Book Description

When King Arthur and his knights get together, the table they have is so long that everyone has to shout to be heard. A rectangular table is too long and a triangular table is too pointy, but somehow they must sit down and discuss the shape of the future. Join a knight called Sir Cumference, his wife, Lady Di of Ameter, and their son Radius as they use different strategies to solve this quandary.

Fanciful illustrations add to the merriment of learning math and will inspire young mathematicians. ... Read more

Reviews (7)

Great for early elementary age kids
I read this to my seven year old. He found the story very interesting and runs about laughing about things like diameter and circumference. He remembers the names, and the way the measurements are presented is very conceptual. Great for young kids, but above third grade or so, I'd say it may not do a whole lot except for help remember the names. It was a *cute* story, not anything very dramatic, and was appropriate for young children.

Lame. Not worth your money.
My amazon record shows that I actually bought this book. I had forgotten, and 'returned' it to the public library, which is where I wish I'd gotten it to begin with. The story is strained and not very informative. Great math books worth buying to read to your child include Math Trek, Math Trek 2, G is for Google, and especially, the Number Devil - reading great math books has been wonderful fun for my family - but not Sir Cumference.

Cute story....the kids love it!
This book creatively exposes the reader to some common geometric terms. Although I would not say it is extremely educational, it is the most mathematically educational story book I've ever seen (because most story books are not about math). My three year old daughter thought the book was for her, and asked to have it read to her every night. She loved it! My 11 year old son is really too old for the book, but he still liked it.

kids loved it
While parts of the story seemed a bit far-fetched, my 5th grade students loved the play on words in the characters' names. They also enjoyed seeing how a rectangle was changed to create other polygons (and finally a circle). At the end of the reading, one student said, "I'll never forget THESE math terms!" I suppose I achieved my objective. The kids didn't much care for the illustrations, however. They created their own!

enjoyable
I'm not a connoiseur of children's literature. But I will be teaching high school math in about a year and used this book for a math methods class. I enjoyed the humor and the art. The reading level is about 4th grade and so is the math content. Most of the book illustrates in a amusing way just how circles are different from other geometric shapes and how these differences affect the design of everyday objects. The book ends by tying the definition of circumference, radius and diameter to the names and actions of the main characters in the story.

However, the story line is marred by a certain tedious political correctness in that the outsiders who appear to threaten the realm are not really a threat - it's all a misunderstanding. This hollows out the content of the ideal of the knight (or soldier or policeman), a hero who employs his strength at the risk of his own life to defend the more vulnerable members of the community from villains and the community itself from outside threats. ... Read more

Reviews (2)

Just the facts, straightaway
This book presents geometry in a straightforward way. Emphasis is on the facts, without getting sidetracked in proofs, (although the purist might object to the fact that proofs are not given). The drawings are relevant and straightforward. The book is well organized and proceeds logically from beginning to end. There are conversational problems with answers in the text, and lots of multiple-choice test questions with answers in the appendix. The test questions are especially good, because they resemble the standardized tests schoolchildren are forced to take these days. This book, along with with a standard school textbook, should make high-school students highly proficient in this subject, and get them ready for more advanced courses such as calculus and trigonometry. Note: I also have the chemistry, physics, and trigonometry books in the Demystified Series and have found them to be of comparable quality.

NOT RECOMMENDED IF YOU NEED HELP IN GEOMETRY
because of the topic coverage. I picked this up because, though a math teacher, I really don't like Geometry. Never have. This book covers some of what a high school student would need in just a few chapters. So, Why aren't I recommending it to students needing help? Because the author completely skips proofs, which is what most students are having trouble with. It also has no chapter dealing with circles and theorems related to them. What it does cover it covers in a highly interesting and original way (why does a stool have three legs instead of four?). It is also filled with topics not covered in a high school geometry course, but which are very interesting on their own. Given the authors other books' titles, this is perhaps "geometry for electricians and hobbyists". If you are someone with bad memories of geometry, but you would like to try revisiting it, then this is highly recommended. It would also be a good outside source for students doing well in Geometry but wanting to read about some higher level topics (including 3 - 4 - and higher dimensional geometry.) The book has loads of multiple choice test questions, so you can see how well you are understanding what you are reading, but it has no detailed solutions in the back--just the correct answers. (techinical point: readers should know that the author teaches polar coordinates "backwards" from the way we teach it in Trigonometry. The form is (r, theta), NOT (theta, r).) ... Read more

Book Description

This accessible introduction to harmonic map theory and its analytical aspects, covers recent developments in the regularity theory of weakly harmonic maps. The book begins by introducing these concepts, stressing the interplay between geometry, the role of symmetries and weak solutions. It then presents a guided tour into the theory of completely integrable systems for harmonic maps, followed by two chapters devoted to recent results on the regularity of weak solutions. A presentation of "exotic" functional spaces from the theory of harmonic analysis is given and these tools are then used for proving regularity results. The importance of conservation laws is stressed and the concept of a "Coulomb moving frame" is explained in detail. The book ends with further applications and illustrations of Coulomb moving frames to the theory of surfaces. ... Read more

Book Description

This book consists of essays that stand on their own but are also loosely connected. Part I documents how numbers and geometry arise in several cultural contexts and in nature: the ancient musical scale, proportion in architecture, ancient geometry, megalithic stone circles, the hidden pavements of the Laurentian library, the shapes of the Hebrew letters, and the shapes of biological forms. The focus is on how certain numbers, such as the golden and silver means, present themselves within these systems. Part II shows how many of the same numbers and number sequences are related to the modern mathematical study of numbers, dynamical systems, chaos, and fractals. ... Read more

Book Description

Quasicrystals and Geometry brings together for the first time the many strands of contemporary research in quasicrystal geometry and weaves them into a coherent whole. The author describes the historical and scientific context of this work, and carefully explains what has been proved and what is conjectured. This, together with a bibliography of over 250 references, provides a solid background for further study. The discovery in 1984 of crystals with 'forbidden' symmetry posed fascinating and challenging problems in many fields of mathematics, as well as in the solid state sciences. Increasingly, mathematicians and physicists are becoming intrigued by the quasicrystal phenomenon, and the result has been an exponential growth in the literature on the geometry of diffraction patterns, the behaviour of the Fibonacci and other nonperiodic sequences, and the fascinating properties of the Penrose tilings and their many relatives. ... Read more

Reviews (1)

Worthwhile with lots of cool pictures
I bought the book on the strength of the pretty pictures scattered throughout the book (much much more interesting than the front cover).The text of the book is as impressive.

"Quasicrystals and Geometry" is a bit between an overview and a cookbook.Sections of the book are historical, parts are practical.Explicit and easy to follow instructions for generation of penrose tilings (and many other really cool tilings) are included.

Despite what the introduction says, this book is not written for a lay audience.A quarter or two of college level math seems to be assumed. ... Read more

Book Description

"An introduction to the ideas of algebraic geometry in the motivated context of system theory." This describes this two-volume work which has been specifically written to serve the needs of researchers and students of systems, control, and applied mathematics. Without sacrificing mathematical rigor, the author makes the basic ideas of algebraic geometry accessible to engineers and applied scientists. The emphasis is on constructive methods and clarity rather than on abstraction. While familiarity with Part I is helpful, it is not essential, since a considerable amount of relevant material is included here.

Part I Scalar Linear Systems and Affine Algebraic Geometry contains a clear presentation, with an applied flavor, of the core ideas in the algebro-geometric treatment of scalar linear system theory. Part II extends the theory to multivariable systems. After delineating limitations of the scalar theory through carefully chosen examples, the author introduces seven representations of a multivariable linear system and establishes the major results of the underlying theory. Of key importance is a clear, detailed analysis of the structure of the space of linear systems including the full set of equations defining the space. Key topics also covered are the Geometric Quotient Theorem and a highly geometric analysis of both state and output feedback.

Prerequisites are the basics of linear algebra, some simple topological notions, the elementary properties of groups, rings, and fields, and a basic course in linear systems. Exercises, which are an integral part of the exposition throughout, combined with an index and extensive bibliography of related literature make this a valuable classroom tool or good self-study resource. ... Read more

Reviews (2)

A nice easy going book.
Can't agree with my friend from Ann Arbor. This is the most accesible book on Riemannian Geometry. Or to be precise, this is not even a book on Riemannian Geometry, It's only a guide! The author doesn't take the universal intrinsic approach to the subject, but only look at everything as a subspace of Eucleadean space, and see how those apparently extrinsically defined concepts indeed have intrinsic meannings. And that gives you a taste of the real part, which you must turn to some other books.

Riemannian Geometry - A Beginner's Guide
Well I am only on page 10 and am considering quiting this book. I have a BSEE with quite a lot of mathematics history, but I know nothing about Riemann geometry. I got bogged down right from the beginning with this book. It's not that the material - so far - is that demanding, but the explanations are terse at best and some formulas seem to pop onto the page from hyperspace. Some variables are undefined, unfamiliar nomenclature is used without explanation and there is no exposition to show from where some rather complicated formula appear. Too much is assumed by the author for a book that claims to be a "beginners guide". You can waste a lot of time trying to guess what is on the authors mind.

I get the impression that by adding another 10 pages or so of elucidating math and text this might be a nice little survey of the subject matter, but as is I can't recommend it. ... Read more

Book Description

One of the most popular artists of the 20th century, M. C. Escher, leaves a rich legacy. The centennial celebration of his birth, held in Rome and Ravello in 1998, gave testimony to the keen interest and new insight into his work, and showcased a number of comtemporary artists and scientists whose work is directly inspired by that of Escher. This book contains 40 of their articles, richly illustrated with original art works in addition to well-known and little-known works by Escher. A CD-ROM complements the articles, containing color illustrations of work by contemporary artists, movies, animations, and other demonstrations. ... Read more

Book Description

Since its original publication in 1990, Kenneth Falconer's Fractal Geometry: Mathematical Foundations and Applications has become a seminal text on the mathematics of fractals. It introduces the general mathematical theory and applications of fractals in a way that is accessible to students from a wide range of disciplines. This new edition has been extensively revised and updated. It features much new material, many additional exercises, notes and references, and an extended bibliography that reflects the development of the subject since the first edition.
* Provides a comprehensive and accessible introduction to the mathematical theory and applications of fractals.
* Each topic is carefully explained and illustrated by examples and figures.
* Includes all necessary mathematical background material.
* Includes notes and references to enable the reader to pursue individual topics.
* Features a wide selection of exercises, enabling the reader to develop their understanding of the theory.
* Supported by a Web site featuring solutions to exercises, and additional material for students and lecturers.
Fractal Geometry: Mathematical Foundations and Applications is aimed at undergraduate and graduate students studying courses in fractal geometry. The book also provides an excellent source of reference for researchers who encounter fractals in mathematics, physics, engineering, and the applied sciences.
Also by Kenneth Falconer and available from Wiley:
Techniques in Fractal Geometry
ISBN 0-471-95724-0

Please click here to download solutions to exercises found within this title:

http://www.wileyeurope.com/fractal ... Read more

Reviews (4)

What every student should know about fractals.
Fractals make headlines from time to time[--are they everywhere?], and and they make lovely color pictures; but they are also part of a substantial mathematical theory, one with an
exciting mathematical history. This very important book presents
the subject in a way that it can be taught to students, and it starts with the basics, systematically, step by step, building up the material. Or it can be used for selfstudy! It has great exercises too! In view of the many applications to geometric analysis, to PDE, and to statistics, it is likely that fractal geometry will soon be a standard math course taught in many (more) math departments. By now it is widely recognized that the selfsimilarity aspects of the wavelet algorithms are key to their sucess. The book came out in 1990, and the author has an equally attractive book on the subject from 1985[The geometry of fractal sets] with a slightly more potential theoretic bent.

Theoretical as well as practical insight
The first part of the book is essentially of a theoretical nature, with a thorough treatment of fractal geometry at a mathematical point of view. The second part on the other hand provides a flavour of the problems of fractal geometry in practice...so mathematicians as well as people interested in applications only should both find this book interesting. The maths are not easy but quite "understandable" for science undergrads...some notions of calculus or topology would help... but the introduction is excellent and allows anyone to follow the course of the book (but for understanding the proofs a good math background is required).

Excellent for understanding the geometrical properties of fractals.

Exposes fractal geometry as a real mathematical discipline.
I appreciate Falconer's books on fractal geometry because they show the topic as it really is: a whole mathematical discipline on its own right and not just a nice temporary fashion.

It begins introducing basic topological concepts and then proceeds to develop the theory for several possible definitions of fractal dimension, showing the relations between them. Then it explores deeply the local geometry of different kinds of fractal objects, and studies some other geometrical situations, like the pojection of fractals (ever thought of a DIGITAL sundial? Here it is described!).

The book also includes a lot of applications to other areas of mathematics and physics, a great amount of graphics, and much more.

The text is suitable from third year undergraduate school and on. It is a larger but lighter version of "The Geometry of Fractal Sets".

One of the best books on fractals to be found anywhere!
The book opens the doors of mathematics: it isn't an easy door, but well worth the effort. It bridges the gap between beginner texts and advanced study and covers the basic material in a hard hitting manner. Those looking for "giltz" should look elsewhere. It is a book of great understanding and depth. Your unique Associates ID is: thefractaltransl. ... Read more

Book Description

Increasingly, researchers in many branches of science are coming into contact with Bayesian statistics or Bayesian probability theory. By encompassing both inductive and deductive logic, Bayesian analysis can improve model parameter estimates by many orders of magnitude. It provides a simple and unified approach to all data analysis problems, allowing the experimenter to assign probabilities to competing hypotheses of interest, on the basis of the current state of knowledge. This book provides a clear exposition of the underlying concepts with large numbers of worked examples and problem sets. The book also discusses numerical techniques for implementing the Bayesian calculations, including an introduction to Markov Chain Monte-Carlo integration and linear and nonlinear least-squares analysis seen from a Bayesian perspective. In addition, background material is provided in appendices and supporting Mathematica notebooks are available, providing an easy learning route for upper-undergraduates, graduate students, or any serious researcher in physical sciences or engineering. ... Read more

Book Description

The theory of Lie algebras and algebraic groups has been an area of active research for the last 50 years. It intervenes in many different areas of mathematics: for example invariant theory, Poisson geometry, harmonic analysis, mathematical physics. The aim of this book is to assemble in a single volume the algebraic aspects of the theory, so as to present the foundations of the theory in characteristic zero. Detailed proofs are included and some recent results are discussed in the final chapters. All the prerequisites on commutative algebra and algebraic geometry are included.

Book Description

This volume contains expanded versions of lectures given at an instructional conference on number theory and arithmetic geometry held August 9 through 18, 1995 at Boston University.Contributor's includeThe purpose of the conference, and of this book, is to introduce and explain the many ideas and techniques used by Wiles in his proof that every (semi-stable) elliptic curve over Q is modular, and to explain how Wiles' result can be combined with Ribet's theorem and ideas of Frey and Serre to show, at long last, that Fermat's Last Theorem is true. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions, modular curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of Wiles' proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serre's conjectures, Galois deformations, universal deformation rings, Hecke algebras, complete intersections and more, as the reader is led step-by-step through Wiles' proof. In recognition of the historical significance of Fermat's Last Theorem, the volume concludes by looking both forward and backward in time, reflecting on the history of the problem, while placing Wiles' theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this volume to be an indispensable resource for mastering the epoch-making proof of Fermat's Last Theorem. ... Read more

Reviews (4)

Yet another application of elliptic curves...
The successful proof of Fermat's Last Theorem by Andrew Wiles was probably the most widely publicized mathematical result of the 20th century. And once again, among their numerous other applications, elliptic curves are employed in the proof. The book is a compilation of articles written by first-class mathematicians, the reading of which will give one a thorough understanding of the proof, along with an overview of some very interesting topics in number theory and algebraic geometry. The reader who undertakes an understanding of the proof already no doubt has a substantial amount of 'mathematical maturity', and no review, no matter how complete, would influence greatly such a reader. Suffice it to say then that this book is excellent, and even a reader interested solely in elliptic curves and modular forms could benefit greatly from the reading of this book.

Great?!?!
This book might be good if you like number theory. But if you're an analyst who hates number theory or a brick-layer, then this book is probably not meant for you. I hope you found this review helpful. Have a nice day.

Not 24 hours
I ordered this book in April and it's May now. It says "Available in 24 hours" but Customer Service says it's a Special Order item that ships in "4-6 weeks".

Highly recommended
This item is very instructively, not only for "real" mathematicians. Of course, sometimes it's very difficult to "read". It gives me pleasure to own the proof of FLT. ... Read more

Book Description

Arising from a summer school course taught by János Kollár, this book develops the modern theory of rational varieties at a level appropriate for graduate study. Kollár's original course has been developed, with his co-authors, into a state-of-the-art treatment of the classification of algebraic varieties. The authors have included numerous exercises with solutions, which help students reach the stage where they can begin to tackle related contemporary research problems. ... Read more

Book Description

This book offers a unique opportunity to understand the essence of one of the great thinkers of western civilization. A guided reading of Euclid's Elements leads to a critical discussion and rigorous modern treatment of Euclid's geometry and its more recent descendants, with complete proofs. Topics include the introduction of coordinates, the theory of area, geometrical constructions and finite field extensions, history of the parallel postulate, the various non-Euclidean geometries, and the regular and semi-regular polyhedra. The text is intended for junior- to senior-level mathematics majors. Robin Hartshorne is a professor of mathematics at the University of California at Berkeley, and is the author of Foundations of Projective Geometry (Benjamin, 1967) and Algebraic Geometry (Springer, 1977). ... Read more

Reviews (4)

This book and course is not for the faint of heart!
This is without exception the hardest math course I have ever taken. Your understanding of the concepts is pertinent. I had to read the 1st chapter over five times just to understand projective geometry. Hartshorne tries to simplify the material but only so much can be done. It is just a hard course, period. The book does contain many example and logical proofs but be ready to burn the midnight candle on this one.

A stunning book
Hartshorne is a leading mathematician known for work in rather abstract geometry (see his book ALGEBRAIC GEOMETRY). He takes Euclid's ELEMENTS as great mathematics, no mere genial precursor, and collates it with Hilbert's FOUNDATIONS OF GEOMETRY.

Of course Harshorne proves that Euclid needed the parallel postulate, by exhibiting a non-Euclidean geometry. He gives a very pretty compass and straight-edge Euclidean theory of circles, which then turns into the Poincare plane model for hyperbolic geometry. He also proves that Euclid needed the method of exhaustion for volumes of solids: he gives the agreeably simple Dehn invariant proof that even a cube and a tetrahedron of equal volumes are not decomposable into congruent parts. It is a famous proof, rarely seen, and a beautiful use of the modern algebraic viewpoint in classical geometry. I had always supposed it must be hard but it is not.

Hartshorne also develops the contested "geometric algebra" of Euclid as a modern axiomatic algebra. Many commentators have shown it is wrong to think Euclid was doing "algebra" in the sense of a disguised theory of the roots of quadratic polynomials. But (unless and until Fowler's THE MATHEMATICS OF PLATO'S ACADEMY changes my mind) I think it is reasonable to say Euclid is doing algebra in this sense.

Well beyond Euclid
Hartshorne is a famous algebraist and one main contribution of this text is to show fascinating interrelations between classical geometries and modern algebra (of course the book contains lots of pure geometry as well). Example 1: Many texts show the impossibility of the classical problems of constructibility by straightedge and compass (by observing that the coordinates of any point so constructed lie in the smallest extension field of the rationals Q closed under taking square roots of positive numbers). Hartshorne's is the only text that goes further, solving the analogous problem when the straightedge is marked (real roots of cubic and quartic equations must also be allowed); Archimedes observed that any angle can be trisected with these tools. Example 2. Dehn's solution to Hilbert's Third Problem is given, whereby any two polyhedra equivalent under dissection must have equal Dehn invariants, and it shown that a tetrahedron has different invariant than a cube. Example 3. In hyperbolic geometry, Hilbert's arithmetic of ends is developed and applied. Example 4. Pejas' algebraic classification of Hilbert planes is discussed.

Hartshorne's text overlaps mine in correcting Euclid's errors, developing rigorous foundations for Euclidean and Non-Euclidean geometries, and covering much history, presented delightfully. He gives a thorough discussion of area and the open problems in that theory. He concludes with a nice chapter on polyhedra.

Hartshorne's New Book, "Geometry: Euclid and Beyond", is a Masterpiece!
I told my wife: "If I have to give up all my books but one, then this is the one I'd keep; no question about it." (More comments later.) ... Read more

Book Description

Algebraic Geometry often seems very abstract, but in fact it is full of concrete examples and problems. This side of the subject can be approached through the equations of a variety, and the syzygies of these equations are a necessary part of the study. This book is the first textbook-level account of basic examples and techniques in this area. It illustrates the use of syzygies in many concrete geometric considerations, from interpolation to the study of canonical curves. The text has served as a basis for graduate courses by the author at Berkeley, Brandeis, and in Paris. It is also suitable for self-study by a reader who knows a little commutative algebra and algebraic geometry already. As an aid to the reader, the appendices provide summaries of local cohomology and commutative algebra, tying together examples and major results from a wide range of topics. ... Read more

Book Description

Complex geometry studies (compact) complex manifolds. It discusses algebraic as well as metric aspects. The subject is on the crossroad of algebraic and differential geometry. Recent developments in string theory have made it an highly attractive area, both for mathematicians and theoretical physicists.

The author's goal is to provide an easily accessible introduction to the subject. The book contains detailed accounts of the basic concepts and the many exercises illustrate the theory. Appendices to various chapters allow an outlook to recent research directions.

Daniel Huybrechts is currently Professor of Mathematics at the University Denis Diderot in Paris.

Book Description

This material is intended to contribute to a wider appreciation of the mathematical words "continuity and linearity". The book's purpose is to illuminate the meanings of these words and their relation to each other. ... Read more

Reviews (9)

fantastic introduction to general topology
The first part of this book that deals with topology is a pedagogical masterpiece. After motivating the key concepts of compactness and continuity in the relatively concrete setting of metric spaces, the book goes on to abstract topological spaces, a beautiful section on compactness including the tychonoff theorem, and an extremely lucid development of the separation axioms and the proof of the urysohn imbedding theorem and the stone-cech compactification. I personally find the chapter on connectedness to be the weak link in this part of the book. Wherever possible, Simmons provides an exhaustive list of examples (especially when introducing the various types of spaces) that aids comprehension. Moreover, some of the central concepts (product topology) and deeper results such as the Stone-Cech compactification are easier to appreciate because the author has a section on topological properties of the relevant function spaces couple of chapters ahead and several exercises along the way. All in all, a highly recommended intro to the subject.

Didactic perfection
In the author's words in the preface, the dominant theme of this book is continuity and linearity, and its goal is to illuminate the meanings of these words and their relations to each other. The book, he says, belongs to the type of pure mathematics that is concerned with form and structure, and such a body of mathematics must be judged by its high aesthetic quality, and should exalt the mind of the reader.

The author's attitude can only be characterized as magnificent, and, if one is to judge his utterances in the preface by what is found after it, one will indeed find perfect evidence of his delight in mathematics and his high competence in elucidating very abstract concepts in topology and real analysis. Indeed, this has to be the best book ever written for mathematics at this level. It is a book that should be read by everyone that desires deep insights into modern real and functional analysis.

After a brief and informal overview of set theory, the author moves on to the theory of metric spaces in chapter 2. His emphasis is on the idea that metric spaces are easy to find, since every non-empty set has the discrete metric, and that metric spaces are good motivation for the more general idea of a topological space. The Cantor set, ubiquitous in measure theory, dynamical systems, and fractal geometry, is constructed as the most general closed set on the real line, i.e. one obtained by removing from the real line a countable disjoint class of open intervals. Continuity of mappings between metric spaces is defined, and also the concept of uniform continuity, the latter of which is motivated very nicely by the author. Then, the author takes the reader to a higher level of abstraction, wherein he asks the reader to consider all of the continuous functions on a metric space, and turn this collection into a metric space of a special type called a normed linear space, and, more specifically, a Banach space. Thus the author introduces the reader to the field of functional analysis.

A lengthy introduction to topological spaces follows in chapter 3. The author motivates well the idea of an open set, and shows that one could just as easily use closed sets as the fundamental concept in topology. And, most important for functional analysis, he introduces the weak topology, and shows how to obtain the weakest topology for a collection of mappings from a topological space to a collection of other topological spaces. The reader can see clearly that the weaker the topology on a space the harder it is for mappings to be continuous on the space.

Compactness, so essential in all areas of mathematics that make use of topology, is discussed in chapter 4. It is motivated by an abstraction of the Heine-Borel theorem from elementary real analysis, and the author shows how well-behaved things are on compact topological spaces. Some important theorems are proved in this chapter, namely Tychonoff's theorem, the Lebesgue covering lemma, and Ascoli's theorem.

Recognizing that the only functions able to be continuous on a space with the indiscrete topology are the constants, and that a space with the discrete topology has continuous functions in abundance, the author asks the reader to consider topologies that fall between these extremes, and this motivates the separation properties of topological spaces. Chapter 5 is an in-depth discussion of separation, and the reader again confronts function spaces, and their ability (or non-ability) to separate the points of a topological space. Spaces that allow such separation to occur are called completely regular, and this property has far-reaching consequences in analysis and other areas of mathematics. The Stone-Cech compactification is discussed as an imbedding theorem for completely regular spaces, analogous to one for normal spaces.

The intuitive idea of a space being connected is given rigorous treatment in chapter 6. Certain pathologies can of course arise when discussing connectedness, and the author shows this by discussing totally disconnected spaces, remarking that such spaces are very important in dimension theory and representation theory. Indeed, computational and fractal geometry is much harder to study because of the existence of these spaces.

Chapter 7 is important to all working in numerical analysis, wherein the author discusses approximation theory. The Weierstrass approximation and the Stone-Weierstrass theorems are discussed in detail.

A slight detour through algebra is given in chapter 8. Groups, rings, and fields are given a minimal treatment by the author, discussing only the basic rudiments that are needed to get through the rest of the book.

Banach spaces make their appearance in chapter 9, with the three pillars of the theory proven: the Hahn-Banach, the open mapping, and the uniform boundedness theorems. These theorems guarantee that the study of Banach spaces is worth doing, and that there are analogs of the finite dimensional theory in the (infinite)-dimensional context of Banach spaces. The theory of Banach spaces is very extensive, but this chapter gives a peek at this very interesting area of mathematics.

Banach spaces with an inner product are considered in chapter 10. These of course are the familiar Hilbert spaces, so important in physics and the subject of a huge amount of research in mathematics. The presence of the inner product allows constructions familiar from ordinary finite-dimensional vector spaces to carry over to the inifinite-dimensional setting, one example being the transpose of a matrix, which is replaced in the Hilbert space setting by a self-adjoint operator.

As a warm-up to the infinite-dimensional theory, finite-dimensional spectral theory is considered in chapter 11. The famous spectral theorem is proven. Then in chapter 12, the reader enters the world of "soft" analysis, wherein topological and algebraic constructions are used to study linear operators on spaces of infinite dimensions. Putting an algebraic structure on a Banach space gives a Banach algebra, and then the trick is deal with the spectrum of an element of this algebra. The reader can see the interplay between algebra, topology, and analysis in this chapter and the next one on commutative Banach algebras. Indeed, the Gelfand-Naimark theorem, that essentially states that elements of a commutative Banach *-algebra act like the functions on its maximal ideal space, has to rank as one of the most interesting results in the book, and indeed in all of mathematics.

Good Classical Introduction to Banach Algebras
This is a fine book, but not quite in the 5-star league. Let me elaborate. The book is divided into three parts: general topology, the theory of Banach and Hilbert spaces, and Banach algebras. The first two parts lead, by way of synthesis, to the last part, where some interesting but elementary results are proved about Banach algebras in general and C*-algebras in particular. I might mention, for example, the Spectral theorem for compact self-adjoint operators, the Stone representation theorem, and the Gelfand-Naimark theorem.

I can attest from personal experience that the book is well-written; indeed I worked through it chapter by chapter. But today there do exist a plethora of other treatments that can at least rival this text in lucidity, organisation and coverage. For example, for general topology, there is an excellent text by Willard titled 'General Topology',as well as Hocking and Young's old 'Topology'. Both of these go much further in the realm of point-set topology than Simmons. Similarly there are any number of well-written texts on functional analysis that cover the subject of Banach spaces, Hilbert spaces and self-adjoint operators very clearly. Indeed in some respects I feel the Simmons book was inadequate by itself and needed to be supplemented by a text on linear algebra; self-adjoint operators -- and by implication, the Spectral theorem -- need to be seen and manipulated in the finite-dimensional version before one examines their infinite-dimensional generalisation. The Simmons book is a bit weak here; one needs to be playing with matrices.

These are, however, minor quibbles. The book can be recommended to a junior- or senior-level undergraduate.

Topology Classic
This book was recommended for our analysis course (final year at Adelaide University). It helped me pass the course but more importantly, gave me an interest in metric spaces and topology. The book is an excellent communicator and nearly 20 years after I have read it I am looking out for a secondhand copy!

One of the best written Mathematics Books ...
I became aquainted with this book many years ago and I still read it ... and send students off to read it. The book is written by an incredible expositor who was and still may be at Colorado College in Colorado. It is always the book that first comes to mind when someone asks for a reference on any of the subjects it covers. These include point set topology, analysis (Not including integration or measure theory), and operator theory. It is introductory. This merely makes you wish the author would have written several advanced sequels to this amazing book.

This book has my highest recommendation. Every mathematics student should own a copy ... ... Read more

Book Description

Teach Yourself Trigonometry is suitable for beginners, but it also goes beyond the basics to offer comprehensive coverage of more advanced topics. Each chapter features numerous worked examples and many carefully graded exercises, and full demonstrations of trigonometric proofs are given in the answer key.

Reviews (3)

Excellent, methodical approach for the self-taught
I teach myself a lot from books. Some books provide little, some a lot. With this book you walk away with a usable knowledge of trigonometry AND invaluable knowledge of how your calculator performs mathematical operations.
I've tok trig in high school, but the subject wasn't presented in such an organized and methodical approach. The overview in Chapter 1 provided insights that eluded me in high school. Chapter 2 provided detailed knowledge of my pocket computer and knowledge of it's use to solve trig problems that I never realized before. The examples and solutions provide feedback that the lessons were learned and owned. The subsequent chapters methodically build consistent, solid, usable knowledge. Mr. Abbot continually enables you to compare table-based calculations with your calculator calculations, building your confidence in your ability to do both. IF you've wrestled with other approaches to learning trig, this is THE book to get you through it.
I have read Mr. Abbott's Calculus book and find his approach conducive to my learning.

reader from texas
The book would ahve been good except before you even get into the material very good there are problems with wrong answers or very confusing explantions. If this much is wrong at the beginning what about later in the book when you get into the stuff you don't know.

Clear discussion but book marred by lots of typos
This book is very well written and a lot of thought was given by the authors as to the presentation of the material and how much background to assume the reader has. I would give the book five stars based on the writing. However, the writers were not well served by the editor, as there are many typos, maybe as many as one every other page. It appears that the proof reading was done by someone without a math background, as many of the errors are obvious, such as an equal sign being omitted, turning an equality into a product. Or the length of the sides of a right triangle being given so that the hypotenuse is not the longest side. Sometimes the answers to questions are messed up, which is very frustrating. Hopefully when the book is republished, it will be given the care it deserves. ... Read more