A First Course in Analysis: (Cambridge Mathematical Textbooks)

This rigorous textbook is intended for a year-long analysis or advanced calculus course for advanced undergraduate or beginning graduate students. Starting with detailed, slow-paced proofs that allow students to acquire facility in reading and writing proofs, it clearly and concisely explains the basics of differentiation and integration of functions of one and several variables, and covers the theorems of Green, Gauss, and Stokes. Minimal prerequisites are assumed, and relevant linear algebra topics are reviewed right before they are needed, making the material accessible to students from diverse backgrounds. Abstract topics are preceded by concrete examples to facilitate understanding, for example, before introducing differential forms, the text examines low-dimensional examples. The meaning and importance of results are thoroughly discussed, and numerous exercises of varying difficulty give students ample opportunity to test and improve their knowledge of this difficult yet vital subject.

Table of Contents:
1. The real numbers; 2. Differentiation; 3. Integration; 4. Sequences of functions; 5. Metric and Euclidean spaces; 6. Differentiation in higher dimensions; 7. Integration in higher dimensions; 8. Curves and surfaces; 9. Differential forms.

About the Author :
John B. Conway is Professor Emeritus of Mathematics at George Washington University, Washington DC. He is the author of eleven books, including Mathematical Connections: A Capstone Course (2010), A Course in Functional Analysis (2010), and the two-volume Functions of One Complex Variable (1995).

Review :
'This is an excellent text for a first course in analysis in one and several variables for students who know some linear algebra. The book starts with the real numbers, does differentiation and integration first in one variable, then in several, and finally covers differential forms and Stokes' theorem. The style is friendly and conversational, and hews to the principal of going from the specific to the general, making it a pleasure to read.' John McCarthy, Washington University, St. Louis 'Conway's previous texts are all considered classics. A First Course in Analysis is destined to be another. It is written in the same friendly, yet rigorous, style that his readers know and love. Instructors seeking the breadth and depth of Rudin, but in a less austere and more accessible form, have found their book.' Stephan Ramon Garcia, Pomona College, California 'This is a beautiful yet practical introduction to rigorous analysis at the senior undergraduate level, written by a master expositor. Conway understands how students learn, from the particular to the general, and this informs every aspect of his text. Highly recommended.' Douglas Lind, University of Washington 'A First Course in Analysis charts a lively path through a perennially tough subject. Conway writes as if he's coaching his reader, leavening the technicalities with advice on how to think about them, and with anecdotes about the subject's heroes. His enjoyment of the material shines through on page after page.' Bruce Solomon, Indiana University, Bloomington 'This year-long undergraduate book carefully covers real analysis 'from sets to Stokes' and is done in a friendly style by an experienced teacher and masterful expositor. There are plenty of examples, exercises, and historical vignettes that both give the student the opportunity to gain technical mastery of the material and to whet their appetites for further study.' William T. Ross, University of Richmond 'A First Course in Analysis is a beautifully written and very accessible treatment of a subject that every math major is required to learn. It will join Conway's other textbooks as a classic in Advanced Calculus. Those who teach and learn analysis through Conway's book will appreciate his cheerful and easy-to-understand style.' Wing Suet Li, Georgia Institute of Technology 'The book has a good selection of exercises, most not very difficult; there are no answers or hints. A very good feature is the inclusion of many short biographies of the mathematicians involved … Bottom line: a well-done 'theory of calculus' text, that is especially useful if you need the theory of multivariate calculus.' Allen Stenger, MAA Reviews 'This book is an excellent textbook on analysis from a well-known author. It is addressed to undergraduate students and will be useful for lecturers as well. The presentation is reader-friendly and a student reading it can feel as if the material is presented by the favorite teacher. The book is well-marked with accents: immediately it is clear if this is an important definition or theorem, or if it is a technical phrase. Most proofs of statements are completely clear and concise. The parts of proofs left as exercises to the reader require a careful study of the material.' Ivan Podvigin, zbMATH