Book Description

Written primarily for readers who have completed the standard first courses in calculus and linear algebra, Elementary Differential Geometry, Second Edition provides an introduction to the geometry of curves and surfaces. Although the popular First Edition has been extensively modified, this Second Edition maintains the elementary character of that volume, while providing an introduction to the use of computers and expanding discussion on certain topics. Further emphasis has been placed on topological properties, properties of geodesics, singularities of vector fields, and the theorems of Bonnet and Hadamard. For readers with access to the symbolic computation programs, Mathematica or Maple, the book includes approximately 30 optional computer exercises. These are not intended as an essential part of the book, but rather an extension. No computer skill is necessary to take full advantage of this comprehensive text.

* Gives detailed examples for all essential ideas
* Provides more than 300 exercises
* Features more than 200 illustrations
* Includes an introduction to using computers, and supplies answers to computer exercises given for both Mathematica and Maple systems ... Read more

Reviews (6)

An Excursion into the Realm of Differential Geometry
My first encounter with this book was during the academic year of 2000-2001, when it was used as the main text for an upper division course on differnetial geometry, at one of the University of California campuses . The class for me --taught by a distinguished scholar-- was only meant to be a brief excursion into the realm of continuous math, beyond analysis and topology. After finishing the class however, I decided to change direction and as time went on, I drifted more and more towards geometry as the field of further concentration. Before we proceed further, let me note that one main complaint that's rather well-known about this text is the issue of numerous typo's therein. What many may not know however is that the first edition from 1966 does not contain any noticeable typo's, unfortunately somehow all of them found their way in the 1997's second edition. This is very likely because of careless typesetting, but one good news is that many of these are noted on the errata sheet available from the author's UCLA web site. Moreover, the book uses a cumbersome section numbering format, and to make things worse, in quite a few places the reader is referred to one or more previous sections. This serves to disrupt the flow of reading by taking up some time and effort to locate the correct previous page number which is being referenced.

Within the eight chapters of the book (seven chapters in the first edition), the reader is first introduced to some preliminaries such as tangent vectors, directional derivatives, and differential forms. In chapter two, the author presents the Frenet frame formulas, covariant derivatives, connection forms, and Cartan's structural equations, which are generalizations of the Frenet frame formulas for surfaces. In chapters three and four, there is a healthy dose of Euclidean geometry and calculus on surfaces. In chapter five, discussion shifts to the study of the shape operators and normal and Gaussian curvatures, where also some useful computational examples have been presented. Geometry of surfaces is the subject of chapter six, where the crucial Gauss' egregium theorem and some global theorems are also discussed, and in chapter seven students are introduced to the basics of the Riemannian geometry, culminating in the famous Gauss-Bonnet theorem. In chapter eight (which is highly topological), the concepts of geodesics, complete surfaces, covering spaces, Jacobi fields, conjugate points, and a couple of constant curvature theorems for surfaces are explored. The appendices include help on using popular computer algebra systems, and another appendix providing solutions to most of the odd-numbered exercises in the book.

Again, looking on the downside, the book lacks a discussion of several essential tools, for example, the Schwarz-Christoffel symbols, tensors, and Lie derivatives, as well as some other important topics such as the first and second fundamental forms, and parallel translation, which only show up in the exercises. Then again, perhaps to keep the level of exposition elemantary and the size limited to less than 500 pages, Dr. O'Neill has preferred to skip some topics. One remedy is to back this text up with Manfredo Do Carmo's 1976 classic, which is mathematically more rigorous, and covers more of the above-mentioned topics (be aware though that Do Carmo is less accessible for the beginning students). Afterwards, one can certainly continue the study of the essentials by reading other advanced books such as Barrett O'Neill's (obscure) graduate-level 1983 treatise on Applications of the Semi-Riemannian Geometry to Relativity, or William Boothby's "An Introduction to Differentiable Manifolds and Riemannian Geometry". One other underrated source which is worthwhile to look into is Richard W. Sharpe's "Differential Geometry: Cartan's Generalization of Klein's Erlangen Program", from the Springer-Verlag GTM series.

Good low dimensional calculation
It's easy to read with enough examples. Suitable for self study after your advanced calculus (inverse function thm/implicit function thm should be covered here) and linear algebra classes. Tons of exercises will help you familiarize yourself with the calculation in low dimension. (Do I love the exercises on minimal surfaces and surfaces of revolution in chapter 5 and 6!) Most of them are workable. This is the strength of the book. Since the author limits the material to low dimensions, some definitions are a bit misleading, such as the definition of exterior derivative of 1-form in chapter 4, where another term to be added happens to be zero. I think there is a big gap in style and level of difficulty between this book and author's "Semi-Riemannian Geometry: With Applications To Relativity".

After this book, probably you want to read Hicks' "Notes on differential geometry", if you can find a copy in some lib. Darling's "Differential Forms and Connections" is also highly recommended. It is modern but not much topological stuff.
Company it with Warner's "Foundations of differentiable manifolds and Lie groups" for topology also much higher algebra.

Solid and Modern Introduction
I worked through the first edition of this book some years back. After finishing this book I was ready for more abstract treatments of Riemannian Geometry. For example, having seen covariant derivatives on 2-surfaces embedded in R^3 motivates the abstract definition of connections on manifolds.

Chapter 1 is a decent introduction to pullbacks and pushforwards of differntial forms and tangent vectors respectively. In fact, all the subsequent geometry is based on pullbacks and pushforwards.This itself motivates the more abstract definition of a differentiable manifold with its coordinate charts. True,tangent vectors are not described in the most abstract fashion (e.g. as derivations on the algebra of functions) but this is not appropriate for a first course.

Chapter 2 describes the language of frame field and connection forms and derives the Frenet-Serret equations in terms of moving frames and structure equations. We associate this with the methods of Elie Cartan, who used moving frames in a systematic manner.

Chapter 3 deals with isometries; frankly speaking I never understood the raison d'etre for such a long chapter on such a topic.

Chapter 4 discusses coordinate patches. Again, this is thoroughly modern, and you won't find this in Struik or Kreyszig. The idea of piecing together coordinate patches to get geometric or topological information is a twentieth-century conception.

Chapter 5 introduces the Shape Operator, which is subsequently used in Chapter 6 to derive the equations of surface theory. This is really moving frames again, in another guise.

Chapter 7 finally tries to put this in a more abstract setting by defining abstract surfaces with an intrinsically defined covariant derivative.Holonomy and the Gauss-Bonnet theorem are discussed.

After reading this book, one would be equipped to handle do Carmo's book on Riemannian geometry, or O'Neill's book on Semi-Riemanninan geometry, or the more recent book by Lee, again on Riemannian geometry.

Very useful, but lacking some abstraction
I like this book very much because it helps me frequently when I need to remember some definitions or formulas, but I think it could be improved if some topics were treated in a more abstract way, as all the material on differential forms, for example.

OK for undergrads, if you can put up with the typos
This book is, I suppose, an acceptable elementary introduction to the topic. However, I found that several important proofs were annoyingly incomplete, with the tag "the proof is too difficult to go into here." Many times, the only ingredient needed to complete the proof is the Inverse Function Theorem or the Implicit Function Theorem, which students taking differential geometry should know. Also, and more significantly, the exercises are riddled with typos; some are minor, others make the exercise incomprehensible. A final minor quibble is that the author uses a cumbersome numbering system for his sections. In sum: If your mathematical background is at least as strong as that of a senior honors undergraduate mathematics major, look elsewhere (perhaps "Differential Geometry of Curves and Surfaces," by do Carmo). If not, and you still would like an introduction to the field, this book may foot the bill--just beware of the typos in the exercises! ... Read more

Book Description

A practical, accessible introduction to advanced geometry

Exceptionally well-written and filled with historical and bibliographic notes, Methods of Geometry presents a practical and proof-oriented approach. The author develops a wide range of subject areas at an intermediate level and explains how theories that underlie many fields of advanced mathematics ultimately lead to applications in science and engineering. Foundations, basic Euclidean geometry, and transformations are discussed in detail and applied to study advanced plane geometry, polyhedra, isometries, similarities, and symmetry. An excellent introduction to advanced concepts as well as a reference to techniques for use in independent study and research, Methods of Geometry also features:

• Ample exercises designed to promote effective problem-solving strategies
• Insight into novel uses of Euclidean geometry
• More than 300 figures accompanying definitions and proofs
• A comprehensive and annotated bibliography
• Appendices reviewing vector and matrix algebra, least upper bound principle, and equivalence relations

Book Description

This classic work is now available in an unabridged paperback edition. The Second Edition retains all the characterisitcs that made the first edition so popular: brilliant exposition, the flexibility permitted by relatively self-contained chapters, and broad coverage ranging from topics in the Euclidean plane, to affine geometry, projective geometry, differential geometry, and topology. The Second Edition incorporates improvements in the text and in some proofs, takes note of the solution of the 4-color map problem, and provides answers to most of the exercises. ... Read more

Reviews (3)

Lots of illustrations and exercises-with answers at the back
This is a wonderful book. It isn't for mathematical beginners, but it isn't opaque either. It requires a student to think, experiment and to learn by puzzling things out in one's mind rather than simple memorization and regurgitation. Nor does it follow the all too common modern method of over simplifying things to allow people to pretend they have learned math while only dabbling in a few basic topics.

This book is amply illustrated with many exercises (answers are provided at the back for all the exercises). The book also has some humor and wit with the quotes it distributes throughout the book to help liven things up.

There is also a list of helpful references and an index. When reading the book, don't be afraid of going to a dictionary or the web or some other math books for clarification of some terms or more basic concepts. It is essential to have everything clear in your mind before moving on or you will stumble. As in all math, it is like a building with the next stage being built on the present one which is built on the previous one. You can't skip steps very successfully very often.

This is a great volume to have in your library, but even better to work through.

this is not an introduction
A better title for this book would be "Advanced Topics in Geometry". The chapters are pretty much self-contained.

This book presumes a thorough, rigorous knowledge of high school geometry such as you might get in a college geometry course designed for future teachers along with considerable mathematical maturity.

Best survey of geometry.
This is Coxeter best book. Introduction to Geometry covers a wide range of topics and is the first book that I will look at for any geometry topic. It is now a little dated but only in the topics that it does not cover. Like all of Coxeter works each topic is clear and to the point. If you only buy one book on geometry this is it. ... Read more

Book Description

Homological algebra first arose as a language for describing topological prospects of geometrical objects. As with every successful language it quickly expanded its coverage and semantics, and its contemporary applications are many and diverse. This modern approach to homological algebra, by two leading writers in the field, is based on the systematic use of the language and ideas of derived categories and derived functors. Relations with standard cohomology theory (sheaf cohomology, spectral sequences, etc.) are described. In most cases complete proofs are given. Basic concepts and results of homotopical algebra are also presented. The book addresses people who want to learn a modern approach to homological algebra and to use it in their work. For the second edition the authors have made numerous corrections. ... Read more

Book Description

When it comes to pinpointing the stuff you really need to know, nobody does it better than CliffsNotes. This fast, effective tutorial helps you master core geometry concepts – from perimeter, area, and similarity to parallel lines, geometric solids, and coordinate geometry – and get the best possible grade.

At CliffsNotes, we're dedicated to helping you do your best, no matter how challenging the subject. Our authors are veteran teachers and talented writers who know how to cut to the chase – and zero in on the essential information you need to succeed. ... Read more

Reviews (3)

Very handy little summary
After several years in a corporate engineering job, I started moonlighting as a math tutor. The Cliff's Quick Review Guides are wonderful to have in my "back pocket" when I need to quickly look something up that is covered in dust in the "archives of my brain."

A Good Start for Geometry Review
This is a helpful tool for practicing concepts of geometry, and stimulation of memory. However, it may not be helpful if you never had any experience with geometry.

A Great Review of Geometry
This review of geometry is exceptional. It is quick, simple, and easy to use. It provides access to all the common postulates and theorems of geometry. This book is designed to be read straight through as a review. In fact, it does not even have an index. Because of this, I found myself searching through the book hunting for small pieces of information. But through the use of the table of contents, I could find the desired information.
Reading this book is an excellent way to prep for any geometry midterm/final or any standardized test (ACT, PSAT, SAT I, SAT II, etc.). It helped me greatly for the SAT, and in a lot less time than studing a text book or reading SAT prep books. If you need to review geometry in a short amount of time, this is the book to get. ... Read more

Book Description

The geometry of lines occurs naturally in such different areas as sculptured surface machining, computation of offsets and medial axes, surface reconstruction for reverse engineering, geometrical optics, kinematics and motion design, and modeling of developable surfaces. This book covers line geometry from various viewpoints and aims towards computation and visualization. Besides applications, it contains a tutorial on projective geometry and an introduction into the theory of smooth and algebraic manifolds of lines. It will be useful to researchers, graduate students, and anyone interested either in the theory or in computational aspects in general, or in applications in particular. ... Read more

Amazon.com

Reinventing the wheel is a terrible waste of time, yet legions of computer programmers do exactly that every day. Geometric Tools for Computer Graphics gives the working graphics programmer a vast collection of programming examples, complex code snippets explained and ready to use. Each chapter is filled with more than just code examples--the explanations needed to understand why these examples work the way they do are given by authors with experience both in writing and in the field.

There is nothing here for the casual graphics programmer and everything for the serious 2-D and 3-D programmer. Thirteen chapters, three appendices, and a three-column index that spans over 30 pages cover everything about computer-graphic geometry, from the basics of using matrices and linear systems to intersecting 3-D objects.

The appendices alone are worth the price: "Numerical Methods," "Trigonometry," and "Basic Formulas for Geometric Primitives" are treasures filled with hard-core examples of the kind that can be put to use right out of the box. Less experienced programmers will find these to be invaluable references, but then there's the rest of the book--nearly 1,000 pages loaded with examples and theory, page after page of information written in a clear, concise voice.

Any hard-core graphics programmer will appreciate the value of the examples presented here, as well as the discussion of theory. After all, there's no need to waste time experimenting with code once the theory is known. Geometric Tools represents the best of both worlds: discussion of theory and code examples built on and culled from years of experience. --Mike Caputo ... Read more

Reviews (4)

Way over my head
I almost gave this book 2 stars but decided that punishing the authors for my stupidity wasn't fair. This book is obviously chucked full of just about every geometric formula you will ever need in computer graphics. But you had better be a darn serious mathematician! I could simply never grasp mathematics at this level. I could never look at those formulas, written in some alien heiroglyphs, and understand them enough to translate them into code. This book is one of those "Written by Professors for Professors." I bought the book because of the reviews that wrote about the code samples. Well, they are there (pseudo-code, but that is better than nothing) but nowhere near what I was expecting. My mistake was thinking it was a programming book. It's not! Bottom line, if you are looking for computer graphics formulas and understand mathematics, this book is for you. If you are a programmer and are expecting a computer graphics library of functions, look somewhere else.

warning: first edition
First of all, I commend the authors on a timely and valuable book. However, a word of caution: this book is full of errors. Every couple of pages I am noting in the margin: did they mean A instead of B? Having encountered so many errors, I am reading every formula with scepticism. The errors are serious enough that I have trouble recommending the book without reservations, but I know of no suitable alternative. I can only hope that the errors will be weeded out of future editions.

A real Treasure,
I just got this book about a week ago and I haven't been able to put it down since. A great resource, a real treasure full of well presented gems. A large number of topics are well presented with mathematical depth that enables you to understand the code. The code is very well written and concise. Truly a great book and a pleasure to read. I hope to see more books like this in the future. I am interested in medical imaging applications and I know I will be using this book for a long time to come. In addition, the authors have a site ... that presents the code and additional very valuable materials.

An incredible amount of material in a single book
This book provides a solid grounding in vectors and matrices, then covers a wide range of 2D and 3D geometric algorithms, such as object/object distance and intersection, boolean operations, BSP trees, convex hulls, and more. It is a comprehensive guide, giving relevant theory, methods, and working code fragments. It's an incredible value for the amount of material it covers. I think it is a must-have for computer graphics professionals (and others in related fields). In the interest of full disclosure, I wrote the Foreword to this book, which I did because I was pleased with how good it is.

Visit Dave Eberly's "Magic Software" site for more about the book's contents. ... Read more

Book Description

Book Series: INTERNATIONAL SERIES IN OPERATIONS RESEARCH AND MANAGEMENT SCIENCE : Volume 51

It is difficult to evaluate an organization's performance when there are multiple inputs and multiple outputs to the system. The difficulties are further enhanced when the relationships between the inputs and the outputs are complex and involve unknown tradeoffs. This book introduces DEA as a multiple-measure performance evaluation and benchmarking tool. The focus of performance evaluation and benchmarking is shifted from characterizing performance in terms of single measures to evaluating performance as a multidimensional systems perspective.

New DEA models and approaches are presented to deal with performance evaluation problems in a variety of contexts. A context-dependent DEA measures the relative attractiveness of similar operations/processes/products. Sensitivity analysis techniques can be easily applied and used to identify critical performance measures. Value chain efficiency models and DEA benchmarking models can be utilized to study the impact of information technology (IT) investments. These models can help organizations better understand the real impact of their IT investments and integrate technology more efficiently and effectively for the future.

Conventional and new DEA approaches are presented and discussed using spreadsheets - one of the most effective ways to analyze and evaluate decision alternatives. The user can easily develop and customize new DEA models based upon these spreadsheets.

This book also provides easy-to-use DEA software - DEA Excel Solver. This DEA Excel Solver is an Add-In for Microsoft® Excel and provides a custom menu of DEA approaches, which include more than 150 different DEA models. It is an extremely powerful tool that can assist decision-makers in benchmarking and analyzing complex operational efficiency issues in manufacturing organizations as well as evaluating processes in banking, retail, franchising, health care, e-business, public services and many other industries. The DEA Excel Solver does not set limit on the number of units, inputs or outputs. With the capacity of Excel Solver, the DEA Excel Solver can deal with large sized performance evaluation tasks. ... Read more

Reviews (2)

GREAT BOOK...THE BEST DEA CD
The book is excellent with a nice mix of theory and applications in DEA. I believe this is the best book on DEA. Of special interest is the CD that looks at many DEA models. The CD has no competitor and is by far the best around. I have used this CD in many of my academic papers on DEA. If you need DEA software this is the book.

very good DEA book and software
This is the DEA book has many DEA models that you can apply with the software supplied. The software is an Excel Add-In which can be easily used. The book also has good intro. on DEA. It is a book for both DEA beginner and advanced users. ... Read more

Book Description

Reviews (1)

math fun
This is a picture book about the Zills family.The family is off on an adventure to hiddenburial site of an ancient pharaoh.The twin brothers in the family get stuck inside a pyramid with their dog.They must use math to figure out a way to escape the pyramid. They must use skills to figure out the riddles written on the wall and locate the hidden burial chamber. Will they ever figure it out?Read Mummy Math to find out!

It's one of those special books which children will read without realizing they are learning a mathematical concept.

This is a recommended book for those teacher or parents that want to make math a little more exciting for children. ... Read more

Book Description

CliffsQuickReview™ Trigonometry mirrors the curriculum for a typical trigonometry course, which includes trigonometric functions, trigonometry of triangles, trigonometric identities, vectors, polar coordinates, and complex numbers. And, like all CliffsQuickReview books, it includes concise, focused review on introductory-level courses, tear-out pocket guide that highlights fundamental concepts, easy-to-navigate design, self-tests and exercises, resource center for recommendations for more books and more! In short, this is the ultimate supplement for studying Trigonometry – compact, portable, and crammed with everything you need to succeed. ... Read more

Reviews (2)

Handy Little Summary
After several years in a corporate engineering job, I started moonlighting as a math tutor. The Cliff's Quick Review Guides are wonderful to have in my "back pocket" when I need to quickly look something up that is covered in dust in the "archives of my brain."

Great series... Great book.
I'm a (returning :P) university Freshman preparing for the College Board CLEP tests. I was already familiar with the material covered in this book, but needed to refresh my memory. This review turned out to be *exactly* what I needed.

The author's ability to explain the material to the student are just shy of enlightening. The discussions & theorem proofs are written in a very concise, clear style.

I'm a big advocate of the Cliff's QuickReview series. Intended as a course supplement, these books are also *GREAT* for students wanting to refine their skills. Most of them are also very accessible to students with less familiarity on the subject; trying to learn it for the first time.

After reading this, I bought the Calculus & Differential Equations QuickReviews & I'm looking forward to reading them! ... Read more

Book Description

Elliptic curves have played an increasingly important role in number theory and related fields over the last several decades, most notably in areas such as cryptography, factorization, and the proof of Fermat's Last Theorem. However, most books on the subject assume a rather high level of mathematical sophistication, and few are truly accessible to senior undergraduate or beginning graduate students.Assuming only a modest background in elementary number theory, groups, and fields, Elliptic Curves: Number Theory and Cryptography introduces both the cryptographic and number theoretic sides of elliptic curves, interweaving the theory of elliptic curves with their applications. The author introduces elliptic curves over finite fields early in the treatment, leading readers directly to the intriguing cryptographic applications, but the book is structured so that readers can explore the number theoretic aspects independently if desired.By side-stepping algebraic geometry in favor an approach based on basic formulas, this book clearly demonstrates how elliptic curves are used and opens the doors to higher-level studies. Elliptic Curves offers a solid introduction to the mathematics and applications of elliptic curves that well prepares its readers to tackle more advanced problems in cryptography and number theory. ... Read more

Reviews (1)

Excellent
Anyone who writes a book on elliptic curves will never do a bad job, for these objects are so beautiful that it would be a sacrilege to do otherwise. Those who study elliptic curves fall under their spell, not only because of their beauty, but also because of their many applications: the spinning top in mechanics, cryptography, exactly solved models in statistical mechanics, precession of the Mercury perihelion in general relativity, the proof of Fermat's Last (Wiles) Theorem, control theory, and string theory, to name a few. This book is an excellent treatment of ECs and would be good for a graduate student starting out in the field. The author gives many concrete examples of the main theorems, and helpful exercises are found at the end of each chapter.

The author begins the book with two neat problems that motivate well the subject of elliptic curves: the pyramid of cannonballs and the right triangle problem, i.e. which integers can occur as areas of right triangles with integer sides? He then immediately begins the elementary theory of ECs in chapter 2. The treatment is pretty standard, although he proves Pascal's and Pappus's theorems using the associativity of the group operation on ECs, which is not usually done in books on ECs. Also somewhat non-standard this early in the game is the discussion of reduction of ECs modulo various primes, and the subsequent definitions of additive, split multiplicative, and non-split multiplicative reduction.

The study of torsion points is done in chapter 3 with the Weil pairing on the n-torsion of an EC taking center stage. A fairly short chapter, the author delays the proof of the properties of the Weil pairing until chapter 11, where it is done with divisors.

Chapter 4 deals with elliptic curves over finite fields, and is one of the most important in the book from the standpoint of cryptographic applications of ECs. Hasse's theorem, giving the bounds for the group of points on an EC over a finite field, is proven in detail. The Frobenius endomorphism is introduced, and a proof of Schoof's algorithm for computing the number of points on ECs over a finite field is given a detailed treatment. There are many symbolic computational software packages in both the open and commerical realm which will do the counting straightforwardly, and anyone interested in cryptography will need to be familiar with some of these. Supersingular curves in characteristic p are introduced, and the author gives a good discussion of the reason why they are named as such.

The discrete logarithm problem, a topic also very important for cryptographic applications, is discussed in chapter 5. The chapter beings with the index calculus, and, recognizing that it does not apply to general groups, the Pohlig-Hellman, baby step-giant step method, and Pollards rho and lambda methods are discussed in details. The author then shows that for supersingular and "anomalous" curves, that the discrete logarithm problem can be reduced to an easier discrete logarithm problem. Along the way, two important concepts are introduced: the p-adic valuation, and the Tate-Lichtenbaum pairing, the latter of which is related to the Weil pairing, but applies to situations where the Weil pairing does not.

Elliptic curve cryptography is then discussed in chapter 6, and the treatment is fairly thorough. The author shows to what extent the Decision Diffie-Hellman problem can be solved using the Weil pairing. He also shows how to represent a message on an elliptic curve, satisfying early on any reader's curiosity on just how this is done. The El Gamal and ECDSA are compared in terms of their computational efficiency. An EC generalization of RSA is also discussed in some detail, along with a cryptosystem based on the Weil pairing. Chapter 7 then gives other applications of ECs, such as factoring and primality testing.

Chapter 8 marks the beginning of the "heavy artillery" in the theory of ECs, for here the author begins the discussion of elliptic curves over the rational numbers, which can be viewed as an example of Diophantine geometry. The famous Mordell-Weil theorem is proved, and as a sign that one is definitely in the arena of modern mathematics, the proof is given in terms of Galois cohomology, which is an abstraction of the Fermat method of descent. The reader gets a taste of height functions, and via some good examples, gets insight into why the rank of the EC is so difficult to compute. A neat example is given of a nontrivial Shafarevich-Tate group.

I did not read the chapters 9, 10, or 11 on ECs over the complex numbers, complex multiplication, and divisors, so I will omit their review. Chapter 12 introduces the famous zeta functions, and their use in obtaining arithmetic information about an EC. Zeta functions motivate the definition of an L-function of an EC, these being tremendously important in modern developments in the theory of ECs, such as the Swinnerton-Dyer and Birch conjecture, the latter of which is motivated rather nicely in this chapter.

The last chapter of the book is an excellent introduction to the proof of Fermat's Last Theorem. Considering the level of the book, the author captures very well the essential ideas. Readers will be well prepared, after studying more algebraic number theory and the theory of Galois representations (which the author only skims in the book), to tackle the full proof if so desired. ... Read more

Reviews (1)

Excellent Electrical Engineering Reference Book
I recently tutored a co-worker for the electrical engineering Professional Engineering license exam and this book greatly helped out in explaining Laplace transforms. The book covers transform circuit analysis by starting out with the basics and then building up to Laplace transforms. After reading this book and working through the examples, I feel much stronger in this area of electrical engineering. I have a B.S. in electrical engineering and a P.E. license since 1995. I have read several other books on this subject, including differential equations text books and I think this book presents the clearest explanations and examples applied directly to electrical engineering. I would recommend this book to anyone with an interest in electrical engineering. ... Read more

Book Description

Featuring rich applications and integrated coverage of graphing utilities, this hands-on trigonometry text guides students step by step, from the right triangle to the unit-circle definitions of the trigonometric functions. Examples with matched problems illustrate almost every concept and encourage students to be actively involved in the learning process. Key pedagogical elements, such as annotated examples, think boxes, caution warnings, and reviews help students comprehend and retain the material. ... Read more

Reviews (1)

Excellent book for all levels
It provides wide range of practical applications, with plain English, colorful pages, step by step from basic to advanced approach. It has got answers at the back. I recommend it`s Instructor`s solutions manual as well... ... Read more

Book Description

Among the many beautiful and nontrivial theorems in geometry found in Geometry Revisited are the theorems of Ceva, Menelaus, Pappus, Desargues, Pascal, and Brianchon. A nice proof is given of Morley's remarkable theorem on angle trisectors. The transformational point of view is emphasized: reflections, rotations, translations, similarities, inversions, and affine and projective transformations. Many fascinating properties of circles, triangles, quadrilaterals, and conics are developed. ... Read more

Reviews (4)

Incredibly helpful
I just worked through this book & I have to say that it's probably the best for "what's next" after high school geometry. It's got a chapter on each of the following things: triangles (with Ceva's & Steiner-Lehmus' theorems, orthic triangles & Euler lines, etc), circles (like power of a point, etc), collinearity & concurrence, transformations (Yaglom's series of 3 books are better for these though but this is a good introduction), and an intro to inversive geometry and projective geometry. One section that blew my mind is the one about the "3-jugs" problem where you've got a jug that holds 8L, one that holds 5L & one the holds 3L and the goal is to get 4L in each of two jugs. Maybe you've heard of it...? This book gives a general method for solving problems like that using reflections. I liked how each section is very short and typically has just 2-3 theorems, so the book is split into manageable pieces. Maybe it's just a psychological thing, but I like that. Depending on the section, there are 3-10 problems at the end of each section; like any book there are easy ones with fairly obvious solutions and there are hard ounes that take a lot of thought. I think the hard ones might have come from olympiads because Sam Greitzer was a USAMO coach at some point. It doesn't matter though, this book gives you a lot of tools you can use to deal with olympiad-level problems. A couple good ones to work on after this book would be Johnson's "Advanced Euclidean Geometry" and Court's "College Geometry", both out of print unfortunately but maybe a nearby library wold have them?

THE BEST SOURCE FOR GEOMETRY
I am a high school student who competes in extracurricular mathematics contests. I've found many great books for trigonometry, algebra, precalculus, calculus, and any other field of math except geometry, i.e. until I found this book. A thorough knowledge of geometry beyond what you learn in high school in necessary needed to be competitive, and the place to get it is Geometry Revisited. This book covers any geometry you could need and is by far the best, if not the only, book out there with what you need to know. This book was also very interesting and thought provoking. The selection and arrangement of topics couldn't be any better. Each lesson even contains exercises (with answers and hints) to show you how to apply what the book teaches you. I would recommend this book to any high school student competing in any contests, to any person interested in extending their knowledge of geometry, or to anyone who with any interest in math. I can tell you that you will learn multitudes and enjoy the book too.

FOR ALL THOSE INTERESTED IN MATHEMATICS
This is 'the book' the typical high school student in America needs to read in order to see what Geometry is really all about. Geometry can be fun and engaging. Unfortunately it has taken a back seat in the high school curriculum. This is unfortunate considering the beauty that the subject has to offer. Many topics are covered in this book: The extended law of cosines, Ptolemy's theorem, transformational geometry, conics, and many, many, many more. This is also a book that is essential for mathematical problem solvers-- particularly for those interested in competing. All that is needed to follow along is the basic geometry learned at the high school level. This book is a must have for any mathematical library.

A++ Highly recommended.

The Best Undergrad Level Geometry Book Ever
Geometry Revisited should be used as a textbook for every geometry class in high school. It offers comprehensive proofs and theorems that every undergrad should know by the time he/she gets through basic geometry. The last sections of this book also offers a slight touch on comcepts such as non-Eucledian and projective geometry, serving as a launchpad to higher levels of education. I highly recommend this book to any high school or undergrad teachers and students. ... Read more

Book Description

This text is designed for a one-quarter or one-semester graduate couse in Riemannian geometry. It focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds. The book begins with a careful treatment of the machinery of metrics, connections, and geodesics, and then introduces the Riemann curvature tensor, before moving on the submanifold theory, in order to give the curvature tensor a concrete quantitative interpretation. The remainder of the text is devoted to proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's Theorem, and a special case of the Cartan-Ambrose- Hicks Theorem. This unique volume will especially appeal to students by presenting a selective introduction to the main ides of the subject in an easily accessible way. The material is ideal for a single course, but broad enough to provide students with a firm foundation from which to pursue research or develop applications in Riemannian geometry and other fields that use its tools. Of special interest are the "exercises" and "problems" dispersed throughout the text. The exercises are carefully chosen and timed so as to give the reader opportunities to review material that hasjust been introduced, to practice working with the definitions, and to develop skills that are used later in the book. The problems that conclude the chapters are generally more difficult. They not only introduce new mateiral not covered in the body of the text, but they also provide the students with indispensable practice in using the ... Read more

Book Description

Practical exercises with step-by-step solutions are designed to make geometric concepts easy to understand and remember.This book may be used as a brush-up for test preparation, as a classroom supplement, or as a self-teaching guide for workers in technical fields. Review high school geometry without intimidation, and practice with examples from the world of work and everyday life. ... Read more

Reviews (1)

Geometry-success in 20 minutes a day
I am a homeschooling mother of 2 high school students. I have been teaching geometry but with great difficulty. It has been 30 years since I have taken geometry and it was not a happy experience then and a struggle now. We do use a regular textbook but they are not always as clear a I would like them to be. I have been using this book as a back up in teaching the basics. They are clearly presented and easy to grasp. Each chapter presents a concept with self tests to make sure you understand before you go on the the next idea. It is a great suppliment to our textbook and a good review for the GED test. ... Read more

Reviews (1)

Too expensive, and depends upon the teacher.
Although I applaud Kay's intentions in presenting geometry inthis fashion, I think that this book would require an excellentteacher to accompany it. In some ways, I might have predicted this, because to preserve the "discovery," you have to be a little coy about what you are trying to teach...you have to hide the point. The book would be an excellent reference text, as Kay provides the historical context of most of the theorems, and the material is presented in a rather dense format ... ... Read more

Book Description

This text is suitable for introductory students, perhaps in programs such as education, art and architecture. The text contains some traditional material from geometry as well as more innovative topics. Throughout the text, the authors place strong emphasis on pedagogy, hands-on model building, a guided discovery method of learning, etc. Much of the material is written in such a way that it can be used in the classroom for enrichment projects, by prospective mathematics teachers. ... Read more

Book Description

The topic of 'circle packing' was born of the computer age but takes its inspiration and themes from core areas of classical mathematics. A circle packing is a configuration of circles having a specified pattern of tangencies, as introduced by William Thurston in 1985. This book lays out their study, from first definitions to latest theory, computations, and applications. The topic can be enjoyed for the visual appeal of the packing images - over 200 in the book - and the elegance of circle geometry, for the clean line of theory, for the deep connections to classical topics, or for the emerging applications. Circle packing has an experimental and visual character which is unique in pure mathematics, and the book exploits that to carry the reader from the very beginnings to links with complex analysis and Riemann surfaces. There are intriguing, often very accessible, open problems throughout the book and seven Appendices on subtopics of independent interest. This book lays the foundation for a topic with wide appeal and a bright future. ... Read more

Book Description

Three million high school students and 172, 000 college students enroll in geometry classes every year. Schaum's Outline of Geometry, Third Edition, is fully updated to reflect the many changes in geometry curriculum, including new terminology and notation and a new chapter on how to use the graphing calculator.

Reviews (3)

NOT AS GOOD AS 2nd Edition!
I think this updated version is not as clear and accurate as the 2nd. Though efforts can be seen trying to improve it and make it more attractive but, simply not as good as the 2nd edition.

Helpful in some cases, but . . .
This is definitely a great review book, but the review problems are very, very easy and not at all like the ones on the actual test. That is definiely my biggest gripe about the book.

However there are many pluses to this book as well. Each theoreom comes with examples, pictures, and definitions so you can easily understand them.

Also, every concept is well defined and in a way that is helpful and concise.

Schaum's Outline of Geometry
This is not only the best elementary geometry (and related subjects) outline, but also the best text in its field, in my opinion. The student will find as many theorems and definitions as in very long and detailed texts, presented much more concisely and in a much more organized fashion here in categories that are easily understood. As with my review of Schaum's algebra, I recommend strongly that the student make flash cards from Schaum's theorems and definitions before trying to work out or even read the numerous solved problems by themselves. Close to 95% of students, in my opinion, make the mistake of doing massive amounts of geometry problems and homework before learning what the theorems and definitions say. This is like trying to learn to play the violin by playing symphonies before one knows how to move the bow and fingers, and is probably the main cause of middle school, high school, and college elementary geometry failure. ... Read more