Book Description

This work is motivated by and develops connections between several branches of mathematics and physics--the theories of Lie algebras, finite groups and modular functions in mathematics, and string theory in physics. The first part of the book presents a new mathematical theory of vertex operator algebras, the algebraic counterpart of two-dimensional holomorphic conformal quantum field theory. The remaining part constructs the Monster finite simple group as the automorphism group of a very special vertex operator algebra, called the "moonshine module" because of its relevance to "monstrous moonshine." ... Read more

Book Description

"The present book is a marvelous introduction in the modern theory of manifolds and differential forms. The undergraduate student can closely examine tangent spaces, basic concepts of differential forms, integration on manifolds, Stokes theorem, de Rham- cohomology theorem, differential forms on Riema-nnian manifolds, elements of the theory of differential equations on manifolds (Laplace-Beltrami operators). Every chapter contains useful exercises for the students."- ZENTRALBLATT MATH ... Read more

Reviews (1)

Another fine text by Klaus Janich
Janich's previous texts on topology and linear algebra are very valuable additions to the library of many existing texts on these subjects. His new book on vector analysis is similarly valuable. It deals with manifolds, differential forms, and the generalized Stokes's theorem. This is the mathematical machinery necessary, for example, for mathematical physics and differential geometry. Janich's chapter 10 discusses classical vector analysis, relating Stokes's theorem in its modern form to the classical integrals theorems of vector analysis. He closes the book with a discussion of De Rham cohomology and differential forms on Riemannian manifolds.

Janich's exposition and mathematical taste are, as always, impeccable. Particularly valuable is a discussion in the second chapter which relates three different ways of defining the tangent space to a differentiable manifold. Each chapter also has a multiple choice test with answers in the back of the book which the reader pursuing self study can use to test his/her knowledge of the subject. Exercises close out each chapter as well.

Though this book is a Springer Undergraduate Text in Mathematics, it assumes some knowledge of topology and a fairly thorough knowledge of differential and integral calculus of several variables (the inverse function theorem, the rank theorem, the regular point theorem, and the regular value theorem.) Even readers not extremely well versed with these subjects will enjoy Janich's enthusiasm for his subject and his clarity of exposition. This is a very nice treatment of an important subject! ... Read more

Book Description

This is the most extensively visual book in the market—highlighted by hundreds of Mathematica and MATLAB generated figures throughout. It now contains a full chapter of material on matrices and eigenvalues up front. All of Multivariable Calculus has been rewritten with matrix notation.Chapter topics include infinite series, vectors and matrices, curves and surfaces in space, partial differentiation, multiple integrals, and vector calculus. ... Read more

Reviews (7)

Nice and Succinct Book on the Subject
If there is one thing I adore about Springer books is that they are cheap, to the point, and very accessible. This book is no exception. I used this book for self study after I took a Calculus III course. My understanding of the concepts I learned in class improved two-fold. There is really nothing negative I can say about this book. It is probably the best buy I've ever made yet. I wish all college texts were like this.

Great book when used in the right context.
This book is very useful as long as you know better than to expect to use it alone. The only way I ever used this book is as an adjunct to my textbook (Stewart's multivariable). Where my big text by Stewart is far, far more rigorous on any given topic, this book by Matthews just punches through to the core concepts. When the big book is making you crazy just relax, take a deep breath and pick up this one. Once you understand the most basic fundamentals of what is happening, go back to your big textbook and get to work figuring out the more subtle mathematical nuances of gradient or curl or whatever. There are also some really useful example problems. Given the low price this book is a definite winner.

Good but Brief
This book is very clear, but is mostly a collection of work with very little explanation. This is good when you understand the concepts; there is less to lead you astray. However, if you don't grasp an idea from what is provided, there is little recourse. I used this book in conjunction with the (ubiquitous) Stewart multivariate calculus book. It's not very expensive so I reccomend it.

not 80% but...
Someone has said in a review that
Its mathmathical rigor is about 80%
but as a student majoring in math I
cannot agree with that. I can't give
it more than 60%.
It's the kind of the book which will
never be met in the math class.
but that does not mean this book is not
good.

it lacks in mathmatical rigor but that's
not problem. it's quite clearly written,
easy to read, and most of all It explains
you the 'meaning' of the equation.
lots of math book you should try youself
very hard to understand its meaning.
they give you the proofs of theorems
definitions of mathmatical objects
but not the meaning nor its context.
that's something you should find out.

but this book gives you that. It explains
you what is the meaning of it.

If you are majoring in math. this can be
a gread secondary text. If you are not.
then this should be your first choice.

Excellent for self-study
I purchased this book to further my self-study of Physics. Simply put, this is the best written book I have read on any math subject (although Spivak's Calculus is also very good, but is on a different topic).

A wonderful feature of this book is the problems - and complete solutions - provided throughout. The problems are not so extensive that you will skip them (there about 10-20 per chapter, as opposed to many other texts which have fifty or more) nor so difficult that you will give up in frustration. Indeed, for self study, the simple fact that there are answers is key; the fact that they are not of the sort where you are asked to derive new math or otherwise are treated as if you are a "student who believes he is to become the next Einstein" is a refreshing change.

This book covers vectors (of course), various vector integrals (line, surface, and volume integrals), and gradient, divergence and curl (if you ever wondered what that odd upside down triangle symbol is, you will learn), before moving into more advanced topics. It briefly covers "suffix notation" and tensors, as well as transformations into different coordinate systems. In sum, about half the book is wonderful and immediately useful, and the other half will become more useful as I encounter it in the course of learning physics.

That said, I wish it went into more detail in the "suffix notation" section, which is quite confusing (although several readings helps) as well as more on tensors, which extensively employs the suffix notation.

I have been told, without confirmation, that Springer is known for it's clarity; this book certainly is extremely clear, well written, and has allowed me to finally begin to understand many parts of mathematical physics which were previously Greek to me. ... Read more

Book Description

Excellent study of sets in topological spaces and topological vector spaces includes systematic development of the properties of multi-valued functions. Topics include families of sets, topological spaces, mappings of one set into another, ordered sets, more. Examples included from different domains. 1963 edition.
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Reviews (3)

Excellent and readable book
I'm not particularly strong in math, and I've never taken a course in topology, but this has been a great book to read thru to learn the tools and topics. The writing is crisp and clear, and the author pushes one thru the material slowly and gradually.

I paid Full Price and don't regret it!
Although this is so much better than the other Dover book by R. Vaidyanathaswamy, I can't say that it is the best topology text that I have read or the best reference, but it covers topics from Banach spaces to functional topology in a pretty readable manner and it is fairly inexpensive!

A classic text.
Claude Berge's Topological Spaces is a classic text that deserves to be in the libraries of all mathematical economists. It contains many of the fundamental underpinnings of modern mathematical economics. This book has been long out of print and its reprinting by Dover is very welcome. Readers who find this book of interest should also look at Dover's edition of Samuel Karlin's two volume book on mathematical programming and games that has been reprinted as a single volume. ... Read more

Book Description

This book is devoted to one of the main questions of the theory of extremal problems, namely, to necessary and sufficient extremality conditions. The book consists of four parts. First, the abstract minimization problem with constraints is studied. The next chapter is devoted to one of the most important classes of extremal problems, the optimal control problem. Next, one of the main objects of the calculus of variations is studied, the integral quadratic form. Finally, local properties of smooth nonlinear mappings in a neighborhood of an abnormal point will be discussed.Audience: The book is intended for researchers interested in optimization problems. The book may also be useful for advanced students and postgraduate students. ... Read more

Book Description

Although presented as a topic of low emphasis in introductory course work, duality plays a central role in optimization theory and application. This comprehensive volume covers a wide range of duality topics ranging from simple ideas in network flows to complex issues in non-convex optimization and multicriteria problems.In addition, it examines duality in the context of variational inequalities and vector variational inequalities, as generalizations to optimization.Duality in Optimization and Variational Inequalitieswill enable the reader to use duality to devise more effective computational methods, and to aid more meaningful interpretation of optimization and variational inequality problems. ... Read more

Book Description

Covers all topics of vector calculus, including vector differentiation and integration, theorems of Green and Stokes, and the divergence theorem.Special topics in tensor notation, linear algebra, differentiation geometry, and curvilinear coordinates are also included. ... Read more

Reviews (1)

They worked hard at it, but it's unacceptable
This book apparently has two primary goals: to cover a large amount of material and to do so at as low a price as possible. Those two goals are astonishingly well achieved, but the deficiencies are so severe that they render the book unacceptable.

First, the printing is substandard: unadorned Courier or something similar, not right-justified, and is so primitive that the publishers cannot use much of the usual notation. Since vector analysis uses mathematical symbolism heavily, this is a major disadvantage. The same on-the-cheap approach applies to the diagrams. There is no color, and the diagrams are composed almost entirely of compass-and-ruler constructions and notation in the same typeface as the text.

Second, the index is inadequate. This book is likely to be used as a reference rather than read sequentially, so the index is important. This index doesn?t include some of the most fundamental terms or topics, such as "component" or "dimension". There are scores (if not hundreds) of references to derivatives in the book, and yet the index includes only five references to derivatives.

There is not a single reader exercise in the book (unless I missed something). The book is intended to be a demonstration of how Vector Analysis is done, so all the problems are worked out by the authors. If the readers have their own textbooks, this is not necessarily a big disadvantage. But this book, in the introduction, actually criticizes other texts for requiring the students to do exercises, and goes on to make the following stunning statement: "The staff of REA considers vector analysis a subject that is best learned by allowing students to view the methods of analysis and solution techniques themselves."

I can't overemphasize what an incredible statement that is. There are not two schools of thought on this issue. The idea that a student can learn Vector Analysis (or any mathematical topic) by watching someone else do it, makes as much sense as recommending that someone learn to play the guitar by listening to Eric Clapton. That one point of educational philosophy alone demonstrates conclusively that the book is incompetent.

The Problem Solver series is apparently produced with another eccentric belief, and one that is annoying for anyone accustomed to reading traditional math books. The publishers apparently believe (according to their own words in the introduction) that one of the shortcomings of other books is that they present material without first explicitly stating what the goal of each paragraph or section is. (In my experience, this criticism is sometimes valid, but usually not.) Their solution to this is to make every word in the text, part of the statement or solution to a "Problem". However, these are not problems in the usual sense. For example, the second "problem" in the book asks for the dimensions of five vectors which are presented in component form. At that point in the book (page 3), the word "component" has not been defined, no vectors have yet been presented in component form, and the equality between dimension and the number of components has not yet been mentioned. So this is not a "problem" that anyone would be able to solve if his knowledge were limited to what has been presented in the book up to that point. The book usually presents those necessary definitions in the course of "solving" the problem (although in this case, they don't define the word component, and as far as I can see they NEVER define that word. As I mentioned earlier, "component" and "dimension" are both missing from the index.) This entire quirky approach reminds me of that TV game where the contestants are required to respond with a question instead of an answer. It requires a pointlessly awkward presentation just to cram the material into their ideological system. The definitions and formulas are not designated separately, as in other books, but imbedded in the problem "solutions".

Most important, the book is full of mathematical errors. Page 22 refers to "vectors of the same magnitude" when they mean "vectors of the same dimension". Page 23 has a problem involving vector addition and subtraction, in which they present a1, b1, c1, a2, b2, and c2 as components of vectors (using subscripts), and proceed to use a1c1 as the DIFFERENCE between those two components (or the distance between the terminal points of the horizontal projections of vectors a and c, which is the same thing). Of course, they should have used c1 - a1. This is not just non-standard notation; it is simply mathematically wrong. And I believe they repeat that same error in other problems later on.

In proving an important theorem (the equivalence of the algebraic and geometric/trigonometric definitions of dot products), they suddenly pull out a formula from analytic geometry that very few calculus students in the US will ever have seen. Since the use of that formula renders the rest of the proof entirely trivial, the student will learn nothing at all about how to prove theorems from reading that proof. The same sort of thing happens in Problem 3-6, where they use a formula for finding the components of a projection - a formula which has never been mentioned before, and which they neither derive nor explain.

The book constantly rounds off intermediate results. This should never be done in these days of 12-digit calculators, and ESPECIALLY when the calculated number is to be used as input into an inverse trigonometric or exponential function. In fact, they don't even really round those numbers off - they just truncate them! This causes their final answers to be wrong in many cases, such as in Problem 3-11, where the answer should be the vector (0.279, 0.885, -0.373). This is amateurish. If the authors don't know better than that then they shouldn't be writing math books.

After reading "Problem" 3-16 on pages 81-82, I finally gave up entirely on this book. The problem statement asks for the cross product of two vectors, while sort of half-defining the cross product within the problem statement itself. Then, in the "solution", they don't find the cross-product at all! They find only the magnitude! They simply forgot to finish the problem!

This book was never competently proofread, peer-reviewed, or edited. Presumably that would have been too expensive. But this amateurish production is unusable for learning Vector Calculus. Not recommended. ... Read more

Reviews (1)

great book!
After a hiatus of many, many years I wanted to resume the study of math that I pursued as an undergraduate, and I wanted to start at the heart of analysis: advanced calculus. I found myself bouncing around among several books because:
1. they were really books about measure theory and real analysis with only a cursory review of advanced calc
2. they were really freshman calc books with a short intro to advanced calc
3. they were 400 pages longer than I wanted to deal with
4. they were, in the words of one Amazon reviewer, "one damn theorem after another, written by mathematicians for mathematicians"

Then I found Guzman, who explains, discusses, illustrates and
gives examples.

Guzman gives examples after definitions and theorems, sometimes concrete ones from the physical sciences. He discusses why certain suppositions are neccessary and gives counterexamples.
The proofs are about as clear as any I've seen, and he avoids technical tricks which make proofs shorter but harder to understand. Diagrams give insight into the ways a mathematician
might think about the theorems and proofs. The book includes the complete solutions to all problems, which makes it even better for self study (in some math books the problems are used to introduce new concepts not covered in the main text, but Guzman's problem set sticks pretty close to the material covered). In short: Guzman is a master expositor. When I don't understand something it's clear that the fault lies in my efforts, not in the book - with many texts I feel the opposite is true. (one of the few other books I feel this way about is "Abstract Algebra" by Dummit and Foote).

I think that in the future math will be taught using interactive software that will allow the reader to experiment with ideas and
visualizations of the material, but in terms of educational materials written on dead trees, it doesn't get much better than this. ... Read more

Book Description

This brief book presents an accessible treatment of multivariable calculus with an early emphasis on linear algebra as a tool. Its organization draws strong analogies with the basic ideas of elementary calculus (derivative, integral, and fundamental theorem). Traditional in approach, it is written with an assumption that the student reader may have computing facilities for two- and three-dimensional graphics, and for doing symbolic algebra.Chapter topics include coordinate and vector geometry, differentiation, applications of differentiation, integration, and fundamental theorems.For those with knowledge of introductory calculus in a wide range of disciplines including—but not limited to—mathematics, engineering, physics, chemistry, and economics. ... Read more

Reviews (1)

Not too explanatory,and not suitable for beginners.
I am doing this book for cal III course in my college and we have almost finished it. But what strikes me as the good part of the book is the questions at the end of the chapter. However there is another thing which also has been very unfortunate about the book, it is very concise and starts from a level which is higher than cal II , so for most people (as i have seen in my class) it has been very rigorous at times. Unsufficient explanations and little application makes it a very dry book. Overall i remember the times when i had to stay up till 3:30 in the morning trying to understand the chapters. ... Read more

Book Description

No one working in duality should be without a copy of Convex Analysis and Variational Problems. This book contains different developments of infinite dimensional convex programming in the context of convex analysis, including duality, minmax and Lagrangians, and convexification of nonconvex optimization problems in the calculus of variations (infinite dimension). It also includes the theory of convex duality applied to partial differential equations; no other reference presents this in a systematic way. The minmax theorems contained in this book have many useful applications, in particular the robust control of partial differential equations in finite time horizon. First published in English in 1976, this SIAM Classics in Applied Mathematics edition contains the original text along with a new preface and some additional references. ... Read more

Book Description

Famed mathematical scholar’s concise exposition of the mathematical basis of tensor analysis, integrated with well-chosen physical examples of the theory, including those involving elasticity, classical dynamics, relativity and Dirac’s matrix calculus. Exercises. Index. Bibliography. Notes.
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Reviews (4)

An Abridged Version of the Author's Treatise
In recent times it has become fashionable to derogate the classical tensor analysis cultivated by such pioneers as Levi-Civita, Schouten and Eisenhart. Modern critics refer to such works as a "sea of indices", the reading of which is likened to "chasing shadows". It is true that tensor analysis predating ~1960 does not uphold the standards of rigor set by modern mathematical analysis, but, in light of the fact that the language has changed so drastically during the intervening years, it would be fair to treat the classical theory as a separate subject, of interest in its own right.

This book offers a valuable, yet not entirely self-contained, introduction to classical tensor analysis. As a beginner, I found the text to be too terse and was forced to consult other sources, such as Levi-Civita's "Absolute Differential Calculus" and Eisenhart's "Riemannian Geometry". Once I had gained some familiarity with the basic notions, Schouten's book became the preferred reference. The author develops an extremely precise notation which he calls the "kernel-index method" and systematically applies it as a problem solving tool throughout the book. Looking back, it is difficult to say how I ever got along without it.

Unfortunately, the book's terseness is due in part to the fact that the first five chapters are basically abridged excerpts from the author's lengthier 1954 treatise "Ricci-Calculus". In nearly every respect, the aforementioned title is better than the present book, for, in the interest of economizing space, the author omitted important details, such as the definition of a manifold and the role of the vector field which generates the infinitesimal transformations used in discussing Lie derivatives.

For classical tensor analysis, Schouten's "Ricci-Calculus" (1954) and "Pfaff's Problem and its Generalizations" (1949, but still in print) are both excellent. For the modern theory, I have found Munkres' "Analysis on Manifolds" and Noll's "Finite Dimensional Spaces" to be exceptionally well written.

This is a no easy book!
First that everything should have present that this it is not an introduction book. Don't hope to learn tensor analysis in this book. From the first chapter it begins to demand you and to tell you what you should know and to understand the subject to follow their reading. Definitively you have to have the clear subject in your mind to enjoy the book. But that doesn't mean that it is bad book, or that the book takes a lie title : the book is for exact science graduates (or science advanced undergraduates). The notation, inclusive, you will notice it heavy, difficult. You can divide the book in three parts: the part corresponding to the chapters 1-5 where it introduces all the elements of the tensor analysis . A second part, the chapter 6, dedicated to the study of the physical objects and their dimensions, and a third part that it includes the remaining chapters, dedicated to applications. It is not an easy book. This is a book on tensors where you won't learn on tensors. It is a beautiful synthesis of the content of the tensor analysis (chapters 1-5). The rest, obviously is impossible to find everything in a single book. Not yet it is enough with to have a general knowledge of the topic for this book. You have to have a solid one.

Possible books that you can read before being faced with this book: A. I. Borisenko (Classic, elementary), Synge, Goldberg, Levi-Civita, Akivis (Elementary, very elementary), Kreiszig (differential Geometry, elementary to intermediate level), etc.

Nice reference, however very incomplete
This book has no introduction! I mean the book has no "preparation" for studying pure tensor analysis, thus it cannot be taken as an introduction to the subject but as a reference for classes (unless you have many other books of geometry and algebra so you can get the "introduction" there). Also, the book has few information of used notations, making it harder to understand and the subject of tensor analysis, according to the author, is not fully explored - its just an "introduction" (Really?). So if you buy this book you rather buy few others together. About the title, I really didn't understand why it is called "for physicists". The author understood that applications make the book truly interesting for physicists. Is that so?

Great Book
I would suggest this book for relativity lovers, who loves to undrestand General relativity deeply. This book is easy to learn and it takes very little time to read but the benefits are very large. ... Read more

Book Description

This book focuses on therequirements of a specific group of readers, structuring the book so thatcalculus is presented as a single subject rather than a collection of topics.With a user-friendly approach that keeps the reader in mind, the material isorganized so that vector calculus is thoroughly covered. Approaches the theoretical aspects of calculus with the belief that, at theintroductory level, it is important to understand the geometric basis fortheorems and develop an intuitive understanding for the statements of thetheorems and their implications. Emphasizes the power of calculus as a toolfor modeling complex physical problems in order to present the methods ofdifferentiation and integration as necessary skills needed to solve problemsthat arise from mathematical models.Excellent as a refresher forthose in fields requiring a strong mathematical background. ... Read more

Book Description

Reviews (2)

An excellent book governing the mathematical beasts; tensors
A must have for all physics majors. Excellent outside source for mechanics, especially upper-division.

A book that returns you to the unbounded rigor of calculus
Not quite a page-turner, but this book presents good applications of tensor math. Sometimes difficult to follow, the book omits many "assumed" steps. However, a fine escape into the world of absolute differential calculus. ... Read more

Book Description

This is the second edition of the well-established text in partial differential equations, emphasizing modern, practical solution techniques. This updated edition includes a new chapter on transform methods and a new section on integral equations in the numerical methods chapter. The authors have also included additional exercises. ... Read more

Book Description

This is an all-encompassing and exhaustive exposition of the theory of infinite-dimensional Unitary Representations of Locally Compact Groups and its generalization to representations of Banach algebras. The presentation is detailed, accessible, and self-contained (except for some elementary knowledge in algebra, topology, and abstract measure theory). In the later chapters the reader is brought to the frontiers of present-day knowledge in the area of Mackey normal subgroup analysisand its generalization to the context of Banach *-Algebraic Bundles. ... Read more