Synopses & Reviews
"""A First Course in Analysis"" is a new approach to the elements of real analysis for post-calculus students. The text outlines the evolution of number systems, then proceeds to introduce the fundamental ideas of completeness, compactness and continuity via a plausible heuristic investigation of the problem of extreme values. The author then presents the standard properties of continuous functions, along with a continuity-focused development of the elementary functions. The book concludes with chapters on the foundations of calculus: differentiation, integration, and infinite series. Applications and looks-ahead to more advanced analysis are pointed out. This text can be adapted not only to courses of different lengths and emphasis, but also to different modes of classroom presentation. In particular, it complements a strictly deductive presentation, that begins, say, with the ordered field axioms."
"This book is a very enjoyable introduction into Analysis. Considering the treatment of the material it should be emphasized that a very valuable feature is that a lot of interesting questions are raised in such a way that definitions, theorems, and proofs are motivated for the readers. This volume is very useful for students and lectures even for bright high school students interested in the introductory theory of functions. ACTA SCIENTIARUM MATHEMATICARUM"
The first course in Analysis, which follows calculus, along with other courses, such as differential equations and elementary linear algebra, in the curricu lum, presents special pedagogical challenges. There is a change of stress from computational manipulation to "proof. " Indeed, the course can become more a course in Logic than one in Analysis. Many students, caught short by a weak command of the means of mathematical discourse and unsure of what is expected of them, what "the game" is, suffer bouts of a kind of mental paralysis. This text attempts to address these problems in several ways: First, we have attempted to define "the game" as that of "inquiry," by using a form of exposition that begins with a question and proceeds to analyze, ultimately to answer it, bringing in definitions, arguments, conjectures, exam ples, etc., as they arise naturally in the course of a narrative discussion of the question. (The true, historical narrative is too convoluted to serve for first explanations, so no attempt at historical accuracy has been made; our narra tives are completely contrived. ) Second, we have kept the logic informal, especially in the course of preliminary speculative discussions, where common sense and plausibility tempered by mild skepticism-serve to energize the inquiry."
This text on advanced calculus discusses such topics as number systems, the extreme value problem, continuous functions, differentiation, integration and infinite series. The reader will find the focus of attention shifted from the learning and applying of computational techniques to careful reasoning from hypothesis to conclusion. The book is intended both for a terminal course and as preparation for more advanced studies in mathematics, science, engineering and computation.
Includes bibliographical references (p. 266-267) and index.
Table of Contents
- Number Systems.
- Approximation: The Real Numbers.
- The Extreme-Value Problem.
- Continuous Functions.
- Infinite Series.