Book Description

With more than 30 million copies sold, Schaum's are the most popular study guide on the planet. Mathematics students around the world turn to this clear and complete guide to general topology for its through introduction to the subject, includingeasy-to-follow explanations of topology of the line and plane and topological spaces. With 650 fully solved problems and hundreds more to solve on your own (with answers supplied), this guide can help you spend less time studying while you make better grades! ... Read more

Reviews (1)

If you are a good researcher, this is a good resource!
If you are taking an introductory Topology course, I recommend using this book. It is a little old, and some of the problems are not solved, but it will offer you an advantage when using your class text. You should use Lipschutz's book together with REA's Topology Problem Solver to give you full study advantage when taking on this most difficult subject. With some moderate study, you should do better on your assignements! ... Read more

Book Description

The second volume of Shafarevich's introductory book on algebraic varieties and complex manifolds. As with Volume 1, the author has revised the text and added new material, e.g. as a section on real algebraic curves. Although the material is more advanced than in Volume 1 the algebraic apparatus is kept to a minimum, making the book accessible to non-specialists. It can be read independently of Volume 1 and is suitable for beginning graduate students in mathematics as well as those in theoretical physics. ... Read more

Reviews (1)

After Hartshorne!!!
This book is very good for the secondary course after learning with Harshorne's Algebraic geometry. ... Read more

Book Description

Written by a computer scientist for computer scientists, this book teaches topology from a computational point of view, and shows how to solve real problems that have topological aspects involving computers.Such problems arise in many areas, such as computer graphics, robotics, structural biology, and chemistry.The author starts from the basics of topology, assuming no prior exposure to the subject, and moves rapidly up to recent advances in the area, including topological persistence and hierarchical Morse complexes. Algorithms and data structures are presented when appropriate. ... Read more

Book Description

Covers topics in plane and solid (space) geometry.Also included are pictorial diagrams with thorough explanations on solving problems in congruence, parallelism, inequalities, similarities, triangles, circles, polygons, constructions, and coordinate/analytic geometry. ... Read more

Book Description

Titles in Barron's Painless Series are textbook supplements designed especially for classroom use by middle school and high school students. The approach of each title is an appeal to students who think that the subject is boring, or too difficult, or both. The authors, all experienced educators, take a light approach, showing kids what is most interesting about each subject, and how seemingly difficult problems can be transformed into fun quizzes, brain-ticklers, and challenging puzzles with rational solutions. Geometry becomes painless—and even fun—once students learn the subject's basic components and see how solving any geometric problem is fitting parts together to solve an intriguing puzzle. They learn the meaning of postulates and theorems, discover angles of all kinds, find the relationships that exist between parallel and perpendicular lines, and discover the characteristics of shapes such as triangles, quadrilaterals, and circles. The author introduces real-world geometry experiments to make concepts less abstract, offers study strategies, and demonstrates how mini-proofs are the first step toward understanding formal geometry proofs. ... Read more

Reviews (2)

Not Very Good
On page 16, it is stated that the area of a circle is pi times the diameter. Is there anybody out there who DOESN'T know that the area of a circle is pi times the square of the radius? That error wouldn't such a big deal, except that there are plenty more to come. I don't recommend this book to anyone.

Definitely NOT for homeschool!
Since I'm homeschooling my high school sophomore this year, I've been spending time looking at math books. "Painless Geometry" seemed like a good bet. Profusely illustrated (albeit with silly monkey pictures) and written in plain English, it looked like just what we'd want.

That's until I started actually using the book. First of all, who ever heard of a 300-page reference book with only three pages of index? How are you supposed to find things that way? It's missing things like the base of a triangle (the index has neither "base" nor "triangle:base") and how to label an angle. The information's in the book, but you certainly can't find it using the index. Not only that, but the pages aren't labeled like a normal book, with the name and number of the chapter at the top or bottom of each page. You can't find your place in a book that way!

There's little depth to the book. There are experiments with pencil and paper, but no real-world examples of where you'd use geometry. Area is calculated in "square units" with no discussion of real units of measure. Pi is introduced with a single paragraph. No explanation is given of its rich history, how it's calculated, or applicability throughout mathematics.

The oversimplifications in this book may make life difficult later. The book states that all angles are measured in degrees, and the degrees symbol is generally omitted. Whatever happened to radians? In one of the problems, she asks for the area of a circle with diameter of ten. The correct answer is 100 times pi. The book states the answer as 314. That's an approximation, not an answer!

Then we started finding the mistakes. Typos like "Computer the area of a circle" (page 184) I can live with. It's hard core mistakes like these I can't tolerate:

The reader is asked to identify what type of triangle has angles of 120, 35, and 35 degrees (page 101). The answer says it's isosceles and obtuse. In reality, it's not a triangle at all, as the angles don't add up to 180 degrees!

How's this for a statement of the Side-Angle-Side postulate (page 126)? "If two sides and the included angle of one triangle are congruent to two triangles and the included angle of a second triangle, then the triangles are congruent." Huh?

There's a "super brain tickler" on page 163 which indicates, according to the answers in the book, that for squares, rhombuses, rectangles, and parallelograms, all four sides are parallel! No. Four parallel line segments wouldn't ever meet. Those four shapes have two sets of parallel sides, not one set of four parallel sides!

.... That tends to leave us with drek like "Painless Geometry."

All in all, I found this book to be poorly proofread, ridded with errors, badly indexed, oversimplified, and disconnected from the real world. It may be good as an adjunct for a student having trouble with a real geometry book, but only if there's someone around to explain what "Painless Geometry" omits or misstates. ... Read more

Book Description

From the reviews: "This is a very interesting book containing material for a comprehensive study of the cyclid homological theory of algebras, cyclic sets and S1-spaces. Lie algebras and algebraic K-theory and an introduction to Connes'work and recent results on the Novikov conjecture. The book requires a knowledge of homological algebra and Lie algebra theory as well as basic technics coming from algebraic topology. The bibliographic comments at the end of each chapter offer good suggestions for further reading and research. The book can be strongly recommended to anybody interested in noncommutative geometry, contemporary algebraic topology and related topics." European Mathematical Society Newsletter In this second edition the authors have added a chapter 13 on Mac Lane (co)homology. ... Read more

Reviews (1)

very comprehensive treatment on the subject
I think that this is probably the only comprehensive book on the subject. There are several lecture notes published, but no other books have collected all up to date recent research results on the subject like this book did. Probably must-read classical book if you are interested in cyclic homology, or if you want to use it for you research. Very unfortunate fact is that this book is very rare, so that very hard to obtain. You may have lots of trouble buying this book. ... Read more

Book Description

Fractal Geometry is a recent edition to the collection of mathematical tools for describing nature, and is the first to focus on roughness. Fractal geometry also appears in art, music and literature, most often without being consciously included by the artist. Consequently, through this we may uncover connections between the arts and sciences, uncommon for students to see in maths and science classes. This book will appeal to teachers who have wanted to include fractals in their mathematics and science classes, to scientists familiar with fractal geometry who want to teach a course on fractals, and to anyone who thinks general scientific literacy is an issue important enough to warrant new approaches. ... Read more

Reviews (1)

See how fractals can be part of the math curriculum
When I teach computer programming, the sections that raise the greatest amount of interest are those that involve the manipulation of images. Therefore, it is natural to me that any mathematics that involves the manipulation of images would generate a great deal of student interest. I was a working mathematician when the concept of fractals hit the world, and like so many others was caught up in the excitement. I wrote programs to generate them and read all of the major books.
That was almost two decades ago, and the field has continued to expand. In this book, you will read about projects where math teachers have incorporated fractals into the curriculum. It is no surprise to me that it was almost universally a success, the sheer beauty of the fractal images guarantees interest and the increase in computing power has made the generation of images much easier. Many of the first images I created required overnight runs, something that can be done in minutes today.
The fact that the images do model much of the natural world also increases the interest. Nature is irregular and unpredictable in the micro sense, and fractals give us a way to describe and maybe understand it. The articles are all well written and easy to follow, and many different types of projects are demonstrated. Some of the papers describe the structure of courses in fractals, so if you have an interest in creating such a course, you can find your point of origin.
While fractals have occasionally suffered from the common malady of being oversold, they do provide a bridge between mathematics and the real world. Therefore, the study of fractals should be part of the mathematics curriculum, and in this book you can read about how many people have successfully done it. Another quality book on contemporary mathematics, it should be part of the next iteration of library purchases everywhere. ... Read more

Book Description

Dale Husemoller is Professor of Mathematics at Haverford College (PA). Most of his research has been in algebraic topology, homological algebra, and related fields. Together with John Milnor, Husemoller is author of Symmetric Bilinear Forms (Springer-Verlag, 1973). He is also the author of Elliptic Curves (Springer-Verlag, 1987) and cyclic homology (Tata Lecture notes, 1991). Visits abroad include the IHES (Institut des Hautes Etudes Scientifiques) in France, Universität Bonne and Universität Heidelberg in Germany, ETH in Zürich, Switzerland, and the Tata Institute of Fundamental Research in Bombay, India. ... Read more

Reviews (2)

Let E be a principal O(n)-bundle...
I enjoyed reading this book, although this wasn't the place where I first started learning about the subject. Husemoller lays out the foundations, starting from scratch, and presupposes a great deal of mathematical maturity. New applications to physics (gauge theory, supersymmetry and all that) are usually better left untouched in a book like this, since Husemoller's secondary concern is about applications to homotopy theory rather than physical applications. To some reviewers, this would leave much else to be desired, but the scope of the book warrants attention being paid to the nitty-gritty of fibres bundles, suitable for a mathematical audience (say, homotopy theorists).

Excellent!

OK book on Fiber bundles
Modern mathematics books are usually written in a formal style that makes for impeccable logic but poor didactic quality. Husemoller in this book gives a good summary of the main results in the theory of fiber bundles but leaves the reader wanting as to just why the techniques used to study bundles work as well as they do. One needs insight and intuition into a subject if one is to apply it or extend its frontiers. The techniques from K-theory and the idea of characteristic classes needs to be explained in detail. It would be have been nice if the author could have explained, and not just expounded, why these techniques are so powerful in the study of fiber bundles. I have read both the 2nd and 3rd edition of this book, went through all of the calculations and proofs, and still was left with a hunger for more understanding of the relevant concepts. For such an understanding I read the original papers in the early part of the 20th century, and read Norman Steenrod's book on fiber bundles. No doubt that understanding of such an abstract formalism does require careful thought, but a good book should be a guide in that effort. Husemoller's book cannot be read as a standalone book in that regard. One must supplement it with a tremendous amount of outside reading. The merits of the book, at least in the 3rd edition, are the discussion of the guage group of the principal bundle, and the inclusion of a chapter on characteristic classes and connections. The physicist reader who is interested in how fiber bundles enter into quantum field theory or superstring theory will welcome this. A good book for reference and worth purchasing, but if you want an in-depth understanding of the theory of fiber bundles, be prepared to read a lot more other than this book....definitely much too formal. ... Read more

Reviews (1)

AN AWESOME READ
Like the Marines who stormed the beaches of Iwo Jima the author, John Tabak, storms the beaches of geometry and ultimately raises the flag on an infinite-dimensional Hilbert space.Buy it.Read it. Love it.
Semper Geometry,
James Tabak USMC
--Geometricians are squared away. ... Read more

Book Description

Here is the first modern introduction to geometric probability, also known as integral geometry, presented at an elementary level, requiring little more than first-year graduate mathematics. Klein and Rota present the theory of intrinsic volumes due to Hadwiger, McMullen, Santaló and others, along with a complete and elementary proof of Hadwiger's characterization theorem of invariant measures in Euclidean n-space. They develop the theory of the Euler characteristic from an integral-geometric point of view. The authors then prove the fundamental theorem of integral geometry, namely, the kinematic formula. Finally, the analogies between invariant measures on polyconvex sets and measures on order ideals of finite partially ordered sets are investigated. The relationship between convex geometry and enumerative combinatorics motivates much of the presentation. Every chapter concludes with a list of unsolved problems. ... Read more

Reviews (1)

a beautiful book
Everyone with any interest in probability or combinatorics should take alook at this book, and at least read Chapter 1, on the Buffon needleproblem.It gives a beautiful conceptual solution, quite different fromthe more well-known solution using integrals to get conditionalprobabilities. I find it hard to imagine anyone reading Chapter 1 and notdeciding to read the entire book.

I heard Rota lecture on this material,and the book has much the same feeling as his lectures: it is clear,elegant, and concise, full of illuminating examples.Relatively littlebackground is required, and it should be easily accessible to beginninggraduate students (or undergraduates with unusually strong backgrounds). ... Read more

Reviews (12)

I tutor and this book is the best
I tutor geometry and am happy to find most california high school students use this book.It is clear and staight forward and I have not seen a better book.

Good addition to any course
We used this book throughout last year in geometry and I found it to be helpful in having plenty of sample problems of how to use the things we learned in class. The problem some of the previous reviewers had, of not being able to understand, have a teacher issue, not a book issue. If the teacher taught instead of the book they would be fine.

horrible book
it teaches some wierd geometry where the actual theorems have been claimed as postulates. if you'll take a look on page 692 where all "postulates" are listed you'll find that many of them are actually theorems (for examples, "postulates" 10-14 are theorems and have proof based on real axioms which for some wierd reason are not listed in this book at all!)

not helpful AT ALL
I used this book for my sophomore year of geometry.The majority of people in my class squeaked by first semester with a B- - C-.Then come 2nd semester, 90% of the class FAILS, including myself.I'm especially disappointed since my report card says straight As for all my classes except for an Incomplete for geometry.This book, like the previous reviewer said, does not provide enough helpful examples to guide you through the problems.The examples help you through the first few problems which are obviously simple but then you go past the fifth problem and you're totally stuck.I'm currently making up the work I left incomplete.Not because I was lazy, it's because I couldn't understand the book's concept.And so this summer, I'm stuck with this cursed book AGAIN.I need to retake the 2nd semester final all because of this book.Now I know not everyone is gifted with mathematical skills as some others may be.But can't there be a book where EVERYONE can understand?Last year in Algebra, I passed with an A.Because the book was easy to comprehend and provided the student with examples thorough enough to guide you through basically all the problems.I forgot the name of it but the initials of the author were H.M.Houghlin...Mc something.Now THAT's a book I'd take home any day.

The Best!
I used this book for a semester geometry course (covering Chapters 1 through 13), and it was marvelous!Besides the fact that there are no typographical errors, the book flows logically from one chapter to the next.In addition, its early introduction of proofs guarantees a successful continuation and comprehension of the essence of geometry.Moreover, the lettered exercises at the end of each section go from easy to difficult--allowing for a gradual increase in skill.Finally, the logic appendix at the end is a great introduction to proofs and to the mathematical field of logic. ... Read more

Book Description

This book combines mathematics (geometry and topology), computer science (algorithms), and engineering (mesh generation) in order to solve the conceptual and technical problems in the combining of elements of combinatorial and numerical algorithms.The book develops methods from areas that are amenable to combination and explains recent breakthrough solutions to meshing that fit into this category. It should be an ideal graduate text for courses on mesh generation.The specific material is selected giving preference to topics that are elementary, attractive, lend themselves to teaching, are useful, and interesting. ... Read more

Book Description

Janos Bolyai (1802-1860) was a mathematician who changed our fundamental ideas about space. As a teenager he started to explore a set of nettlesome geometrical problems, including Euclid's parallel postulate, and in 1832 he published a brilliant twenty-four-page paper that eventually shook the foundations of the 2000-year-old tradition of Euclidean geometry. Bolyai's "Appendix" (published as just that--an appendix to a much longer mathematical work by his father) set up a series of mathematical proposals whose implications would blossom into the new field of non-Euclidean geometry, providing essential intellectual background for ideas as varied as the theory of relativity and the work of Marcel Duchamp. In this short book, Jeremy Gray explains Bolyai's ideas and the historical context in which they emerged, were debated, and were eventually recognized as a central achievement in the Western intellectual tradition. Intended for nonspecialists, the book includes facsimiles of Bolyai's original paper and the 1898 English translation by G. B. Halstead, both reproduced from copies in the Burndy Library at MIT. ... Read more

Book Description

First published in 1952, this book has proven a valuable introduction for generations of students. It provides a clear and systematic development of projective geometry, building on concepts from linear algebra. ... Read more

Book Description

This book is an attempt to give a systematic presentation of both logic and type theory from a categorical perspective, using the unifying concept of fibred category. Its intended audience consists of logicians, type theorists, category theorists and (theoretical) computer scientists. ... Read more

Reviews (1)

Excellent book.
Excellent book. The best in its field. I would recommend it, particularly for students. ... Read more

Book Description

Master Math: Basic Math and Pre-Algebra teaches you in a very user-friendly and accessible manner the principles and formulas for establishing a solid math foundation. ... Read more

Reviews (2)

Table of Contents
Master Math: Pre-Calculus Table of Contents

Introduction

Chapter 1 Geometry

1.1. Lines and angles 1.2. Polygons 1.3. Triangles 1.4. Quadrilaterals (four sided polygons) 1.5. Circles 1.6. Perimeter and area of planar two-dimensional shapes 1.7. Volume and surface area of three-dimensional objects 1.8. Vectors

Chapter 2 Trigonometry

2.1. Introduction 2.2. General trigonometric functions 2.3. Addition, subtraction and multiplication of two angles 2.4. Oblique triangles 2.5. Graphs of cosine, sine, tangent, secant, cosecant and cotangent 2.6. Relationship between trigonometric and exponential functions 2.7. Hyperbolic functions

Chapter 3 Sets and Functions 3.1. Sets 3.2. Functions

Chapter 4 Sequences, Progressions and Series

4.1. Sequences 4.2. Arithmetic progressions 4.3. Geometric progressions 4.4. Series 4.5. Infinite series: convergence and divergence 4.6. Tests for convergence of infinite series 4.7. The power series 4.8. Expanding functions into series 4.9. The binomial expansion

Chapter 5 Limits

5.1. Introduction to limits 5.2. Limits and continuity

Chapter 6 Introduction to the Derivative

6.1. Definition 6.2. Evaluating derivatives 6.3. Differentiating multivariable functions 6.4. Differentiating polynomials 6.5. Derivatives and graphs of functions 6.6. Adding and subtracting derivatives of functions 6.7. Multiple or repeated derivatives of a function 6.8. Derivatives of products and powers of functions 6.9. Derivatives of quotients of functions 6.10. The chain rule for differentiating complicated functions 6.11. Differentiation of implicit vs. explicit functions 6.12. Using derivatives to determine the shape of the graph of a function (minimum and maximum points) 6.13. Other rules of differentiation 6.14. An application of differentiation: curvilinear motion

Chapter 7 Introduction to the Integral

7.1. Definition of the antiderivative or indefinite integral 7.2. Properties of the antiderivative or indefinite integral 7.3. Examples of common indefinite integrals 7.4. Definition and evaluation of the definite integral 7.5. The integral and the area under the curve in graphs of functions 7.6. Integrals and volume 7.7. Even functions, odd functions and symmetry 7.8. Properties of the definite integral 7.9. Methods for evaluating complex integrals; integration by parts, substitution and tables

Index

Appendix Tables of Contents of First and Second Books in the Master Math Series

Great book. Lots of good trig.
This book is pretty small but it gives great explanations of geometric shapes, angles, and trig functions like tan, sin, cos, and others. It's straight to the point and I learned from it very quickly. I highly suggest getting this book before moving on to a more advanced, or even just a regular geometry or trig book. ... Read more

Book Description

Geometry is hard. This book makes it easier. You do the math. This is the fourth title in the series designed to help high school and college students through a course they'd rather not be taking. A non-intimidating, easy-to-understand companion to their textbook, this book takes students through the standard curriculum of topics, including proofs, polygons, coordinates, topology, and much more. ... Read more

Book Description

This book is meant to be an introductory text, albeit at an upper graduate level. Themain prerequisite for reading this book is some familiarity with the basic theory of ellipticcurves as described, for example, in the first volume. Numerous exercises have beenincluded at the end of each chapter. A list of comments and citations for the exerciseswill be found at the end of the book. ... Read more

Reviews (1)

The most fascinating objects in all of mathematics
This book is a continuation of the authors earlier book on elliptic curves, which was also an excellent book, and treats the more specialized topics in elliptic curves. I cannot think of a branch of physics or engineering that has not made use of some facet of the theory of elliptic curves, and they have myriads of applications in other fields also, such as cryptography and financial engineering. The book is very organized, straightforward to read, the author summarizes well his intentions at the beginning of each chapter, and recommends several references for topics left out of the main discussion. Space does not allow a detailed chapter by chapter review, so I will confine my review to the first two chapters, which were of main interest to me. In summary, Chapter 1 discusses how to study elliptic curves by taking a collection of them, each member being isomorphic, and studying the properties of modular functions and differential forms on this collection, now thought of as an algebraic curve, called the moduli space. The famous linear operators, called the Hecke operators, act on the the space of modular forms, and they and their eigenfunctions satisfy the same set of relations. One then attaches the well-known L-series to the modular forms that has very interesting algebraic and analytic properties. In more detail, the author does the following in the chapter. The set of lattices in the complex plane modulo non-zero multiplication L/C* is considered, along with the set of elliptic curves over the complex plane modulo complex isomorphism. These collections are proven to be bijective by showing that L/C* is isomorphic to C by first putting a complex structure on it. This leads to a surjective map from the upper-half plane H to L/C*. Proving this to be injective leads to a bijection from SL2(Z)\H to L/C*. Since the matrix -1 acts trivially on H, one can quotient out +1 and -1 and obtain the modular group. The quotient space modular group\H is a 2-sphere minus a point, but can be made into a Riemann surface by extending the upper half-plane (called H*). The modular curve X(1) = modular group\H* results and is compact and Hausdorff. A complex structure is put on it, making it into a a Riemann surface of genus 0. Meromorphic functions on X(1) are rational functions of the j function, but more interesting functions are defined on X(1), namely the modular functions, such as the Eisenstein series. These considerations lead to a proof of the uniformization theorem for elliptic curves over C. For a given elliptic curve E, a study of the set of all isogenies to E of degree n is the same as that of studying degree n maps from E to other elliptic curves, which is called the dual isogeny, and leads to the Hecke operator. The Hecke operator and the homothety operator both map the divisor group of the lattice to itself, and generate a commutative algebra, called the Hecke algebra. Hecke operators can act on modular forms of weight 2k, and modular forms exist which are simultaneous eigenfunctions for the Hecke operator of weight 2k. It can be proven, but the author does not do so, that the normalized eigenfunctions form a basis for the space of cusp forms of weight 2k. The Fourier coefficients of the eigenfunction have an Euler product decomposition of a Dirichlet series attached to f, called the L-series. In the next chapter, the author considers elliptic curves that have extra endomorphisms, called complex multiplication. The collection of endomorphisms is usually taken to be the real numbers R, or R(K), which is the ring of integers of R tensored with the rational numbers. And, just as in chapter 1, he studies collections of elliptic curves, but here ones with the same endomorphism ring., called ELL(R) in the book. Asking the question of how to construct an elliptic curve with complex multiplication by a particular R(K) leads him to studying the ideal class group of R(K), and this group is shown to act transitively on ELL(R(K)). The author also shows that every elliptic curve with complex multiplication is defined over an algebraic extension of Q. Several interesting examples of ellipti curves with complex multiplication are given. After a brief review of class field theory, the author proves that K(j(E)) is the Hilbert class field H and shows how the Galois group of H/K acts on j(E). The torsion points of E are then used to generate abelian extensions of K, using the Weber function for E/H, thus generalizing the usual cyclotomic extensions of number theory. Very interesting examples are given of these constructions and it is also shown that j(E) is an algebraic integer. Then after a brief review of cyclotomic class field theory, the author proves what he calls the main theorem of complex multiplication, which says that an automorphism of the torsion subgroup is essentially analytic multiplication by an idele of K. This theorem allows one to define a Grossencharacter associated to an elliptic curve with complex multiplication. For such a curve one can then define an L-series and show that it can be expressed as a Hecke L-series with Grossencharacter. ... Read more

Book Description

This introductory text develops the geometry of n-dimensional oriented surfaces in Rn+1. By viewing such surfaces as level sets of smooth functions, the author is able to introduce global ideas early without the need for preliminary chapters developing sophisticated machinery. the calculus of vector fields is used as the primary tool in developing the theory. Coordinate patches are introduced only after preliminary discussions of geodesics, parallel transport, curvature, and convexity. Differential forms are introduced only as needed for use in integration. The text, which draws significantly on students' prior knowledge of linear algebra, multivariate calculus, and differential equations, is designed for a one-semester course at the junior/senior level. ... Read more

Reviews (3)

Introduces differential geometry to advanced-calc students
As a math undergrad at Kent State University some twenty-odd years ago, I took a course in differential geometry. This was the text; I still have my copy. (Autographed by the author, in fact; I met him on a visit to his university, where I subsequently attended grad school.)

The title of this book states, accurately, that its subject matter is 'elementary topics _in_ differential geometry'. This is one of those 'transition' books that introduces students familiar with Subject A to a more-or-less-systematic smattering of elementary topics in Subject B. Here, Subject A is multivariate calculus and Subject B is, of course, differential geometry.

Since that's what this book is for, there are way more numbers and pictures in it than you'll ever see in a modern graduate-level differential geometry text. The idea is to show the student the geometric meaning behind all the advanced calculus and help him/her understand _both_ words in the name 'differential geometry'. In short, much of the motivation here is geometric.

I liked it a lot and I am still grateful for its highly accessible introduction to a fascinating field. However, I must also add that its approach is not representative of any graduate-level math course I ever took. Of course this is an undergraduate text and isn't supposed to represent graduate-level coursework. Nevertheless, it _may_ give a student the wrong idea about what to expect in more advanced treatments. (Is there some personal history lurking behind that remark? You guess.)

An excellent 'transitional' book, then, and highly recommended to readers who want to connect their knowledge of multivariate calculus to the geometry of Euclidean space. It's also a fine example of an expository work on mathematics that remembers its target audience. However, as other reviewers have commented, it needs some answers to the exercises in order to be really useful for self-study.

A good start
This book could be considered as the second semester of an advanced calculus course and serves as an excellent introduction to differential geometry. The approach is rigorous, but the author does employ a great deal of illustrations to explain the relevant concepts. The first five chapters cover vector fields on curves and surfaces. The many concrete examples given by the author illustrate effectively the normal and tangent vector fields. The Gauss map is then appropriately introduced in Chapter 6 and shown to be onto for compact, connnected, oriented n-dimensional surfaces in n+1-dimensional Euclidean space.

This is followed by a discussion of geodesics and parallel transport in the next two chapters. The important concept of holonomy is introduced in the exercises along with the Fermi derivative. These ideas are extremely important in physical applications and must be understood in depth if the reader is to go into areas such as general relativity and high energy physics.

The next chapter considers the local behavior of curvature on an n-surface via the Weingarten map. The important concept of the covariant derivative is introduced. The concept of a geodesic spray, so important in the theory of differential equations, is introduced in the exercises.The curvature of plane curves is treated in Chapter 10 with the circle of curvature introduced. The Frenet formulas, which relate the tangent and normal vectors to the curvature and torsion, are discussed in the exercises. The curvature of surfaces is discussed later in Chapter 12 with the first and second fundamental form introduced, along with the very important Gauss-Kronecker curvature. And in this chapter the author introduces the idea of local and global properties of an n-surface. Although not rigorous, the discussion is helpful for students first introduced to these concepts.

After a nice overview of convex surfaces, the parametrization of surfaces is discussed in the next two chapters, where the inverse function theorem for n-surfaces is proved. This is followed by a consideration of focal points with Jacobi fields discussed in the exercises.

More measure-theoretic concepts are discussed in the next chapter on surface area and volume. Partitions of unity are brought in so as to define the integral of an n-form over a compact oreinted n-surface. Exterior products of forms are introduced in the exercises.

Soap bubble enthusiasts will appreciate the discussion on minimial surfaces in Chapter 18. Although very short, the author's treatment does bring out the important ideas. Minimal surfaces have taken on particular important in the new membrane theories in high energy physics recently. This is followed by a detailed treatment of the exponential map in Chapter 19. Once again, techniques with a variational calculus flavor are used to characterize geodesics as shortest paths.

After a discussion of surfaces with boundary in Chapter 20 the Gauss-Bonnet theorem is proved in Chapter 21 using Stoke's theorem. The discussion of this important result is crystal clear and should prepare the reader for more advanced statements of it in the general context of differentiable manifolds. This is followed by a brief discussion of rigid motions and isometries in the next two chapters. The book ends with ta discussion of Riemannian geometry, a topic of upmost importance in physics and discussed here with care.

A very good book and one that will be useful to beginning students of differential geometry, and also physics students going into the areas of gravitational physics or high energy physics.

Another Differential Geometry Book
I bought this book as a supplement, and I wish I hadn't. It's more archaic and has a large amount of 'hidden' steps than most mathematical books. It has problems, but no solutions. Not recommended for the physics, applied physics or self-learner. It's really aimed at the 'hard-core' mathematicians, and even they would have to have some experience/guidance in differential geometry.

I have an MS in physics, and found this book to be very difficult to get information out of. It has a few nuggets, but can only be seen after going through other books.It might go well with a good lecturer, but as a self-studied person, this is not the way to go. ... Read more