Michael Spivak

Calculus, 3rd Edition

Publish or Perish Inc, Houston, Texas
Hardback (ISBN-13: 9780521867443 | ISBN-10: 0521867444)
April 2006

Courses: Analysis, Mathematical Analysis, Introduction to Real Analysis, Calculus II, Advanced Calculus, Calculus
Spivak's celebrated textbook is widely held as one of the finest introductions to mathematical analysis. His aim is to present calculus as the first real encounter with mathematics: it is the place to learn how logical reasoning combined with fundamental concepts can be developed into a rigorous mathematical theory rather than a bunch of tools and techniques learned by rote. Since analysis is a subject students traditionally find difficult to grasp, Spivak provides leisurely explanations, a profusion of examples, a wide range of exercises and plenty of illustrations in an easy-going approach that enlightens difficult concepts and rewards effort. Calculus will continue to be regarded as a modern classic, ideal for honours students and mathematics majors, who seek an alternative to doorstop textbooks on calculus, and the more formidable introductions to real analysis.

*One of the most celebrated texts of its type now readily available outside of the US: combines the rigor of more formidable books with the leisurely explanations, profusion of examples, exercises and illustrations associated with ‘doorstops’

*Ideal for students; clear, crisp explanations of what analysis and mathematics are really about

*Full range of exercises, from the straightforward to the challenging that deepen understanding; solutions available in book form via http://www.mathpop.com/bookhtms/cal.htm

Contents

Preface; Part I. Prologue: 1. Basic properties of mumbers; 2. Numbers of various sorts; Part II. Foundations: 3. Functions; 4. Graphs; 5. Limits; 6. Continuous functions; 7. Three hard theorems; 8. Least upper bounds; Part III. Derivatives and Integrals: 9. Derivatives; 10. Differentiation; 11. Significance of the derivative; 12. Inverse functions; 13. Integrals; 14. The fundamental theorem of calculus; 15. The trigonometric functions; 16. Pi is irrational; 17. Planetary motion; 18. The logarithm and exponential functions; 19. Integration in elementary terms; Part IV. Infinite Sequences and Infinite Series: 20. Approximation by polynomial functions; 21. e is transcendental; 22. Infinite sequences; 23. Infinite series; 24. Uniform convergence and power series; 25. Complex numbers; 26. Complex functions; 27. Complex power series; Part V. Epilogue: 28. Fields; 29. Construction of the real numbers; 30. Uniqueness of the real numbers; Suggested reading; Answers (to selected problems); Glossary of symbols; Index.

Ivan Niven
University of Oregon

Maxima and Minima Without Calculus

Series: Dolciani Mathematical Expositions (No. 6)
Hardback (ISBN-13: 9780883853061 | ISBN-10: 088385306X)

The purpose of this book is to put together in one place the basic elementary techniques for solving problems in maxima and minima other than the methods of calculus and linear programming. The emphasis is not on the individual problems, but on methods that solve large classes of problems. The many chapters of the book can be read independently, without references to what precedes or follows. Besides the many problems solved in the book, others are left to the reader to solve, with sketches of solutions given in the later pages.

* A new printing of this MAA classic first published in 1981

* Each chapter can be read independently of the rest

* Many exercises included with sketch solutions provided at the end of the book

Contents

Preface; 1. Background material; 2. Simple algebraic results; 3. Elementary geometric questions; 4. Isoperimetric results; 5. Basic trigonometric inequalities; 6. Polygons inscribed and circumscribed; 7. Ellipses; 8. The bees and their hexagons; 9. Further geometric results; 10. Applied and miscellaneous problems; 11. Euclidean three-space; 12. Isoperimetric results not assuming existence; Postscript on calculus; Solutions of problems; References; Index.

Reviews

‘A book superb in its exposition and teeming with a variety of elegant mathematical results... All instructors in calculus courses should keep this book within easy reach and administer generous doses of its contents to students who are seized by a fit of differentiation frenzy when confronted with any kind of extremum problem.’ Mathematical Reviews

‘As befits a volume in the Dolciani series, the expository style is above reproach: it is particularly impressive that the author manages, without frightening his readers, to be honest about the difference between a proof which assumes the existence of an extremal solution and one which does not. This is a book which deserves to be read very widely.’ Times Higher Education Supplement

‘… An endless source of fascinating problems which appear at first thought to be elementary, but usually turn out to be unresponsive to normal methods, with elegant and easily understandable solutions by the methods given in the text.’ The Mathematical Gazette

Etienne Bezout
Translated from the French by Eric Feron

General Theory of Algebraic Equations

Cloth | January 2006 | ISBN: 0-691-11432-3
362 pp. | 6 x 9

This book provides the first English translation of Bezout's masterpiece, the General Theory of Algebraic Equations. It follows, by almost two hundred years, the English translation of his famous mathematics textbooks. Here, Bezout presents his approach to solving systems of polynomial equations in several variables and in great detail. He introduces the revolutionary notion of the "polynomial multiplier," which greatly simplifies the problem of variable elimination by reducing it to a system of linear equations. The major result presented in this work, now known as "Bezout's theorem," is stated as follows: "The degree of the final equation resulting from an arbitrary number of complete equations containing the same number of unknowns and with arbitrary degrees is equal to the product of the exponents of the degrees of these equations."

The book offers large numbers of results and insights about conditions for polynomials to share a common factor, or to share a common root. It also provides a state-of-the-art analysis of the theories of integration and differentiation of functions in the late eighteenth century, as well as one of the first uses of determinants to solve systems of linear equations. Polynomial multiplier methods have become, today, one of the most promising approaches to solving complex systems of polynomial equations or inequalities, and this translation offers a valuable historic perspective on this active research field.

Etienne Bezout (1730-1783) is credited with the invention of the determinant (named Bezoutian by Sylvester) as well as several key innovations to solve simultaneous polynomial equations in many unknowns. By the time of his death, he was a member of the French Academy of Sciences and the Examiner of the Guards of the Navy and of the Corps of Artillery. Eric Feron Dutton/Ducoffe Professor of Aerospace Engineering at Georgia Institute of Technology, and Visiting Professor of Aerospace Engineering at Massachusetts Institute of Technology, where he is affiliated with the Laboratory for Information and Decision Systems and the Operations Research Center. He is also an Adviser to the French Academy of Technologies. His interests span numerical analysis, optimization, systems analysis, and their applications to aerospace engineering.

Contents

Daniel D. Bonar / Denison University, Ohio
Michael J. Khoury, Jr. / Ohio State University

Real Infinite Series

Series: Classroom Resource Material
Hardback (ISBN-13: 9780883857458 | ISBN-10: 0883857456)

April 2006

This is a widely accessible introductory treatment of infinite series of real numbers, bringing the reader from basic definitions and tests to advanced results. An up-to-date presentation is given, making infinite series accessible, interesting, and useful to a wide audience, including students, teachers, and researchers. Included are elementary and advanced tests for convergence or divergence, the harmonic series, the alternating harmonic series, and closely related results.

One chapter offers 107 concise, crisp, surprising results about infinite series. Another gives problems on infinite series, and solutions, which have appeared on the annual William Lowell Putnam Mathematical Competition. The lighter side of infinite series is treated in the concluding chapter where three puzzles, eighteen visuals, and several fallacious proofs are made available.

Three appendices provide a listing of true or false statements, answers to why the harmonic series is so named, and an extensive list of published works on infinite series.

* An up-to-date presentation is given, making infinite series accessible, interesting, and useful to a wide audience, including students, teachers, and researchers

* As well as the many exercises, puzzles and worked examples the text contains 101 true or false questions concerning infinite series

* 18 visuals (what Martin Gardner calls ‘look-see’ diagrams) provide insight

Contents

Preface; 1. Introduction to infinite series; 2. More sophisticated techniques; 3. The harmonic series and related results; 4. Intriguing results; 5. Infinite series and the Putnam competition; 6. Final diversions; Appendix A. 101 true or false questions; Appendix B. Harmonic series article; Appendix C. References.

Simon Wood / University of Bath, England, UK

Generalized Additive Models: An Introduction with R

Series: Texts in Statistical Science Series Volume: 67
ISBN: 1584884746
Publication Date: 2/27/2006
Number of Pages: 416

Provides a complete resource for the penalized regression spline approach to GAMs, supported by the R package mgcv
Covers linear, generalized linear, generalized additive, and corresponding mixed models within a single volume
Develops skill in the practical applications of the models discussed while imparting a solid understanding of the underlying theory
Includes copious illustrations, worked out examples, and exercises and solutions

Now in widespread use, generalized additive models (GAMs) have evolved into a standard statistical methodology of considerable flexibility. While Hastie and Tibshirani's outstanding 1990 research monograph on GAMs is largely responsible for this, there has been a long-standing need for an accessible introductory treatment of the subject that also emphasizes recent penalized regression spline approaches to GAMs and the mixed model extensions of these models.

Generalized Additive Models: An Introduction with R imparts a thorough understanding of the theory and practical applications of GAMs and related advanced models, enabling informed use of these very flexible tools. The author bases his approach on a framework of penalized regression splines, and builds a well-grounded foundation through motivating chapters on linear and generalized linear models. While firmly focused on the practical aspects of GAMs, discussions include fairly full explanations of the theory underlying the methods. Use of the freely available R software helps explain the theory and illustrates the practicalities of linear, generalized linear, and generalized additive models, as well as their mixed effect extensions.

The treatment is rich with practical examples, and it includes an entire chapter on the analysis of real data sets using R and the author's add-on package mgcv. Each chapter includes exercises, for which complete solutions are provided in an appendix.

Concise, comprehensive, and essentially self-contained, Generalized Additive Models: An Introduction with R prepares readers with the practical skills and the theoretical background needed to use and understand GAMs and to move on to other GAM-related methods and models, such as SS-ANOVA, P-splines, backfitting and Bayesian approaches to smoothing and additive modelling.

Contents