Pure and Applied Undergraduate Texts, Volume: 1
1998; 240 pp; hardcover
ISBN-13: 978-0-8218-4787-9
Introduction to Analysis is designed to bridge the gap between the intuitive calculus usually offered at the undergraduate level and the sophisticated analysis courses the student encounters at the graduate level. In this book the student is given the vocabulary and facts necessary for further study in analysis. The course for which it is designed is usually offered at the junior level, and it is assumed that the student has little or no previous experience with proofs in analysis. A considerable amount of time is spent motivating the theorems and proofs and developing the reader's intuition. Of course, that intuition must be tempered with the realization that rigorous proofs are required for theorems. The topics are quite standard: convergence of sequences, limits of functions, continuity, differentiation, the Riemann integral, infinite series, power series, and convergence of sequences of functions. Many examples are given to illustrate the theory, and exercises at the end of each chapter are keyed to each section. Also, at the end of each section, one finds several Projects. The purpose of a Project is to give the reader a substantial mathematical problem and the necessary guidance to solve that problem. A Project is distinguished from an exercise in that the solution of a Project is a multi-step process requiring assistance for the beginner student.
Undergraduate students interested in learning analysis.
Pure and Applied Undergraduate Texts, Volume: 2
2002; 788 pp; hardcover
ISBN-13: 978-0-8218-4788-6
This book introduces students with diverse backgrounds to various types of mathematical analysis that are commonly needed in scientific computing. The subject of numerical analysis is treated from a mathematical point of view, offering a complete analysis of methods for scientific computing with appropriate motivations and careful proofs.
In an engaging and informal style, the authors demonstrate that many computational procedures and intriguing questions of computer science arise from theorems and proofs. Algorithms are presented in pseudocode, so that students can immediately write computer programs in standard languages or use interactive mathematical software packages.
This book occasionally touches upon more advanced topics that are not usually contained in standard textbooks at this level.
An instructor's manual with complete solutions to all the problems as well as supplementary material is available to teachers using the book as the text for the class. To receive it, send e-mail to textbooks@ams.org.
Undergraduate and graduate students interested in applied mathematics and scientific computing.
Pure and Applied Undergraduate Texts, Volume: 3
2004; 424 pp; hardcover
ISBN-13: 978-0-8218-4789-3
As the title indicates, this book is intended for courses aimed at bridging the gap between lower-level mathematics and advanced mathematics. The text provides a careful introduction to techniques for writing proofs and a logical development of topics based on intuitive understanding of concepts. The authors utilize a clear writing style and a wealth of examples to develop an understanding of discrete mathematics and critical thinking skills. While including many traditional topics, the text offers innovative material throughout. Surprising results are used to motivate the reader. The last three chapters address topics such as continued fractions, infinite arithmetic, and the interplay among Fibonacci numbers, Pascal's triangle, and the golden ratio, and may be used for independent reading assignments. The treatment of sequences may be used to introduce epsilon-delta proofs. The selection of topics provides flexibility for the instructor in a course designed to spark the interest of students through exciting material while preparing them for subsequent proof-based courses.
The book includes a large number of problems of varying difficulty. A student manual with solutions to selected problems is available. For more information regarding the student manual, please contact AMS Member and Customer Services at cust-serv@ams.org.
An instructor's manual with complete solutions to all the problems as well as supplementary material is available to teachers using the book as the text for the class. To receive it, send e-mail to textbooks@ams.org.
Undergraduate students interested in becoming mathematicians.
Sets and logic
Proofs
Number theory
Combinatorics
Relations
Functions and cardinality
Graph theory
Sequences
Fibonacci numbers and Pascal's triangle
Continued fractions
Answers or hints for selected exercises
Bibliography
Index
Pure and Applied Undergraduate Texts, Volume: 4
1992; 433 pp; hardcover
ISBN-13: 978-0-8218-4790-9
This book presents the theory and applications of Fourier series and integrals, eigenfunction expansions, and related topics, on a level suitable for advanced undergraduates. It includes material on Bessel functions, orthogonal polynomials, and Laplace transforms, and it concludes with chapters on generalized functions and Green's functions for ordinary and partial differential equations. The book deals almost exclusively with aspects of these subjects that are useful in physics and engineering, and includes a wide variety of applications. On the theoretical side, it uses ideas from modern analysis to develop the concepts and reasoning behind the techniques without getting bogged down in the technicalities of rigorous proofs.
Undergraduate and graduate students interested in studying the Fourier transform
Pure and Applied Undergraduate Texts, Volume: 5
2006; 590 pp; hardcover
ISBN-13: 978-0-8218-4791-6
Advanced Calculus is intended as a text for courses that furnish the backbone of the student's undergraduate education in mathematical analysis. The goal is to rigorously present the fundamental concepts within the context of illuminating examples and stimulating exercises. This book is self-contained and starts with the creation of basic tools using the completeness axiom. The continuity, differentiability, integrability, and power series representation properties of functions of a single variable are established. The next few chapters describe the topological and metric properties of Euclidean space. These are the basis of a rigorous treatment of differential calculus (including the Implicit Function Theorem and Lagrange Multipliers) for mappings between Euclidean spaces and integration for functions of several real variables.
Special attention has been paid to the motivation for proofs. Selected topics, such as the Picard Existence Theorem for differential equations, have been included in such a way that selections may be made while preserving a fluid presentation of the essential material.
Supplemented with numerous exercises, Advanced Calculus is a perfect book for undergraduate students of analysis.
An instructor's manual with complete solutions to all the problems as well as supplementary material is available to teachers using the book as the text for the class. To receive it, send e-mail to textbooks@ams.org.
Undergraduate students interested in teaching and learning undergraduate analysis.