Undergraduate texts in mathematics

This book presents a concise unified view of mathematics and its historical development. It is aimed at senior undergraduates - or any other mathematicians - who have mastered the basic topics but wish to gain a better grasp of mathematics as a whole. Reasons for the emergence of the main fields of modern mathematics are identified, and connections between them are explained, by tracing the course of a few mathematical themes from ancient times down to the 20th century. The emphasis is on history as a method for unifying and motivating mathematics, rather than as an end in itself, and there is more mathematical detail than in other general histories. No historical expertise is assumed, and classical mathematics is rephrased in modern terms whenever it seems original sources, and readers wishing to explore the classics for themselves will find it a useful guide. An advantage of the unified approach is that it ties up loose ends and fills gaps in the standard undergraduate curriculum. Thus readers can expect to add to their mathematical knowledge as well as gaining a new perspective on what they already know.

x, 192 p. : 24 cm

This is a reprint of "A First Course in Calculus," which has gone through five editions since the early sixties. It covers all the topics traditionally taught in the first-year calculus sequence in a brief and elementary fashion. As sociological and educational conditions have evolved in various ways over the past four decades, it has been found worthwhile to make the original edition available again. The audience consists of those taking the first calculus course, in high school or college. The approach is the one which was successful decades ago, involving clarity, and adjusted to a time when the students'background was not as substantial as it might be. We are now back to those times, so it's time to start over again. There are no epsilon-deltas, but this does not imply that the book is not rigorous. Lang learned this attitude from Emil Artin, around 1950.

"This is a new, revised, edition of this widely known text. All of the basic topics in calculus of several variables are covered, including vectors, curves, functions of several variables, gradient, tangent plane, maxima and minima, potential functions, curve integrals, Green's theorem, multiple integrals, surface integrals, Stokes' theorem, and the inverse mapping theorem and its consequences. The presentation is self-contained, assuming only a knowledge of basic calculus in one variable. Many completely worked-out problems have been included."--Back cover.

The purpose of this book is to show what mathematics is about, how it is done, and what it is good for. The relaxed and informal presentation conveys the joy of mathematical discovery and insight and makes it clear that mathematics can be an exciting and engrossing activity. Frequent questions lead the reader to see mathematics as an accessible world of thought, where understanding can turn opaque formulae into beautiful and meaningful ideas. The text presents eight topics that serve to illustrate the unity of mathematical thought as well as the diversity of mathematical ideas. Drawn from both "pure" and "applied" mathematics, they include: spirals in nature and in mathematics; the modern topic of fractals and the ancient topic of Fibonacci numbers; Pascal's Triangle and paper folding -- two topics where geometry, number theory, and algebra meet and interact; modular arithmetic and the arithmetic of the infinite. The final chapter presents some ideas about how mathematics should be done, and hence, how it should be taught; these ideas are referred to throughout the text, whenever mathematical strategy and technique are at issue. Presenting many recent discoveries that lead to interesting open questions, the book can serve as the main text in courses dealing with contemporary mathematical topics (for mathematics students or for prospective or in-service mathematics teachers) or as enrichment for other courses. It can also be read with pleasure on its own by anyone interested in the intellectually intriguing aspects of mathematics.

This book, the third of a three-volume work, is the outgrowth of the authors' experience teaching calculus at Berkeley. It is concerned with multivariable calculus, and begins with the necessary material from analytical geometry. It goes on to cover partial differention, the gradient and its applications, multiple integration, and the theorems of Green, Gauss and Stokes. Throughout the book, the authors motivate the study of calculus using its applications. Many solved problems are included, and extensive exercises are given at the end of each section. In addition, a separate student guide has been prepared.

"This unique book is a guided tour through number theory. While most introductions to number theory provide a systematic and exhaustive treatment of the subject, the authors have chosen instead to illustrate the many varied subjects by associating recent discoveries, interesting methods, and unsolved problems. In particular, we read about combinatorial problems in number theory, a branch of mathematics cofounded and popularized by Paul Erdos. Janos Suranyi's vast teaching experience successfully complements Paul Erdos's ability to initiate new directions of research by suggesting new problems and approaches. This book will surely arouse the interest of the student and the teacher alike."--Jacket.