Book Description

Riemannian geometry has today become a vast and important subject. This new book of Marcel Berger sets out to introduce readers to most of the living topics of the field and convey them quickly to the main results known to date. These results are stated without detailed proofs but the main ideas involved are described and motivated. This enables the reader to obtain a sweeping panoramic view of almost the entirety of the field. However, since a Riemannian manifold is, even initially, a subtle object, appealing to highly non-natural concepts, the first three chapters devote themselves to introducing the various concepts and tools of Riemannian geometry in the most natural and motivating way, following in particular Gauss and Riemann. ... Read more

Reviews (1)

A fine book.
This book is a good reference for anyone who wants to know about the development of riemannian geometry. It covers many active topics in modern differential geometry for every graduated students who can choose some interested fields for the thesis. ... Read more

Book Description

This text on analysis on Riemannian manifolds is a thorough introduction to topics covered in advanced research monographs on Atiyah-Singer index theory. The main theme is the study of heat flow associated to the Laplacians on differential forms. This provides a unified treatment of Hodge theory and the supersymmetric proof of the Chern-Gauss-Bonnet theorem. In particular, there is a careful treatment of the heat kernel for the Laplacian on functions. The author develops the Atiyah-Singer index theorem and its applications (without complete proofs) via the heat equation method. Rosenberg also treats zeta functions for Laplacians and analytic torsion, and lays out the recently uncovered relation between index theory and analytic torsion. The text is aimed at students who have had a first course in differentiable manifolds, and the author develops the Riemannian geometry used from the beginning. There are over 100 exercises with hints. ... Read more

Reviews (3)

Fascinating book - I couldn't put it down
Steve Rosenberg's writing style is fascinating.I was up until 3 in the morning finishing this book.

Pithy plot, great characters
A real page turner!It was unbelievable how all the pieces fit together at the end.Rosenberg is a genius.

Good introduction but not detailed enough
Although a good introduction is given certains ideas are not really worked out in detail (proofs,...) the level may be put a little bit higher ... Read more

Book Description

This book introduces the basic ideas of mathematical proof to students embarking on university mathematics. The emphasis is on helping the reader to understand and construct proofs and write clear mathematics. The authors achieve this by exploring set theory, combinatorics and number theory, which include many fundamental mathematical ideas. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the all time great classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. The over 250 problems include questions to interest and challenge the most able student but also plenty of routine exercises to help familiarize the reader with the basic ideas. ... Read more

Reviews (2)

Essential for Calculus Students!
If you are using the textbook, _Calculus and Analytic Geometry_, it is imperative to get a copy of this solutions manual.The solutions manual comes in handy whenever you run into a difficult problem, and the solutions are very clear and thorough for every problem in the book.This book really helped me in college calculus.

great accompaniment to an excellent book
McGraw Hill's 'Calculus and Analytic Geometry' is an excellent calc bookin itself, and this guide is an excellent resource for any student usingthat book. It contains solutions for all of the problems in the textbook,step by step in an easy to understand manner. I definitely recommend thisbook and it deserves 5 stars, but I'm only giving it 4 stars because theprice is a bit steep. ... Read more

Book Description

In this book, Professor Kempf gives an introduction to the theory of algebraic varieties from a sheaf theoretic standpoint. By taking this view he is able to give a clean and lucid account of the subject, which will be easily accessible to all newcomers to algebraic varieties. ... Read more

Reviews (3)

A Sophisticated (but introductory) treatment
The title of the book is misleading. This book is about abstract algebraic varieties (as defined by Serre). Kempf claims that his book is accesible to anyone with a background in basic commutative algebra and this is true. However, a short exposure to examples of classical varieties (say, from the book of Harris) would be quite helpful to the reader who wishes to learn algebraic geometry from this book. For instance, an affine variety is defined as a space Y with a sheaf of k-algebras such that Mor(X,Y) is in natural bijection with k-Alg-Hom(k[Y],k[X]) for all other such spaces X which have a sheaf of k-algebras. An algebraic vareity is defined as a space that has an open covering of affines.

Having said that, I must add that this book is extremely well written. It's slightly dense but the assiduous reader is rewarded heavily by an exposition which proves the Riemann-Roch theorem and covers sheaf cohomology in a relatively small exposition. Moreoever, the abstract approach is finally helpful for the reader that wishes to learn about schemes. Indeed, the reader only has to drop some assumption on the structure sheaf of the topological space to get a scheme.

Coming back to pre-requisites, there are very few of them. A standard graduate course in Algebra covering basics of Noetherian rings and some module theory should be enough. Kempf proves the Nullstellensatz -- that should give some idea of the pre-requisites.

I will also take this opportunity to warn the reader that Kempf's usage of common words does not match the accepted meaning. For instance, consider a finitely generated nilpotent free k-algebra A over an algebraically closed field k. Kempf defines Spec(A) to be the set of all k-algebra homomorphisms from A into k. Clearly, Kempf's Spec(A) is the conventional Max-Spec(A) (the set of all maximal ideals of A). The conventional Spec(A) is the set of all prime ideas of A which has many more points than Max-Spec(A) in most cases.

A full-result book,but not for beginner....
In my opinion,that I have read this book,this is really a book on elementary ALGEBRAIC Geometry!,I mean it treats elementarily subjects that you may find them after a graduate course in algebraic geometry... thisbook begins with preliminary results on affine varieties,but in advancedview points, for example it defines an affine varieties asa variety(!)that is a represetative functor by a affine algebra over an algebraicallyclosed field. it continues up to classical results on affine varieties.G.Kempf defines a product diagramin the category of spaces withfunctions. after this you will see definitions of sheaves, and socohomologies, this is not eassy for a first-stage reading in algebraicgeometry,but for a second one is funy! at last you have some results of thetheory ofA. GROTHENDIECK and some interesting theorem on picard group anddivisors. this book may be viewed as a solotion book for some naturalquestions after a course in algebraic geometry.

A full-result book,but not for beginner....
In my opinion,that I have read this book,this is really a book on elementary ALGEBRAIC Geometry!,I mean it treats elementarily subjects that you may find them after a graduate course in algebraic geometry... thisbook begins with preliminary results on affine varieties,but in advancedview points, for example it defines an affine varieties asa variety(!)that is a represetative functor by a affine algebra over an algebraicallyclosed field. it continues up to classical results on affine varieties.G.Kempf defines a product diagramin the category of spaces withfunctions. after this you will see definitions of sheaves, and socohomologies, this is not eassy for a first-stage reading in algebraicgeometry,but for a second one is funy! at last you have some results of thetheory ofA. GROTHENDIECK and some interesting theorem on picard group anddivisors. this book may be viewed as a solotion book for some naturalquestions after a course in algebraic geometry. ... Read more

Book Description

This new edition is written for students preparing for technical, engineering technology or scientific careers. It begins with thorough coverage of topics in precalculus, calculus and differential equations, with an emphasis on how they relate to technical applications. The student has the opportunity to solve problems, much as they will in their future career, through the text's extensive applications and integrated use of technology. The text and exercises are designed to help students learn mathematical concepts and skills. Once these skills have been learned, students are encouraged to use graphing calculators to solve difficult problems with greater ease. Hints, notes and cautions are found throughout the text provide problem-solving techniques. ALSO AVAILABLE Student Solutions Manual, ISBN: 0-8273-7417-8 INSTRUCTOR SUPPLEMENTS CALL CUSTOMER SUPPORT TO ORDER Instructor's Guide, ISBN: 0-8273-7416-X ... Read more

Book Description

A revision of McGraw-Hill's leading calculus text for the 3-semester sequence taken primarily by math, engineering, and science majors. The revision is substantial and has been influenced by students, instructors in physics, engineering, and mathematics, and participants in the national debate on the future of calculus. Revision focused on these key areas: Upgrading graphics and design, expanding range of problem sets, increasing motivation, strengthening multi-variable chapters, and building a stronger support package. ... Read more

Reviews (10)

Brilliant method
This book is literally the best basic calculus text you can possibly get. Anyone wanting to start learning calculus NOW should get this. No real previous mathematical knowledge is necessary. There are several appendices on algebra and series etc. The book discusses trigonometry, so you can learn the book practically without knowing a thing. The "feeling" of the book is inexplicable. Reading this book really gave me an true understanding of basic calculus. Excellent for people like me who need proofs (especially visual ones) have a solid grasp on concepts. If your only goal in learning calculus is to do well on examinations, this book is definately not for you. You should get "Calculus for Dummies" or something like that. The great thing about the text is that it appeals to almost everyone. If there is a certain chapter you don't care for or doesn't matter to you, for example on methods of graphing, you can just skip it, and it will not do any harm. Highly intelligently organized. If you want some help in you physics class on basic vector algebra, just turn to chaper 18 and just read! This book is full of applications, which is great. It also has several historical notes. The colors make the book very engaging to read. Unless your colorblind, this will help engage your interest. Very adequate spacings on the paper as to keep you clearheaded and focused. The drawings rival those of Picasso. They show calculus to be a LIVELY subject. The examples (inside the chapters) are very helpfull. Stein offers several suggestions on how to solve certain problems. Its a shame; this book does not attract the amount of attention it deserves. I did get stuck a couple times; but that is inevitable. Definately get this one: its a gem to have. I can understand how some people would hate this book; its not very concise. That should not be a hindrance. If you feel there is no need to read on about a subject, skip some pages. At the end of chapters it all comes together with a summary of the most important concepts. The book prepares you for study of calculus-based sciences such as physics, and for more advanced mathematical topics as well. I worship this book.

One of the best books on calculus I have read
I am reading the translated into spanish edition of the book and have found it very well written, with many examples and exercises, I would say that was written with love for the subject and for the student. I think that the author succeeds in making mathematics very interesting and beautiful for the reader.

excellent for basic calculus ....
This book is the best place to start to learn calculus. It starts from very basic principles and also contains some more advanced stuff like Stokes Theorem etc. Some readers may find that the book contains too many basic trivial explanations, but I see this as a strength. When you start learning calculus, I think it is a good idea to explain even the trivial, to make sure that you have a good understanding of really everything. I am sure that most readers will benifit from this, even those already having some more advanced math knowledge. Yes : even this latter group will appreciate the benifits of this book, they can always skip some explanations but will benifit from the very clear exposition of more advanced concepts like Stokes Theorem etc ... Myself for instance, I like the more rigorous and abstract math like "real mathematical analysis", but when I need to refresh some calculus and geometry techniques, this book is really the best to sharpen my intuition and understanding of calculus.
Another excellent feature of the book : this book should serve as an example for the layout of math books : it contains a lot of spacing (handy to make personnal annotations), contains a lot of examples, and contains a lot of excellent pictures illustrating a concept... Also some nice anecdotes are added to keep the reader interested. I wish all math books were like this.
If math is not your strongest skill and you need to learn some higher calculus this book will be your excellent companion helping you to gain the insight and intuition you need. If you are busy with more advanded and abstract math, this book also has something to offer to you : this book serves as a fallback point for sharpening your mathematical intuition and refreshing some concepts that you might have forgotten.
Small drawbacks are : -Some more advanced concepts (like Stokes Theorem) are very well explained, but others are explained without proof (convergence of series...) or with simplified proofs (for instance limited to two dimensional cases, though excellent again to gain mathematical intuition). Maybe this is acceptable for a calculus course, but may disappoint the reader who is looking for rigour.

Conclusion : perfect book to gain insight in calculus, it suits well on the shelf of everybody busy who needs calculus...

The worst math book ever
If you have to buy it, buy it. Otherwise, avoid it like the plague. I was forced to buy it because this ass was a professor at UC Davis and made all Davis math students buy it. Even the other professors hated it, but we were stuck with it becuase of his tenure.

Hated this book
I have been doing calculus for a very long time and I am appalled by the ugly approach to mathematics by these two authors. Math is so beautiful because it is not phoney and the outcome is useful, exciting, tangible, palpable. But not here. The authors simply love wasting time and energy with useless stuff like proving limits that are instinctively so obvious. What next, prove that 1+1=2? You might get sick of math reading this book, but it's not maths fault, really. ... Read more

Book Description

At the heart of this monograph is the Brunn-Minkowski theory. It can be used to great effect in studying such ideas as volume and surface area and the generalizations of these. In particular the notions of mixed volume and mixed area arise naturally and the fundamental inequalities that are satisfied by mixed volumes are considered in detail. The author presents a comprehensive introduction to convex bodies and gives full proofs for some deeper theorems that have never previously been brought together. Many hints and pointers to connections with other fields are given, and an exhaustive reference list is included. ... Read more

Reviews (6)

Geometrical Illustrations of Fourth Dimension
Living in a world of three dimensional space makes it hard for us to conceive fourth dimension and it gets even harder to visualize the fifth and higher dimension. Superstring theorists predict the existence of 10th and 26th dimensions in universe; hence it seems reasonable for many of us to understand how it would be like to be living in fourth dimension. Thomas Banchoff is one of the leaders in the study of higher dimension using computer graphics; he has illustrated fourth dimension using basic geometrical approach such as slicing the spatial dimension, observing the shadows of structures, comparing the folded and foldout versions of polytops and description of configuration of spaces. This book is useful for someone who appreciates geometry, but for a reader who likes to visualize the fourth dimension he/she may read Clifford Pickover's Surfing through Hyperspace, which does a better job in illustrating fourth dimension.

Concise Well-Written And Beautifully Illustrated Work
Mathematical ideas, when first learned, tend to undergo a curious inner transformation. At the outset, some tangible representation is necessary to effectively latch onto the concept. Thereafter, the symbolic elaboration using the language of mathematics is sufficient to encompass not only that particular figure, but limitless others like it as well. The underlying geometry is still there, but there are simply too many possibilities to illustrate in any amount of time.

The first step of illustrating must be manifest, using ink or chalk or sand or digital pixels. In this way, even the finest geometric illustrations can be considered extremely crude and innacurate in comparison to rigorous mathematical precision. Consider, however, how extraordinarily difficult it would be to grasp trigonometric functions, vector spaces, or even the basic Cartesian coordinate system, without first observing supporting representative illustrations. Even if later forgotten, those initial images are crucial for understanding.

This work provides a wide range of richly color-illustrated examples of the abstract geometric structures dealt with regularly in mathematics and the sciences. It is unique in its quality and affordability, and is supported with excellent prose, briefly describing the developmental history, and frequently how to reconstruct the figures from a sparse handful of assumptions. From an introductory description of dimension, this book then branches into numerous and diverse major topics: scaling, slices, regular polytopes, perspective, coordinate geometry, and non-euclidean geometry. While sparing in its level of mathematical description and precision, it never diverges into a fully artistic exposition on the subjects either. There is a careful balance, to guide the reader into better understanding the particular system under discussion.

Certainly reading this book is merely the first step of a far longer term process. Symbolic computing programs, such as Mathematica, Maple or MatLab, will assist in visualization, as well as in understanding the pragmatic relation between the graphical and set-theoretic descriptions of the figures. Other books will also assist in this. Many of Rucker's works provide further descriptions of certain topics, specifically Geometry Relativity & The Fourth Dimension is admirable in its brevity and profundity. Abbott's classic Flatland is the foundational book on non-technical description of dimensions. The venerable What Is Mathematics? by Courant and Robbins combines illustration and mathematics as well as any work written since. Design science touches on these topics frequently as well, Kappraff's Connections is an extraordinary example of this. Deeper mathematical topics include set theory, algebraic groups, vector analysis, and too many others to list.

However abstract the concepts diagrams and illustrations in this book may seem initially, most if not all have been utilized for practical application in recent times. You may very well be using devices on a daily basis, which have these concepts as a basis for their functionality. Keep this in mind while reveling in what the individual imagination can conjure.

The royal road to geometry!
This book is a jewel! It contains a wide collection of visual geometry. Professor Banchoff is able to link geometry to many aspects of life. It's a treasure trove for anybody teaching geometry at any level. It's a book that can be read at many levels. If you're willing to skip a bit here and there, you can get a very good general idea. But if you want to really understand all the details, it can make for hours of challenging reading. I'm still reading it! :-)

What is a dimension?
A comfortable introduction to modern geometry for the general reader, with emphasis on the concept of the dimension. This reference concludes with an introduction to non-euclidean geometry.

A must reading for any geometry enthusiast!
After reading this book, no doubt Thomas Banchoff is a deeply experienced geometry enthusiast.Unlike many schoolbooks, his book shows the main ideas underlying a multi-faceted geometry with minimal technical complication nonsense, using simple concepts and a bright argumentation, almost without losing insight! He never misses an opportunity to connect geometry to other sciences like algebra, relativity, optics, mechanics and to arts. It is not only the 'Everything you desired to know about the 4th dimensions' but also a bunch of 2D and 3D geometry 'master tricks' as well as a historical narration (including recent discoveries).

Examples: - how to find yourself the polytopes (4D 'polyhedra') with 3D representations - how to easily calculate vertex coordinates of the 5 polyhedra - how to draw a torus on a hypersphère! -.. and many others

No way you could escape this reading with the same vision of geometry! ... Read more

Reviews (1)

A good review of analytic geometry
Advertised as a review for those experienced with the material, this work delivers the goods. Short, concise explanations with many worked examples are followed by a large number of problems, and brief solutions to most of them are found in the back of the book. All are ordinary, with nothing to distinguish them from those found elsewhere. Which is the way it should be when the goal is to recap material that has already been learned.
The primary focus is on the conic section. All types are covered, as well as the manipulations of the second-degree equations needed to identify the form. Curve sketching is included as well as optional chapters on polar forms and parametric equations.
A good example of a book designed for review, it will do the job if that is what you are looking for.

Published in Journal of Recreational Mathematics, reprinted with permission. ... Read more

Book Description

This book presents a unified mathematical treatment of diverse problems in mathematics, physics, computer science and engineering using geometric algebra. This text is a practical resource for professionals, researchers, and practitioners, cyberneticists, computer scientists, engineers, applied physicists and applied mathematicians. Several examples are presented to clarify the importance of geometric algebra in signal and image processing, filtering and neural computer, computer vision, robotics and geometric physics. A useful resource to gain a greater understanding of the potential of geometric algebra for the design and implementation of real time artificial systems. ... Read more

Reviews (4)

not so bad, IMO
I didn't think this book was so bad. There are enough examples & lots & lots of problems ranging from simple computations to actual proofs (like that e is irrational, for example). The explanation is fairly clear also. Off the top of my head, topics for 1st year in this book include functions, continuity, limits, derivatives, applications of the derivative (lots of word problems), Riemann integration, applications of integration (like separable ODEs, volumes), techniques of integration (partial fractions, parts, trig substitution, etc). The stuff I did in 2nd year was vectors, differential geometry, partial derivatives, spherical & cylindrical coordinates, applications of partial derivatives & multiple integrals, line & surface integrals.I didn't think this book was as bad as the other reviewers. If this one isn't good enough though, there's certainly no shortage of other calculus texts to choose from.

Not good
It is a safe to assume that a book isn't good if the author's(') name(s) is/are larger than the title. Many textbooks suffer from this egomania which can be easily cured with an exacto knife: Just cut around the name(s) and peel.

The book suffers from a poor use of the english language with ideas being presented with too many unnecessary words. Images are labelled with three numbers making the layout confusing i.e. "Fig. 14.9.5" This book has a sloppy finish to it and will only waste your time. Integrated Physics and Calculus is a much better buy.

A Former Penney Student's Perspective
If you are expecting to learn calculus on your own, this is probably not the book for you. However, if used by a competent lecturer, this is a very good textbook to learn the basics of calculus from. The problems assigned to each section are EXCELLENT for driving home the concepts presented.

A Student's perspective
I took a math class using this book and it was frustrating. The problems were challenging which was good, but the support offered by the book was poor. If you needed to understand something you couldn't use the book to help you out. All it did was spit out a few equations and inconsistent derivations of them. I found the book very frustrating and if you're looking to learn math skills I'd recommend something else. ... Read more

Book Description

Elliptic equations of critical Sobolev growth have been the target of investigation for decades because they have proved to be of great importance in analysis, geometry, and physics. The equations studied here are of the well-known Yamabe type. They involve Schrödinger operators on the left hand side and a critical nonlinearity on the right hand side.

A significant development in the study of such equations occurred in the 1980s. It was discovered that the sequence splits into a solution of the limit equation--a finite sum of bubbles--and a rest that converges strongly to zero in the Sobolev space consisting of square integrable functions whose gradient is also square integrable. This splitting is known as the integral theory for blow-up. In this book, the authors develop the pointwise theory for blow-up. They introduce new ideas and methods that lead to sharp pointwise estimates. These estimates have important applications when dealing with sharp constant problems (a case where the energy is minimal) and compactness results (a case where the energy is arbitrarily large). The authors carefully and thoroughly describe pointwise behavior when the energy is arbitrary.

Intended to be as self-contained as possible, this accessible book will interest graduate students and researchers in a range of mathematical fields.

Book Description

Surgery theory, the basis for the classification theory of manifolds, is now about forty years old. The sixtieth birthday (on December 14, 1996) of C.T.C. Wall, a leading member of the subject's founding generation, led the editors of this volume to reflect on the extraordinary accomplishments of surgery theory as well as its current enormously varied interactions with algebra, analysis, and geometry.

Workers in many of these areas have often lamented the lack of a single source surveying surgery theory and its applications. Because no one person could write such a survey, the editors asked a variety of experts to report on the areas of current interest. This is the second of two volumes resulting from that collective effort. It will be useful to topologists, to other interested researchers, and to advanced students. The topics covered include current applications of surgery, Wall's finiteness obstruction, algebraic surgery, automorphisms and embeddings of manifolds, surgery theoretic methods for the study of group actions and stratified spaces, metrics of positive scalar curvature, and surgery in dimension four.

In addition to the editors, the contributors are S. Ferry, M. Weiss, B. Williams, T. Goodwillie, J. Klein, S. Weinberger, B. Hughes, S. Stolz, R. Kirby, L. Taylor, and F. Quinn.