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A carefully revised edition of the well-respected ODE text, whose unique treatment provides a smooth transition to critical understanding of proofs of basic theorems. First chapters present a rigorous treatment of background material; middle chapters deal in detail with systems of nonlinear differential equations; final chapters are devoted to the study of second-order linear differential equations. The power of the theory of ODE is illustrated throughout by deriving the properties of important special functions, such as Bessel functions, hypergeometric functions, and the more common orthogonal polynomials, from their defining differential equations and boundary conditions. Contains several hundred exercises. Prerequisite is a first course in ODE. ... Read more

Customer Reviews (5)

If you have a class which uses this book, drop it.
I can't believe that this is the fourth edition of this book. Considering the lack of organization and typos, I'd think it was an unpublished draft of a book. It's honestly the worst book I have ever used. I found it harder to learn anything from than rudin. It's unclear what knowledge is a prerequisite for the material here, as the authors seeem to do some sort of review of first order linear differential equations in the beginning but then go off on different tangents. Often terminology will be left undefined, especially in the exercises where it actually matters the most. One highlight of this happening is the chapter on limit cycles, where no formal definition "limit cycle" is actually given. Perhaps this wouldn't be too bad, but there are exercises asking to PROVE things about limit cycles. And I assure you that this is not at all an isolated incident.

My impression is that the authors attempted to write a terse, rigorous account of differential equations. Instead we get an unmotivated disorganized potpourri of topics whose significance is never made clear.

Worst math textbook I have ever had the misfortune of using.
This is a terrible textbook.After being forced to use it for a class, I can safely say that in every set of problems there are at least 5 errors.I'm not talking about small typos--these errors disrupt the entire problem.Often, the author screws up equations entirely in the problems and randomly changes what he has named each variable.Also, it appears as if the authors didn't bother to solve out the problems they were doing.Some of the problems lead to long and tedious computation: if you're in an Ordinary Differential Equations class, you should know how to take a derivative.There's no point in forcing people to solve a fifth order linear Differential Equation leading to a system of 5 equations, it just wastes time and serves no purpose in illuminating the ideas the authors were trying to teach (try taking the fourth derivative of 4 variations of c*e^((sqrt(2)/2)*x)*sin((sqrt(2)/2)*x) and solving for the constants and you'll see what I mean).The book is also painfully outdated (Look forward to computers that can calculate to 7 decimal places!!!) rendering entire sections of the book useless.Half of the formulas are wrong anyway and can't be trusted without verification from someone who is already knowledgeable about the subject.For a book that is so expensive, and in paperback no less, I expected more examples and at least a reasonable editor.Unless you're forced to buy this book, or enjoy reading an unedited copy that the authors clearly didn't even look at after they typed, I would highly recommend against wasting so much of your money.Overall, just a terrible terrible textbook all around and not worth even buying as a reference text.

About those mistakes...
Possibly the best, and certainly the most realistic, mathematics exam question is a statement that you are to prove, if it is true, or for which you're to provide a counterexample, if it's false.Since you're not told whether the statement is true, the first, and frequently hardest, step is to figure out whether you believe it, thus determining whether you next seek a proof or a counterexample.Extra credit is awarded, if you can elegantly amend the hypotheses of a false statement to produce one that's true and proceed to a proof, which can sometimes seem like an afterthought, once you've understood the situation.(For what it's worth, Birkhoff loved this type of question, when he taught from this book.)

I say this type of question is realistic because you find yourself in the same place, when trying to solve a real problem.For example, suppose you know that a certain proposition is true, but some calculations you're doing would be made easier, if its converse were also true.Your first step isn't (or shouldn't be!) to jump in and try to prove the converse.Rather, you ask yourself whether the converse makes sense--i.e., would its truth contradict some other statement that you know to be true, or do other things you know suggest its truth?If it passes this test, you proceed to construct a simple example in which the converse holds, then try to make the example a little more realistic and probably harder, because you've weakened its hypotheses.Only after constructing a few examples whose hypotheses don't seem artificially strong do you finally try to prove the converse.(If your examples are good ones, they may suggest how a proof works but unfortunately not always, especially if the most likely proof is nonconstructive.They will in any case give you an idea of what the hypotheses and any auxiliary conditions should look like.)

My point is that in good exam questions and real problems, you almost never know whether what you're trying to prove is actually true, despite the best evidence you've accumulated. Mathematical experience entails time spent trying to prove things later realized to be false--or, almost as-bad, true but uninteresting--because that's how we learn in any logically rigorous subject.Regrettably, knowledge absent the real frustration of framing appropriate questions is beginner's luxury soon left behind in the practice of pure or applied science.Indeed, the balance-of-frustration, as it were, not infrequently tilts toward posing the right questions, walking the maze sometimes being less problematic than mapping it in the first place.

To my mind this is the best high-end ODE book around, assuming you've already worked your way through a book like Braun that emphasizes calculation.Among the more applied texts its main competitor is Carrier and Pearson, but that text is in many ways idiosyncratic because of the authors' very Socratic approach.If your bent is a bit more theoretical, at this level there are Arnold; Devaney, Hirsch, and Smale; and its predecessor, Hirsch and Smale, which Amazon incorrectly attributes to Smale alone.But the geometric (dynamical-system) approach of these books is sufficiently different from B&R that they're complements, rather than substitutes.Pretty much the same is true of Hurewicz's superb older text.At an advanced level Coddington and Levinson remains the best reference, for my purposes at least, despite its age.Arnold's advanced text is also excellent but again has a geometric emphasis, as its title suggests.(Full citations are in Listmania; see my Amazon profile.I should stress that I'm no doubt ignoring many fine favorites, a few of which appear at the end of my citations, either through unfamiliarity, or because I haven't actually used them.)

The errors in B&R obviously shouldn't be there.But having been warned, you'll learn a great deal, if you think of them as practice for problem-solving by making sure you actually believe a statement before you try to prove it.We do this in some measure all the time, of course, but the errors in B&R mean that the reader should sometimes make explicit what's usually an implicit step.For example, think through the statement's implications, try to construct counterexamples, and so forth.If you conclude that it''s wrong, ask how the statement's hypotheses need to be modified to make it work.

To wit, my fellow reviewer did finally conclude that an infinite sequence ofSturm-Liouville eigenfunctions is bounded, and his bringing this up in his review suggests that he's never forgotten it.Thus, for the thoughtful the exercise was successful, if not quite in the way intended, and even if "frustrating [for] a student," as our learned friend complains.Catching just a few errors in this way will make you so gloriously self-confident that you'll start probing the questions and problems in every math book you use, which is a very good thing.

As the saying goes, "When life gives you lemons [especially the juicy ones, like this book], make lemonade!"

A WORD ABOUT COUNTEREXAMPLES.Talented analysts, like mathematicians in other fields, tend to keep a a relatively small number of very carefully constructed counterexamples in mind and try to adapt them to situations that arise as a step toward their understanding.One of my professors, a mathematician of some note (not Birkhoff or Rota but easily in their league), had a counterexample he called the "wandering, shrinking interval" that wrecked many incorrect conjectures, much to his delight.It takes astonishingly few of these counterexamples to be really effective--a dozen, say, may be on the high side, if the counterexamples are well-chosen and evolve over time--and each analyst seems to have his or her own list, with a much larger, less subtle, core of interesting pathologies that all good analysts seem to know.

The subtleties ultimately derive from an oral tradition that's handed down from professor to student, which is the real purpose of advanced instruction at any level.However, a number of books talk about the well-known core in various areas of mathematics--e.g., analysis, topology, topological vector spaces (functional analysis), differential equations, and probability.They can get you started in constructing your own counterexamples, not least by getting you into the conversations where the more sophisticated concepts are discussed; this review's Listmania page on my profile has pointers.

POSTSCRIPT (added 18 Mar 2009). When I read this review after almost a year, I'm struck that there's no errata in later printings of the Fourth Edition.This edition first appeared in January 1989; Birkhoff died in November 1996; and Rota in April 1999.It seems highly likely that the authors became aware of many errors before their deaths, colleagues and students being what they are, so there probably was ample opportunity to issue sucha sheet, if they'd so desired.

They may have concluded, however, that corrections are superfluous, because a careful student of mathematics--the type that they tried to reach, and of which there's never been any shortage in Cambridge--would proceed as I've suggested without further ado.Indeed, an instructor can see the uncertainty as beneficial, as it is in the exam questions I've described.There's no knowing at this point, of course, but I would find it quite plausible that they reasoned in this way. --PJES

Full of mistakes
This text is based upon lecture notes (written either by Rota and then bound by Birkhoff or vice versa) and it is VERY apparant that the authors spent little time on checking for mistakes.It seems as though every fourth problem has a typographical error or is completely incorrect.This textbook does not take an intuitive approach to the subject and the many errors gives me reason to rate it low.

Nothing is more frustrating to a student than to have a text make a claim, ask you to prove it, and then you find out the claim is false.This is not a unique occurance.Steer clear of this textbook.

Good text but watch for errors
This is a good text for a course in ODE after the computational course is completed.This book covers many topics in ODE, but be careful and watchfor errors; especially near the end of the text.Most of the errors comein the excercises.For instance: you are asked to show that "everyfinite sequence of eigenfunctions of a S-L system is bounded".Duh! This is trivial until one realizes that every infinite sequence is alsobounded. ... Read more

Product Description
Now available in paperback, this classic, written by two young instructors who became giants in their field, has shaped the understanding of modern algebra for generations of mathematicians and remains a valuable reference and text for self study and college courses. ... Read more

Customer Reviews (4)

too concise for its own good
Any college level math text should have AT LEAST one of the following, but hopefully both:

1) plenty of examples illustrating the use of the taught proofs, theorems, algorithms, procedures, etc.

2) Answers to AT LEAST some of the exercises in the back of the text.

Without one of these, how would you ever know you were grasping the material?

Unfortunately, this book comes up pretty short on #1 and completely empty on #2. there are many sections that are lacking examples, and there are NO solutions to the exercises.

the book reads more like an ordered collection of theorems and corollaries. Not so great if you wanna teach yourself. Probably more ideal for decorating a professor's bookshelf.

A classic algebra text!Wonderful book...
This is a classic book on Algebra.There is much that I like about it.It is exactly what the name suggests--a survey course.It briefly introduces all sorts of topics, including rings, fields, groups, galois theory, vector spaces, lattices, boolean algebras, and much more.It is written at a fairly elementary level and it generally doesn't go into a great amount of depth in each subject.Interestingly, many (more modern) algebra texts omit a number of rather basic topics in this book.Also, many modern books separate "linear algebra" from "abstract algebra", whereas this book takes a more integrated approach.

I find it exceptionally clear and easy to read.Many of the subjects are made particularly easy; there is a strong concrete flavour to the text.The authors provide good motivation for the material.

I think this book would make excellent reading material for someone who is planning to study algebra.I did not pick it up until early in graduate school, and I wish I had had access to it earlier, when I was first studying ring and field theory.It is a fantastic reference for intermediate students, since it covers just about all the basics of algebra, and does so in a very understandable way.I think this book would make a fine textbook for an undergraduate course as well.

This is how algebra texts ought to be written
I have just started reading this book, and already I am
enthralled by the beauty and elegance of the authors'
exposition. Assuming nothing more than an acquaintance with
school algebra and a little geometry, they develop
the basic properties of central algebraic structures, including
rings, groups and fields. These are treated by reference to
familiar examples, such as the ring of integers and the
rational, real and complex fields. Everything that one learned
in school algebra is to be found here, though, as is to be
expected, each topic is treated at a rigorous, mathematically
sophisticated level. In the first two chapters, the properties
of the integers and rational numbers are gradually examined,
ultimately down to the definition of addition and multiplication
on the basis of Peano postulates. The authors then consider
polynomials, the real and complex numbers, vector spaces, linear
algebra and other topics.
The writing style is clear, concise and elegant, with each new
concept being carefully defined as it is introduced. The proofs
achieve a satisfying balance between detail and brevity. Indeed,
reading the proofs and completing the exercises would do much, I
am sure, to enhance a reader's mathematical facility.

If you are interested in acquiring a deeper understanding of
algebra, this book should serve as an excellent introduction.

A smorgousborg of symmetries of the square
Modern algebra is an extraordinary topic and Birkhoff and MacLane do a superb job of exploring it. However, as is often the case with mathematical texts, the material can be somewhat dry. ... Read more

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The purpose of the third edition is threefold: to make the deeper ideas of lattice theory accessible to mathematicians generally, to portray its structure, and to indicate some of its most interesting applications. This 1996 reprint includes expanded and updated Additional References. ... Read more

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This book is largely devoted to two special aspects of fluid mechanics: the complicated logical relation between theory and experiment, and applications of symmetry concepts. ... Read more

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A concise survey of the current state of knowledge in 1972 about solving elliptic boundary-value eigenvalue problems with the help of a computer. This volume provides a case study in scientific computing -- the art of utilizing physical intuition, mathematical theorems and algorithms, and modern computer technology to construct and explore realistic models of problems arising in the natural sciences and engineering. ... Read more

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A SHORTER VERSION OF "A Survey of Modern Algebra, REVISED EDITION " ... Read more

Customer Reviews (10)

very solid introduction
i used this text as my first algebra book. if you just began math and want to see a solid introduction to modern math this is it; self-contained and introduces many important ideas that are commonly used in modern math but typically not taught in undergraduate math courses.

Rock solid
I had the first edition of this book as a student about 30 years ago and found that although the text was for the most part readable, the notion of universal element notion from category theory was introduced in a baffling way very early on and this was the stumbling block.
I've been rereading mathematics books out of pleasure of late and wanted to read this one, so I looked up the Amazon reviews. It was written that you should get the third edition. I ordered it and was not disappointed. The stumbling block has been recognized by the authors and pushed to much further in the book. A new chapter on Galois Theory has been added.
The contents of this book is more basic in general than say Lang's book and the reader is not left to fill in the missing pieces and it is generally more readable. What's more the exercises are well chosen to consolidate the learning.
The book is not encyclopedic, but to my mind constitutes a very solid modern grounding for any mathematical student.

Amazing Book
This is a fabulous book.I have had basically one year of effectively 'algebra for non-mathematicians' at a graduate level, and have a background in Physics and Engineering.I find this book to be the perfect next step in terms of understanding things a bit more deeply than Dummit&Foote, in particular because they bring in some wider issues and connect them to the basic material of group theory, ring theory, modules vector spaces etc.In other words it covers the same or similar material as Dummit & Foote, but adds in some wider examples (more lattice theory, operator theory) as well as a higher level of sophistication because it brings in a category theory perspective for example.It seems that this book is really oriented toward seeing the big mathematical picture implicit in lots of physics.So in that sense, for me, it's a fantastic mathematical physics book, without explicitly being so.

Graduate-Level Algebra emphasizing categorical ideas and applications outside algebra
Garrett Birkhoff and S. MacLane's _A Survey of Modern Algebra_ introduced U.S. undergraduates to the (axiomatic) algebra of Emmy Noether and Emil Artin, with elementary topics useful in applications in science and engineering. Birkhoff-MacLane has a place for algebraic number theory, but puts it in its place---Chapter 14! Birkhoff-MacLane features Birkhoff's interests in congruence relations (c.f., universal algebra), partially ordered sets (c.f., lattice theory), and linear algebra and geometry.

MacLane-Birkhoff's Algebra strives to teach algebra using the spirit and the ideas of category theory. Thus module theory is central to the text. However, this text is in theory accessible to undergraduate students, because the level of abstraction increases gradually, the examples are elementary, proofs are given in detail, and most problems can be solved easily (in the beginning chapters). These features make MacLane-Birkhoff a complement to Lang's Algebra, which uses category theory.

(Also, MacLane-Birkhoff does use ideas from lattice theory and universal algebra more than other texts and has a particularly detailed development of linear and multilinear algebra.)

For a more comprehensive graduate textbook, I would recommend Grillet's "Algebra" (which should replace Lang's book except in isolated populations of algebraic number theory).

Superb, if read with the right outlook
Birkhoff and MacLane collaborated for much of their careers, and their "A Survey of Modern Algebra", first published in 1941, was an easy-to-read, easy-to-teach-from, easy-to-learn-from early fruit of their collaboration. This jointly written book "Algebra", first published in 1967 and vastly improved in the 3rd Edition, can be far more difficult to tackle unless one goes at it with understanding of how to approach it. It mostly reflects MacLane's approach, rather than Birkhoff's, and MacLane was not only brilliant, but unusual among pure mathematicians, perhaps even idiosyncratic; he finally died at an advanced age a few months ago, and his passion for his field is reflected in the fact that he continued to advise graduate students well into his 90s, just as he had advised me (and criticized my thinking incessantly) as a graduate student more than 50 years before.

MacLane was far less interested in any particular topic in mathematics, although he was a master of many, than he was in how one should think about mathematics to understand it, do it on one's own, extend it, and most important of all, recognize when one had fully though through a problem and solved it, as contrasted to having merely produced a plausible discussion of it.

I know of no book on pure mathematics more worth reading than this one, but in contrast to some other reviewers who are probably clearer thinkers than I, I have to tackle it with great patience and care. The secret of grasping it without getting bogged down is to keep constantly in mind that MacLane filled in details without being much interested in them except as necessary completion of exposition. So, when you read it, do not concentrate on details; concentrate on overall structure of thought and exposition and then, later, come back to absorb details. That was how MacLane worked, and that was how he tried to teach his students to work. The key question always in his mind was: what formulation of axioms and structure is fruitful for attacking the topic at hand, and how can we use that formulation to create an inexorable train of thought leading to important results? This book, "Algebra" is very much a reflection of that way of thinking.

So, when you first read this book, skip freely over much of the development of particular topics. Instead, spend a great deal of time thinking about definitions, and about the precise way in which key theorems are stated. Spend time and effort exploring the question of why seemingly trivial variations of these would be less fruitful, or could even lead one into error. Skip from one part of the book to another, without getting bogged down in any one part. Ask yourself also why certain topics and certain cases are excluded. E.g. right at the beginning of the discussion of quadratic forms is a simple definition which begins: "If V is a finite dimensional vector space over a field F of characteristic not 2, ..." Pause right there and ponder over why fields of characteristic 2 are excluded from this definition; just skim the next ten pages without studying them. If you think hard enough to see why fields of characteristic 2 must be excluded from the discussion, the entire ensuing discussion of quadratic forms becomes crystal clear.

Once you have mastered the style in which the material is presented, you can quite easily come back and follow the details. And if you do that, I hope you will find this ook as rewarding as I have. ... Read more

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A study of the art and science of solving elliptic problems numerically, with an emphasis on problems that have important scientific and engineering applications, and that are solvable at moderate cost on computing machines. ... Read more

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Chapters: Thoralf Skolem, Garrett Birkhoff, Henry Wallman, Øystein Ore, Robert P. Dilworth, Alfred Horn, Bjarni Jónsson, Richard J. Wood. Source: Wikipedia. Pages: 33. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Thoralf Albert Skolem (23 May 1887 23 March 1963) (Norwegian pronunciation: ) was a Norwegian mathematician known mainly for his work on mathematical logic and set theory. Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skolem attended secondary school in Kristiania (later renamed Oslo), passing the university entrance examinations in 1905. He then entered Det Kongelige Frederiks Universitet to study mathematics, also taking courses in physics, chemistry, zoology and botany. In 1909, he began working as an assistant to the physicist Kristian Birkeland, known for bombarding magnetized spheres with electrons and obtaining aurora-like effects; thus Skolem's first publications were physics papers written jointly with Birkeland. In 1913, Skolem passed the state examinations with distinction, and completed a dissertation titled Investigations on the Algebra of Logic. He also traveled with Birkeland to the Sudan to observe the zodiacal light. He spent the winter semester of 1915 at the University of Göttingen, at the time the leading research center in mathematical logic, metamathematics, and abstract algebra, fields in which Skolem eventually excelled. In 1916 he was appointed a research fellow at Det Kongelige Frederiks Universitet. In 1918, he became a Docent in Mathematics and was elected to the Norwegian Academy of Science and Letters. Skolem did not at first formally enroll as a Ph.D. candidate, believing that the Ph.D. was unnecessary in Norway. He later changed his mind and submitted a thesis in 1926, titled Some theorems about ...More: http://booksllc.net/?id=453765 ... Read more

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An understanding of the developments in classical analysis during the nineteenth century is vital to a full appreciation of the history of twentieth-century mathematical thought. It was during the nineteenth century that the diverse mathematical formulae of the eighteenth century were systematized and the properties of functions of real and complex variables clearly distinguished; and it was then that the calculus matured into the rigorous discipline of today, becoming in the process a dominant influence on mathematics and mathematical physics.

This Source Book, a sequel to D. J. Struik's Source Book in Mathematics, 1200-1800, draws together more than eighty selections from the writings of the most influential mathematicians of the period. Thirteen chapters, each with an introduction by the editor, highlight the major developments in mathematical thinking over the century. All material is in English, and great care has been taken to maintain a high standard of accuracy both in translation and in transcription. Of particular value to historians and philosophers of science, the Source Book should serve as a vital reference to anyone seeking to understand the roots of twentieth-century mathematical thought.