everyone.
One of the best numerical analysis books I ever came across. This describes the theory behind numerical analysis, so if you expect to find a lot of numerical examples and written algorithms, this is NOT the book you're looking for.
Though there are some examples and algorithm, this is a math book, not a computer science oriented book. So buy this book if you are interested in the mathematical theory and ideas behind numerical analysis. Algorithms come and go, but the theory is always the same.
In my work as a computational physicist I use this book extensively and find it invaluable.
It takes some time to get used to, but little effort in understanding math never killed anyone.

1-0 out of 5 stars Worst Math Book I've ever had to use
As a math major here at the University of Michigan I must say that this is the worst Math book I have ever had to use in a class. The book is full of archaic proofs and lots of general solutions, but no practical examples on how these solutions could be used. This book is basically just a big group of equations, without any explanation as to how or why you would want to use any of them. Theres lots of numerical analysis books out there, you could do better than buying this potential paper-weight.

2-0 out of 5 stars thorough, but thoroughly unreadable
This is a standard textbook by a leading authority. There is little hand-waiving here. However, this is hardly a book to learn by.

The typesetting could have been a bit better. I wish the proofs had been set off from the examples and the text a little more. There is also too much referencing to earlier equations. Rather than referring me over and over to equatin (6.2.1), just re-write the equation.

Also, this book is starting to show its age. It is now 11 years old, so its bibliography is a bit outdated, as are references to computer programs.

My most severe criticism of this book is that it is sorely lacking in explanations. There is little intuition provided here. Definately not an undergrad book. A much better text to learn from--but not as useful as a reference as this book is--is Burden and Faires. B&F make lots of use of pseudo-code and I applaud them for it. It helps detangle some of the math.

5-0 out of 5 stars Excellent introduction to numerical analysis
Out of some 7 or 8 numerical analysis texts from which I have learned or taught, this is easily the best. Its organization is standard, its exposition is excellent, it is comprehensive in its coverage of introductory topics, it has a very good bibliography, and its problems are very good. It is a good introduction for graduate students; it is a little advanced for most undergraduates, though strong undergraduates would benefit from its use. No computer coding is supplied though coding from the book's explanations is straightforward. ... Read more

12. Numerical Analysis
by Richard L. Burden, J. Douglas Faires
list price: $140.95
our price: $134.95
(price subject to change: see help)
Asin: 0534382169
Catlog: Book (2000-12-29)
Publisher: Brooks Cole
Sales Rank: 85773
Average Customer Review: 2.85 out of 5 stars
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Book Description

The new Seventh Edition of Burden and Faires' well-respected Numerical Analysis provides a foundation in modern numerical-approximation techniques.Explaining how, why, and when the techniques can be expected to work, the Seventh Edition places an even greater emphasis on building readers' intuition to help them understand why the techniques presented work in general, and why, in some situations, they fail.Applied problems from diverse areas, such as engineering and physical, computer, and biological sciences, are provided so readers can understand how numerical methods are used in real-life situations.The Seventh Edition has been updated and now addresses the evolving use of technology, incorporating it whenever appropriate. ... Read more

Reviews (20)

5-0 out of 5 stars Very moderate calculus is all it takes
Anyone who thinks this book is too difficult and/or requires a Ph.D. in mathematics has simply never learned any math, such as calculus and linear algebra. In that case, it's indeed easier to simply buy software that implements all the necessary numerical algorithms. This book is not a set of instructions for using a calculator, it is a book for an intelligent reader who thinks creatively and wants to understand the logic behind classical numerical methods.

Very transparent, clear, and straight to the point this book is all I needed to quickly learn about the Gaussian quadrature and understanding both the algorithm itself as well as WHY IT WORKS AND DOES SO EFFICIENTLY. Please disregard the previous author's review, as its poisonous tone alone should suggest that he is trying to blame his own mathematical deficiencies upon the authors of this very worthwhile text.

2-0 out of 5 stars Numerical Analysis for Dummies its not...
This book covers all the topics a reader would expect of numerical analysis and comes with a CD of pre-built code for many of the analysis techniques. From my perspective, the authors' present theorem and proof with relatively few examples. I found myself referring to Gerald and Wheatley's Applied Numerical Analysis (among others) for the duration of my college course to attain the level of understanding expected by the university. Gotta love libraries! At $.., this is the most expensive math book I've purchased, and I can say that I wouldn't value it at this price if it had not been selected by the university. Best of luck to those who read it...

3-0 out of 5 stars full of errors
I normally don't write reviews for books, but I felt compelled to say that this book has quite a few errors that I've personally found quite annoying. The errors aren't mentioned in the authors' online errata either, which covers only the 1st printing. I'd think you could iron out most bugs after 7 editions, but apparently not. The coverage of material itself, while not great, is acceptable, but there are random errors scattered throughout that threw me off. At least a few of the algorithms, when implemented, don't work properly. Some of the solutions in the back aren't accurate or are just wrong (e.g., some ask for what h you need to be below a certain error bound, then proceed to give a larger h than is really necessary). Just my two cents.

4-0 out of 5 stars Review of Numerical Analysis, 7th edition
This is a numerical analysis book written from a mathematician's point of view, and requires from the reader a good background in calculus and linear algebra.

Even though the book has an initial chapter ("mathematical preliminaries"), reading this chapter is not enough if the student has not a good previous mathematical knowledge.

The book introduces modern approximation techniques and explains how, why and when these techniques are expected to work, and allows the reader to understand why one algorithm works better than other for a given problem.

The text contains many examples as well as application problems in various areas of science and engineering.

The book uses Maple as the standard software for symbolic and approximate calculus, even though Mathematica and Derive are mentioned too and could be used instead with small modifications.

The original English edition (7th edition) includes a CD-ROM with all the algorithms, expressed in different formats (C, Fortran, Pascal, Maple, Mathematica and MATLAB), although the Spanish translation (edited by Thomson Learning) does not include the CD-ROM. However, there is an Internet address in which the CD-ROM contents can be accessed.

To conclude, the book is a good text that requires a mathematical background from the reader and covers a broad range of modern approximation techniques. It is not a mere numerical methods cookbook, but a text that analyzes and applies the numerical methods instead.

2-0 out of 5 stars Wordy, poor algorithms, worse code
Like other reviewers, I'm still struggling to find a decent advanced mathematics textbook. Some of the problems with Burden's book includes insufficient examples and explanations. He introduces strange and unnecessary notation in his algorithms; for example in chapter 7 (Iterative techniques for solving linear systems) many of his index loops run from 1 to n. If he set them from 0 to n-1, it would clean up much of his logic. He also apparently loves the variable XO to represent the initial approximation x naught.

Maybe due to my physics background, but his notation of representing indexes of variables as a _power_ is confusing:
Burden represents the i-th index of x as x^(i), not to be confused the i-th power of x: x^i. Modern typesetting includes subscripts, why not use them instead? Heck, use LaTeX and do the same thing (x_i)!

Finally, several of the codes on the included CD refused to run, and some of them didn't give correct answers. You will need some programming experience to edit, as none of the codes (at least all of the Matlab and possibly all of the C) adhere to any programming standards or formatting. Mr. Burden (or his programmer) is invited to purchase and use Steve McConnell's "Code Complete"--or hire someone who knows how to write maintainable code well. What is the purpose of supplying code if it cannot be used in other projects? "Gee Wiz, the book includes Code!" one might exclaim. "But what good is it?" is the inevitable response. ... Read more

13. Topics in Matrix Analysis
by Roger A. Horn, Charles R. Johnson
list price: $50.00
our price: $50.00
(price subject to change: see help)
Asin: 0521467136
Catlog: Book (1994-06-24)
Publisher: Cambridge University Press
Sales Rank: 323255
Average Customer Review: 5 out of 5 stars
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Book Description

Building on the foundations of its predecessor volume, Matrix Analysis, this book treats in detail several topics with important applications and of special mathematical interest in matrix theory not included in the previous text.These topics include the field of values, stable matrices and inertia, singular values, matrix equations and Kronecker products, Hadamard products, and matrices and functions. The authors assume a background in elementary linear algebra and knowledge of rudimentary analytical concepts.This should be welcomed by graduate students and researchers in a variety of mathematical fields and as an advanced text and modern reference book. ... Read more

Reviews (1)

5-0 out of 5 stars A great reference source for advanced matrix analysis
Horn and Johnson's MATRIX ANALYSIS AND TOPICS IN MATRIX ANALYSIS are true classics (like Knuth's Art of Computer Programming). You will find classic theorems and lemmas in matrix theory and linear algebra here along with their proofs (some of these are not found elsewhere).

TOPICS IN MATRIX ANALYSIS contains a lot of stuff including LMI's, Kronecker and Hadamard products of matrices and their properties etc. I found this book indispensible when I was studying Semidefinite Programming.

Both these books are now available in paperback (cost around 30+) dollars each. I have recently purchased both copies and can only strongly recommend them to anyone else. ... Read more

14. Real and Complex Analysis (Higher Mathematics Series)
by WalterRudin
list price: $135.31
our price: $135.31
(price subject to change: see help)
Asin: 0070542341
Catlog: Book (1986-05-01)
Publisher: McGraw-Hill Science/Engineering/Math
Sales Rank: 80292
Average Customer Review: 4.6 out of 5 stars
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Book Description

This is an advanced text for the one- or two-semester course in analysis taught primarily to math, science, computer science, and electrical engineering majors at the junior, senior or graduate level. ... Read more

Reviews (15)

5-0 out of 5 stars Best (math) book ever written
This text is a model of mathematical style. The usual Rudin stuff: concise and elegant proofs, great chanllenging exercises and that undefinable sense of quality -mathematical taste- pervading all the book.

The book covers the standard material on 'real variable' (measure theory') in a masterful and compact way; then it goes through the standard complex analysis to a level deeper than usual and showing in a very original way its intertwining with real variable. The final third of the book is devoted to more specialized topics.

Just a warning: the construction of Lebesgue measure is based on Riesz representation theorem, whose lengthy proof is imposed to the reader in chapter 2. It is really tough, and makes this chapter much harder to read than the rest of the book.

If you want to learn REAL mathematics, this is the book for you, you'll learn not only the subject matter, but a great style as well.

5-0 out of 5 stars Excellent, often intriguing treatment of the subject
The first part of this book is a very solid treatment of introductory graduate-level real analysis, covering measure theory, Banach and Hilbert spaces, and Fourier transforms. The second half, equally strong but often more innovative, is a detailed study of single-variable complex analysis, starting with the most basic properties of analytic functions and culminating with chapters on Hp spaces and holomorphic Fourier transforms. What makes this book unique is Rudin's use of 20th-century real analysis in his exposition of "classical" complex analysis; for example, he uses the Hahn-Banach and Riesz Representation theorems in his proof of Runge's theorem on approximation by rational functions. At times, the relationship circles back; for example, he combines work on zeroes of holomorphic functions with measure theory to prove a generalization of the Weierstrass approximation theorem which gives a simple necessary and sufficient condition for a subset S of the natural numbers to have the property that the span of {t^n:n in S} is dense in the space of continuous functions on the interval. All in all, in addition to being a very good standard textbook, Real and Complex Analysis is at times a fascinating journey through the relationships between the branches of analysis.

5-0 out of 5 stars Welcome to the self-service analysis center!
This year we have been using Walter Rudin's treatise as the main text for a standard first-year graduate sequence on real analysis, backed up by Wheeden/Zygmund's book on Measure and Integral, and the two seem to complement each other quite nicely. Rudin writes in a very user-friendly yet concise manner, and at the same time he masterfully manages to maintain the high level of formality required from a graduate mathematics text. To be totally honest, a few years ago my very first attempt at learning graduate-level real analysis in a classroom setting (via Folland's book) was not successful, as I found the exposition in Folland very dense and rigid, and the homework problems too difficult to do. Rudin's book however is a lot more accessible for the beginning graduate students who may not have had any more than some basic exposure to measure theory in their upper division analysis classes. One point to keep in mind is that Rudin developes the measure in the more formal axiomatic way, instead of in the more concrete constructive approach. In the constructive approach, one first introduces the "subadditive" outer measure as a set function which is defined on the power set P(X) of a nonempty set X. One then proceeds by showing that the restriction of the domain of the outer measure to a smaller class of subsets of X (a sigma algebra M), obtained via applying the Caratheodory's criterion, results in a "countably additive" set function which is called a measure on (X,M). (The latter is the approach taken in both H.L. Royden and Wheden/Zygmund). The formal approach is not very intuitive and is less natural for a beginning graduate student who might not have developed a certain level of mathematical maturity yet.

Also, Rudin does not discuss some of the more advanced or interdisciplinary topics such as distribution theory (Sobolev spaces, weak derivatives, etc.) or applications of measure theory to the probability theory, both explored in the book by Folland. Last but not least, it's worth noting that contrary to the common practice, Folland includes many end-of-chapter notes where he outlines some important historical aspects of the development of the topics, and also gives a few references for further study. For example, in the notes section at the end of the chapter on Lebesgue integration, he mentions --and briefly outlines-- the basics of the theory of "gauge integration" (also called Henstock-Kurzweil theory) which serves to construct a more powerful integral than that of the Lebesgue's. As another instance, having already defined and used "nets" within the chapter on topology, in the end-notes Folland also introduces "filters" and "ultrafilters". These are all machineries which have been developed to play the role of the metric space sequences in general (locally Hausdorff) topological spaces, but for some historical reasons, ultrafilters have nowadays taken a backseat to the nets (the older general topology books usually prove the Tychonoff theorem using ultrafilters). All said, I can recommend taking up Royden as your very first approach to measure theory, then based on how well you think you have learned the first course, move on to either Rudin or Folland for a more advanced treatment. Please note that the other books I have mentioned above do not discuss complex analysis, a subject which is also masterfully presented in Rudin. There are however a few other equally well-written complex analysis books to pick from, for example John B. Conway's classic from the Springer-Verlag graduate series, or L.V. Ahlfors' masterpiece, to name just a couple.

5-0 out of 5 stars A Comprehensive Guide to Analysis
Rudin's Real and Complex Analysis is an excellent book for several reasons. Most importantly, it manages to encompass a whole range of mathematics in one reasonably-sized volume. Furthermore, its problems are not mere extensions of the proofs given in the text or trivial applications of the results- many of the results are alternate proofs to major theorems or different theorems not proved. With that in mind, this book is not appropriate for a course where the instructor wants students to merely understand the theorems well enough to develop applications- the structure of the book is far better suited for a more theoretical course.

For example, the construction of Lebesgue measure is considered one of the most important topics in graduate analysis courses. After this construction, more abstract measures are developed, and then one proves the Riesz Representation Theorem for positive functionals later.

Conversely, Rudin develops a few basic topological tools, such as Urysohn's Theorem and a finite partition of unity, to construct the Radon measure needed in a sweeping proof of Riesz's Theorem. From this, results about regularity follow clearly, and the construction of Lebesgue measure involves little more than a routine check of its invariance properties.

Another example of where Rudin takes a more theoretical approach to provide a more elegant, yet less intuitive proof, is the Lebesgue-Radon-Nikodym theorem. Other books generally introduce signed measures with several examples, and use this result, along with properties of measures to derive the proof. On the other hand, since the first half of the book contains an intermission on Hilbert Space, Rudin uses the completeless of L^2 and the Riesz Representation Theorem for a more sweeping proof.

In the real analysis section, Rudin covers advanced topics generally not covered in a first course on measure theory. The chapters on differentiation and Fourier analysis are key examples of this. Rudin uses maximal functions to develop the Lebesgue Point theorem and results from complex analysis, and provides an incredibly thorough proof of the change-of-variables theorem. The ninth chapter, on Fourier transforms, relies heavily on convolutions, which are developed as a product of Fubini's theorem. This, in turn, is used to prove Plancherel's theorem and the uniqueness of Fourier transforms as a character homomorphism.

The tenth chapter, on basic complex analysis, essentially covers an entire undergraduate course on the subject, with added results based on a solid knowledge of topology on the plane. Once a solid foundation on the topic is laid, Rudin can develop more advanced topics from Harmonic analysis using general results from real analysis like the Hahn-Banach theorem and the Lebesgue Point theorem (for Poisson integrals).

Most of the basic results from the power series perspective are covered in the text, but while the geometric view is examined, it is still done in a very analytic, formula-based way that does not allow the reader to gain too much intuition. Nonetheless, all the basic results are covered, and Rudin uses these to develop the main theorems, such as the Mittag-Leffler and Weierstrass theorems on meromorphic functions, and the Monodromy Theorem and a modular function used to prove Picard's Little Theorem.

As an introductory text, even for advanced students, Rudin should probably be accompanied by more descriptive texts to develop better intuition. In fact, I would recommend Folland's Real Analysis and Ahlfors' Complex Analysis for self-study, because the problems are easier and one can learn better through those. With a good instructor, though, Rudin's text is concise and elegant enough to be both useful and enjoyable. It is also a good test to see how well one REALLY knows the subject.

5-0 out of 5 stars The Holy Bible of Real Analysis
This book has no doubt about it. It is a HOLY BIBLE of Real Analysis Course. Trully speaking, if you know this book means you know mathematics. The book is written in very nice form, the excercises is extremely hard and challenging. Rudin put a lot of efford to make this book became clear in analysis. There is no joke stuff in his book, everything you step to another chapter, it must be related one and another. This book is designed truly for only who wants to become MASTER PIECE IN mathematics course, no doubt about it. Another Suggested reading H L Royden,this book is pretty similar to Rudin. No doubt about it also. Both of them truly are designed for MASTER PIECE. ... Read more

15. Numerical Mathematics and Computing
by E. Ward Cheney, David R. Kincaid
list price: $111.95
our price: $111.95
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Asin: 0534389937
Catlog: Book (2003-07-25)
Publisher: Brooks Cole
Average Customer Review: 2.8 out of 5 stars
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Book Description

Authors Ward Cheney and David Kincaid show students of science and engineering the potential computers have for solving numerical problems and give them ample opportunities to hone their skills in programming and problem solving. The text also helps students learn about errors that inevitably accompany scientific computations and arms them with methods for detecting, predicting, and controlling these errors. A more theoretical text with a different menu of topics is the authors' highly regarded NUMERICAL ANALYSIS: MATHEMATICS OF SCIENTIFIC COMPUTING, THIRD EDITION. ... Read more

Reviews (5)

1-0 out of 5 stars From a student's view......Garbage
I had to use this book for an undergraduate Numerical Analysis class. I'm a Computer Science major with a math minor and this is my last semester. I found this book to be horrible when coupled with an instructor that is equally as horrible. The explainations are too short and lack examples, the problems in each chapter are hard to solve based on the chapter's explaination; they seem to deviate far beyond what was explained in the corresponding chapter. There are some formulas and theorem's mentioned that have no examples to show how they work.

The book is not totally at fault in my case. I also have a horrible instructor and have to rely soley on this book to learn the material. This book just makes it very, very hard to teach myself. My only praise of the book is it's pseudocode for implementing the methods explained. They can easily be used to program them in C++ or other languages.

Overall the book is very confusing but it is still far better than my instructor who doesn't explain anything or answer questions.

4-0 out of 5 stars Basic but Good
This is a good, basic, undergraduate text covering scientific computing. It gives a nice, broad overview of some basic topics, problems for the student to solve, and is generic as far as programming languages are concerned.

That being said, for my use this book was not detailed enough and failed to go into sufficient detail into many different areas (such as the eigenproblem). It is definantly an undergrad text and would be an excellent choice for a 300-level math or computer science class, it also provides a good general background in numerical computing. In that regard this book is a fine choice.

4-0 out of 5 stars Reader from Belgium compares to the wrong book
The description "it looks like they cut the 2nd edition in half and labeled it '4th edition'" would be approximately correct if you talk about the 2nd edition of the much more advanced text by approximately the same authors, "Numerical Analysis: Mathematics of Scientific Computing" by Kincaid, Cheney and Cheney, which was published in 1995 and does conform to the description "covered so much in detail". If you expect graduate level coverage, that is the book to get, not this one, which is an undergraduate text, and aimed at students that don't major in math.

4-0 out of 5 stars A concise text for introductory numerical analysis
I am a mathematics professor at a liberal art college. This is one of the few textbooks that is suitable for our students who are mathematics majors or related sciences majors. The approach is mathematical and computer science oriented, rather than cookbook type or overwhemingly applied. I and my colleages use this text and recommend it.

1-0 out of 5 stars Total piece of crap
Our professor for numerical analysis at the university told us to get this book because it covered so much in detail. He has the 2nd edition for some time now. I just got my 4th edition and it looks like they cut the 2nd edition in half and labeled it '4th edition'. Get another book instead. ... Read more

16. An Introduction to the Finite Element Method (Mcgraw Hill Series in Mechanical Engineering)
by J. N.Reddy
list price: $140.31
our price: $140.31
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Asin: 0072466855
Catlog: Book (2005-01-11)
Publisher: McGraw-Hill Science/Engineering/Math
Sales Rank: 169173
Average Customer Review: 4.5 out of 5 stars
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Book Description

******Text coming late December 2004!******

J.N. Reddy's, An Introduction to the Finite Element Method, third edition is an update of one of the most popular FEM textbooks available.The book retains its strong conceptual approach, clearly examining the mathematical underpinnings of FEM, and providing a general approach of engineering application areas.

Known for its detailed, carefully selected example problems and extensive selection of homework problems, the author has comprehensively covered a wide range of engineering areas making the book approriate for all engineering majors, and underscores the wide range of use FEM has in the professional world.

A supplementary text Web site located at contains password-protected solutions to end-of-chapter problems, general textbook information, supplementary chapters on the FEM1D and FEM2D computer programs, and more! ... Read more

Reviews (9)

5-0 out of 5 stars a very good book
I learnt FE from this book. It is clear, coincise, full of well-worked examples. It covers almost all of the aspectes of FE programming. The book contains two educational codes, thought both are FORTRAN77 code, one can easily understand what's going on, so as to rewrite it in C or, say, FORTRAN90.

The first s chapters of the book regard the FEm method in general, the 4th focuses on structural mechanics, 5th is about errors in FE analysis, 6th and 7th about numerical integration and 8th is on 2 and 3 dimension FE problem.

Actually this is an introductory book, so the 2 and 3 D problems are not deeply trated.

The book is never hard to understand, and it's suitable (waw!!) for self study.

4-0 out of 5 stars It is one of my best list.
Just completing his second and third chapter in FE formulation, you cannot deny Dr.Reddy's ingenius in FE. His solid background in Variational Calculus will enlight you the real nature of Finite Element approximation and bring you understand how can fomulate FE statement from BVP. Don't be bored in his mathematical written styles, because you could not understand FE in other ways.Absolutely, this book is one in my best list of FE textbook. His enlightment could not be found in a plenty of FE textbooks sold in your bookstore. Just read it!!! You may think like me.

5-0 out of 5 stars THE BOOK
This book is a nice starting point for FEM modeling.I have had experience with contol volume and finite difference and am trying to catch on FEM.It helps me where i need help. There is this other
book by Schaums that complements this book. He goes into finite elements from the very basic engineering calculus. This book taught me finite elements. And i have published in FEM since. Quite a few papers as a matter of fact.

I have bought a second book on Finite Elements authored by him. I am in love with the book. The mathematical formulations....the very fact that it looks so abstruse that is the reason i love it..clarity of thought i respect like mad! And he has it all!!!

5-0 out of 5 stars One of the Best Textbooks in FE
Not only was I introduced to the infinite world of Finite Elements through this book, but I have also used it in the FE class I teach at UCSD. I have found the variational formulations approach most helpful as it fits anyone in the various disciplines of engineering and applied mathematics. "Give me the field equation and I shall develop the FE model" is the motto I have used to build a career in FE; and Dr. Reddy's Book was certainly the cornerstone. OK..I am biased...Dr. Reddy was my doctoral advisior; he is the best in the business.

Samuel K. Kassegne, PhD, PE San Diego, California

3-0 out of 5 stars Great Introductory Book
This book is quite good. It will really teach you the basics of Finite element Method. But one problem i encountered is that if you use this book to learn, the next time you need to study deeper FEM, you will have a hardtime understanding the very different approaches and also notation. Butstill this book will give you many things you need to know. ... Read more

17. Calculus Demystified : A Self Teaching Guide (Demystified)
by Steven G. Krantz
list price: $19.95
our price: $13.57
(price subject to change: see help)
Asin: 0071393080
Catlog: Book (2002-08-01)
Publisher: McGraw-Hill Professional
Sales Rank: 120711
Average Customer Review: 2 out of 5 stars
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Book Description

* Explains how to understand calculus in a more intuitive fashion
* Uses practical examples and real data
* Covers both differential and integral calculus
... Read more

Reviews (6)

1-0 out of 5 stars Better look elsewhere...
I consider myself a pretty smart kid. I had straight As throughout high school and I'm currently on a 1 year homestay in Japan. I've always been good at Math and science, able to pick up abstract concepts very easily.

I decided to learn Calculus on my own before I start Cegep, as something to do in my spare time. That being said, I'm having a very hard time with this book. I CAN get through it, but it seems needlessly difficult.

The explanations in here aren't the best for the everyday reader. Many concepts and ideas seem to have been omitted, leaving the reader wondering if he's skipped something important or didn't understand well. And the things which are explained are rigorous definitions that just aren't accessible.

The book is littered with very hard 'Try it yourself' questions to which the book doesn't give any answers, and even the answers provided for the excersises and final exam are not developped in a way so that you can understand if you get it wrong.

There are many other Calculus books out there, so stay away from this one. I guess this book may be alright for someone who just needs a quick revision, but if you're starting from scratch, this book is terrible

And I'd like to point out I was releived to see that other people hated this book too ^_^ I'm glad it's not just me...

1-0 out of 5 stars Demystified? Not likely.
I'm currently in my junior year of college and I have not taken a math class since my junior year of high school. This semester, however, I will be taking calculus. I'm a fairly intelligent person (not a genius by any means) and was able to teach myself college algebra and pre-calc from the "Demystified" series texts without too much trouble (the algebra and pre-calc texts are phenomenal by the way). However, reading this book was like hitting a brick wall crotch first. In fact, I couldn't read it: I gave up after the first few pages and desperately flipped through the remainder of the tome to see if I could find some kind of daylight at the end of the tunnel. None was found. The text was very techinal and seemed to really stress rigor. The book was downright teaming with rigorous definitions and wretched mathematical symbolism: it was more difficult to understand than my Calc text (Single Variable Calculus: Concepts and Contexts by Stewart). I literally thought Calculus was going to be the death of me until I picked up "Claculus for Dummies" and "How to Ace Calculus" (both of which are hilarious). Wow! Calculus can be interesting and fun and understandable. Its almost as if the joy of claculus is a secret that "Calculus Demystified" won't let you in on. Obviously, I recommend that you check out one of the other texts mentioned above unless of course, you have already taken Calculus and are looking for a CHALLENGING refresher. This book was no help to a beginner like me.

2-0 out of 5 stars Calculus made Difficult
This is not a book for learning calculus.

It seriously confuses and complicates simple principles by explaining them in notation only a mathematician could love.

Then it gets worse. The author periodically presents the reader with a "Try it yourself problem" that ought to be called a "Puzzle" because it is more complex than any previous example, and no correct solution is tendered. This only leaves the reader wondering if he got it right after all. It shakes the student's confidence, which leads to second guessing destroying any positive reinforcement crucial to learning.

Not only is it frustrating, it is detrimental to progress since the student is left on his own to confirm that he has properly grasped the concept and is ready to move on. And what if if the student has taught himself incorrectly....?

Some teachers enjoy showing off how well they have mastered a difficult subject. Steven Krantz (the author) seems more interested in demonstrating his own grasp of mathematic notation than teaching the subject of calculus to his students.

If I wanted to discourage a student from ever taking any more math, I would send them a copy of this book. On the other hand, the information is all there and it is complete and correct. The format is one that serves well for a for the highly advanced student seeking a challenging brush up, or the casual mathematic genius looking for the equivalent of a crossword puzzle book in calculus. For us mere mortals, don't even open the book. It will do more harm than good.

2-0 out of 5 stars Calculus Re-mystified
If you think calculus is a difficult subject, this book will only convince you further. This book may be good for someone trying to brush up on calculus skills, but for those trying to learn it for the first time, it will do nothing but confuse. If you want a more easy to follow introduction to calculus, try Silvanus P Thompson's 'Calculus Made Easy'

2-0 out of 5 stars Demystified? Don't Think So.
There comes a time for many instructors when they have been studying upper level esoterica for so long that they forget what it's like to not know the basics of a subject. They assume that since the subject is obvious to them, it must be obvious to everyone. Unfortunately, this seems to be the case with "Calculus Demystified". While this book may be solid review for a former math major, a good introduction it's not.

Some may say that calculus is an inherently boring subject - this book will do nothing to disabuse you of that notion. The writing style is almost a parody of the stultifying math teacher who puts everyone to sleep. I have quite a few math texts that I use for my tutoring business, and this is by far the least accessible of the lot.

For the calculus novice, I'd recommend "How to Ace Calculus: the Streetwise Guide" as a far more readable text. For the more advanced student who wishes to review the basics or to supplement his or her Calc II or III course, "Calculus Demystified" would fit the bill. ... Read more

18. Matrix Analysis
by Roger A. Horn, Charles R. Johnson
list price: $40.00
our price: $40.00
(price subject to change: see help)
Asin: 0521386322
Catlog: Book (1990-02-23)
Publisher: Cambridge University Press
Sales Rank: 226162
Average Customer Review: 4.2 out of 5 stars
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Book Description

Linear algebra and matrix theory have long been fundamental tools in mathematical disciplines as well as fertile fields for research. In this book the authors present classical and recent results of matrix analysis that have proved to be important to applied mathematics. Facts about matrices, beyond those found in an elementary linear algebra course, are needed to understand virtually any area of mathematical science, but the necessary material has appeared only sporadically in the literature and in university curricula. As interest in applied mathematics has grown, the need for a text and reference offering a broad selection of topics in matrix theory has become apparent, and this book meets that need.This volume reflects two concurrent views of matrix analysis. First, it encompasses topics in linear algebra that have arisen out of the needs of mathematical analysis. Second, it is an approach to real and complex linear algebraic problems that does not hesitate to use notions from analysis. Both views are reflected in its choice and treatment of topics. ... Read more

Reviews (5)

1-0 out of 5 stars Not for the uninitiated
I bought this book hoping to learn about matrix analyis. I did not. This book is simply a reference manual with plenty of theorems, axioms etc. with little explanation. They give it to you rough and row. NOT A SINGLE SOLVED EXAMPLE, and not even solutions for the exercises given in the book are provided. If you intend to learn about matrix analysis, as I did, let not the 5 stars review mislead you. Don't make the same mistake, this book is not for you.

5-0 out of 5 stars An encyclopedic reference for matrix analysis and linear alg
Horn and Johnson's MATRIX ANALYSIS is simply a masterpiece. You can find each and every result in matrix analysis along with it's proof in this book. Look at their companion volume "Topics in Matrix Analysis" too. Some of these results cannot be found elsewhere.

5-0 out of 5 stars Excellent book.... for the initiated
Horn and Johnson has written an excellent reference book on somewhat-advanced linear algebra (from the point of view of an engineer). There's a lot of treasures in this book, but this book is NOT for beginning linear algebra. Rather, it is written as a handy reference to review and learn certain topics in linear algebra.

Nonetheless, I really like their take on linear algebra. They motivate you in every subject and problem (for example, the relations between eigenvectors, eigenvalues, and optimization problems). These insights are invaluable and definitely worth the admission price.

Together with Golub and Van Loan's "Matrix Computations", this is THE linear algebra book to get, although only if you're sufficiently initiated.

5-0 out of 5 stars The most comprehensive and up-to-date reference/textbook.
If you need to know anything about matrices - this is the book. It cover all the basic theory. Perfect for graduate-level courses. The book also has lots of material relevant to optimization theory and can be used in Operations Research courses. One of the nice features is a comprehensive index that makes the book a perfect reference source. Nicely written and easy to read.

5-0 out of 5 stars Excellent treatment of the subject matter
Horn and Johnson are clear, interesting, and accurate. The book does a good job of both teaching and intriguing.USA ... Read more

19. Advanced Calculus: A Course in Mathematical Analysis
by Patrick M. Fitzpatrick
list price: $134.95
our price: $134.95
(price subject to change: see help)
Asin: 0534926126
Catlog: Book (1995-08-21)
Publisher: Brooks Cole
Sales Rank: 59256
Average Customer Review: 2.75 out of 5 stars
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Book Description

Advanced Calculus is designed for the two-semester course on functions of one and several variables.The text provides a rigorous treatment of the fundamental concepts of mathematical analysis, yet it does so in a clear, direct way.The author wants students to leave the course with an appreciation of the subject's coherence and significance, and an understanding of the ideas that underlie mathematical analysis. ... Read more

Reviews (12)

4-0 out of 5 stars It's not THAT bad...
OK, it has a few typos, it is kinda hard to read and not very rigorous at some points, but all in all it's still an useful book for an intro to intermediate level analysis course--and I found it concise and very thorough, especially since it has a couple chapters on one variable analysis.

3-0 out of 5 stars Good treatment: Unfortunately this is a textbook
The treatment of analysis in several variables in this book is solid. Fitzpatrick does a decent job explicating the theory while mainting a very rigorous presentation. Unfortunately, this book is a textbook. His examples are trivial while his sample problems are significantly more difficult. Quite frankly, Fitzpatrick should incorporate more examples that require multiple applications of concepts in dealing with proofs other than those of theorems. For students unaccustomed to proof based mathematics, this work is a disastrous introduction to multivariable calculus. Finally, his chapter treating the Hessian Matrix is an abhorration. He fails to distinguish his variables and the confusion is discouraging. Nevertheless, for students with some background in proof-based mathematics, this is a good treatment of the subject.

4-0 out of 5 stars Good for streamlined Intro to Multivariable Analysis
Background: B.S. Econ, B.S. Math, Master's Econ classwork, starting Ph.D. Econ Fall 2004. Took undergrad metric spaces and single-var analysis and some graduate real analysis in the past.

Sat in on a quarter of undergrad Multivariable Analysis (a previous offering of the same class was cancelled last year before I graduated with the math degree). This book was the assigned text. I did all the class homework. The class covered the Differential part of the multivariable chapters, not the integral part, but skipped the metric spaces chapter.

I liked the book despite its several but ususally easily discernable typos. It covered the Multivariable stuff in Rn efficiently for me as an introduction. I can envision many uses in math and econ for the material and approaches used in the book. Now I'll go on to other texts that cover the same material and more but with a higher level of mathematical sophistication (like the Dover published, C.H. Edwards, Calc of Several Variables book they tried two years ago). Also, I'll return to some Optimization texts to get more out of them.

In order to get the most out of the multivar part of the book, you definitely need to have good comfort with concepts and proof techniques used in Single-var analysis and in basic set theory.

3-0 out of 5 stars Not as bad as everyone says
I feel like I have to defend this book a little having used it during my undergraduate years. Sure there are some valid complaints, Fitzpatrick's notation can be poor sometimes and his development of one variable analysis is needlessly cluttered with unnecessary machinery.

Those complaints aside, this text is quite thorough and does a good job motivating and explaining most of the big ideas (which is something that many analysis texts often refrain from doing unfortunately).

If you read the reviews on this page you'll see many complaints that Fitzpatrick doesn't baby his readers by cramming tons of examples into the text to illustrate each concept to death. He will also often omit the details of a proof, only giving a sketch and challenging the reader to complete the proof on his or her own. I agree that this can be a bit aggravating if you use this in your first class in rigorous mathematics. But if you've got a few upper level math classes under your belt and these things still bother you, then perhaps mathematics is not the field you should be specializing in.

OVERALL OPINION: this is not a bad book for a second undergraduate coarse in analysis. If you are looking for a good single variable analysis text and have not done much in term of rigorous math before, then there are plenty of more user friendly texts out there. If you are looking for a thorough and challenging overview of undergraduate analysis, then this text is one of the many possibilities you should consider.

3-0 out of 5 stars it's fair, not great tho
Fitz's Advanced calculus is a fair textbook. It's not great and the Rn is somewhat ponderously developed. Furthermore, there are typo errors that one can occasionally find in the book reading each section. Excercises vary in difficulty, number (some sections only have 10 problems and others have 30 problems) and quality. From my experience with this text, as a TA, the problems really don't develop the skills neccesary for learning higher math; rather, it bludgeons the reader to recall simple ideas and rehash them out in a proof (the Principle of Mathematical Induction is one example, a beautiful tool, but poorly used). Furthermore, certain proofs, like the Triangle Inequality, aren't rigorously laid out in the text. They simply sketch a proof and leave it at that. There are also small, but not inconsequential holes, in other proofs. If one does not have a brilliant lecturer to go along with this mediocore book, then the student could leave the class with terrible skill, or lack thereof, in proofs. Lastly, if one must purchase a Real Analysis text, go with Walter Rudin's Principles of Mathaematical Analysis. If that seems like a very big jump in mathematical difficulty and maturity level, then try Spivak's "Caluclus," Serge Lang's "Undergraduate Analysis," or Apostol as well. This text is a fair book, but certainly not outstanding or worth the price they're asking for.

BTW, if Real Analysis is the reader's first introduction to proof based mathematics, then he might do well to purchase a copy of "An Introduction to Mathematical Reasoning." It's a small book for roughly $30, but it's a wonderful piece to properly develop the skills needed in theoretical math. ... Read more

20. A Mathematician's Apology (Canto)
by G. H. Hardy
list price: $16.99
our price: $11.89
(price subject to change: see help)
Asin: 0521427061
Catlog: Book (1992-01-31)
Publisher: Cambridge University Press
Sales Rank: 34439
Average Customer Review: 4.53 out of 5 stars
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A Mathematician's Apology is a profoundly sad book, the memoir of a man who has reached the end of his ambition, who can no longer effectively practice the art that has consumed him since he was a boy. But at the same time, it is a joyful celebration of the subject--and a stern lecture to those who would sully it by dilettantism or attempts to make it merely useful. "The mathematician's patterns," G.H. Hardy declares, "like the painter's or the poet's, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics."

Hardy was, in his own words, "for a short time the fifth best pure mathematician in the world" and knew full well that "no mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game." In a long biographical foreword to Apology, C.P. Snow (now best known for The Two Cultures) offers invaluable background and a context for his friend's occasionally brusque tone: "His life remained the life of a brilliant young man until he was old; so did his spirit: his games, his interests, kept the lightness of a young don's. And, like many men who keep a young man's interests into their sixties, his last years were the darker for it." Reading Snow's recollections of Hardy's Cambridge University years only makes Apology more poignant. Hardy was popular, a terrific conversationalist, and a notoriously good cricket player.

When summer came, it was taken for granted that we should meet at the cricket ground.... He used to walk round the cinderpath with a long, loping, clumping-footed stride (he was a slight spare man, physically active even in his late fifties, still playing real tennis), head down, hair, tie, sweaters, papers all flowing, a figure that caught everyone's eyes. "There goes a Greek poet, I'll be bound," once said some cheerful farmer as Hardy passed the score-board.

G.H. Hardy's elegant 1940 memoir has provided generations of mathematicians with pithy quotes and examples for their office walls, and plenty of inspiration to either be great or find something else to do. He is a worthy mentor, a man who understood deeply and profoundly the rewards and losses of true devotion. --Therese Littleton ... Read more

Reviews (15)

5-0 out of 5 stars A classic of the mathematical literature
Every discipline has a list of items that must be read if one is to be considered educated in that field. There is no doubt that this book should be required reading for any degree in mathematics. Most of the soul of mathematics is contained in the 91 pages of the 'Apology' (the first 58 pages consists of the foreword by Snow). Written in his later years when Hardy knew his mathematical powers were failing, this is a superb exposition by a brilliant, eccentric personality. He not only captures the grandeur of mathematical discovery, but also clearly articulates the feelings of a man who knows that his time has passed. First published in 1940, the twin messages are timeless.
Clearly distinguishing between the real mathematician and the puzzle solver, Hardy is exceptional in declaring what the real beauty of mathematics is. Among all the beautiful things that exist, the percentage of individuals that can truly appreciate an elegant theorem is among the smallest. However, anyone who can read this work and not see at least some of the poetic qualities of mathematics has a blind spot in their soul. One of the masterpieces of literature, this book can be understood and appreciated by anyone with an eye for the beautiful things that life has to offer.

Published in Journal of Recreational Mathematics, reprinted with permission.

5-0 out of 5 stars A unique book, should be read by everybody
Hardy was a man that comes along rarely in life, and this book is an even rarer portrait of how men like him think. As a mathematician, Hardy was excellent, his collaboration produced much fruitful work, and he is perhaps most renowned for discovering the young protege Ramanujan. But this book is not really about his work, but about his views on life, and mathematics, as a whole. Considering how little people in American society know about mathematics and its practitioners, this book, which is emminently readable, will give all people a unique view of what some mathematicians think like. The book is short, but interesting from first page to last. Hardy was past his mathematical prime when he wrote this book, but this book probably is his most influencial he ever wrote.

The introduction by C.P. Snow is more like a short biography about Hardy, and it's about the same length as Hardy's actual text. It gives us insights into what one of Hardy's friends thought of him, and it also frames the life Hardy was living in as he wrote this book.

Hardy's opinions are strong, and undoubtedly every reader will disagree here and there with him. But he shows the reader some of the gems of mathematics, and perhaps the reader will be able to appreciate those even without formal mathematical training. He also talks about war and what he thinks of it. Whatever the reader thinks about Hardy's opinions, this book gives us the opportunity to glimpse into the mind of an artist - one different than the usual meaning attached to the word, but one nonetheless - and experience a part of human life not experienced by many - the wonders of mathematics.

4-0 out of 5 stars One should not need an apology
What is discomforting to me about this book is simply that Hardy decided to write it. Hardy describes that people do what they do because they do it well (the so called standard apology), that they do it because they don't do anything else well, or they didn't have a chance to do anything else. Hardy also exclaims that pure mathematics has no utilitarian value, and does not benefit society.

But why can't mathematicians study math for the love of the subject? What is so bad about not caring about utilitarian value? Why should we have to justify our existence to others? The fact that Hardy seems so compelled to justify his existence, and all he comes up with is benefiting a pool of knowledge in the platonic realm, is almost pathetic. Why couldn't he have studied math for his own self-interest? This book would have been all the more refreshing if he stated he loved working with Ramanujan and Littlewood, and that this in itself is a justification, and not some means to some cloudy end.

Other areas of the book are equally disappointing. We hear the commonplace notion that after 40, one's mathematical abilities are pretty much over. Yet this is a self-fulfilling prophecy. Without confidence in your abilities, how do you expect to get anywhere? What about Erdos, who still actively did math up until his death? Wiles was over 40 when he finally resolved Fermat's Last theorem.

You may wonder with this criticism, why I chose 4 stars. This is because, for all its drawbacks, it is at least an interesting account of Hardy's relationship with math. It is however, disheartening how fatalistic he is. It would have been refreshing to hear something of pride, not pretentious or sneering, but pride exclaiming that nobody should ever feel the need to write an apology for their existence.

5-0 out of 5 stars Motivation for anyone questioning being a Mathematician
Hardy, in just a few short chapters, crafted a beautiful book that is nothing less than a devotional for mathematicians. Reading one chapter a day for the rest of my life would keep anyone in a state of mathematical bliss. In my own educational journey, Hardy gave me the feul for the fire to persist in my degree program. Every sentence of this short memoir is quotable, and I would recommend this book for anyone considering entering the field of math as well as anyone who feels stale in their choice of becoming a mathematician. In A Mathematician's Apology, Hardy clearly shows that he is the Knute Rockne of math.

4-0 out of 5 stars Interesting book for the mathematics lover.
I think this is overall a worthy read for any true lover of mathematics. Firstly, its a very short book and can be covered in one or two sittings easily. The book isn't very detailed in a discussion of 'pure' mathematics, so with that said, this is a good book to compare and contrast a person's own ideas about mathematics to. It is NOT a good book to solely get reasons why it is is worthy to study pure mathematics. I also disagree with a thing or two in the book(the major one being that Hardy believes mathematics exists itself in reality and that mathematicians are only observers to this)and because of the historical time the book was written in, some of the statements in the book are completely inaccurate (ie that quantum mechanics and relativity have no 'practical use' and that pure mathematics is of no 'practical' use in wartime and is an 'innocent' or 'immaculate' science because of this). To sum it up, if you have personal ideas about 'pure' mathematics and its justification, then this is a good book to get a few ideas on, but if you have no clue what 'pure' mathematics is and want to know more about it and/or want a justification of its uses, then read a book on the history of mathematics. ... Read more

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