Undergraduate Texts in Mathematics

By: George Pedrick

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Hardcover | 29 March 1994

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308 Pages

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24.77 x 16.51 x 1.91

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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.

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Preface
Notations and Conventions
Introduction
Background: Number Systems	p. 1
Counting: The Natural Numbers	p. 1
Measurement: The Rational Numbers	p. 13
The Axioms of Ordered Fields	p. 19
Decimal Representation. Irrationals	p. 25
Approximation: The Real Numbers	p. 37
Least Upper Bound	p. 38
Completeness. Nested Intervals	p. 40
Bounded Monotonic Sequences	p. 42
Cauchy Sequences	p. 47
The Real Number System	p. 50
Countability	p. 52
Appendix. The Fundamental Theorem of Algebra. Complex Numbers	p. 55
The Extreme-Value Problem	p. 64
Continuity, Compactness, and the Extreme-Value Theorem	p. 65
Continuity of Rational Functions. Limits of Sequences	p. 72
Appendix. Completion of the Proof of the Fundamental Theorem of Algebra	p. 76
Sequences and Series of Reals. The Number e	p. 78
Sets of Reals. Limits of Functions	p. 90
Continuous Functions	p. 97
Implicit Functions. [actual symbol not reproducible] The Intermediate-Value Theorem	p. 97
Inverse Functions. [actual symbol not reproducible]	p. 101
Continuous Extension. Uniform Continuity. The Exponential and Logarithm	p. 104
The Elementary Functions	p. 109
Uniformity. The Heine-Borel Theorem	p. 112
Uniform Convergence. A Nowhere Differentiable Continuous Function	p. 118
The Weierstrass Approximation Theorem	p. 124
Summary of the Main Properties of Continuous Functions	p. 128
Appendix. A Space-Filling Continuous Curve	p. 128
Differentiation	p. 133
Differential and Derivative. Tangent Line	p. 135
The Foundations of Differentiation	p. 140
Curve Sketching. The Mean-Value Theorem	p. 146
Taylor's Theorem	p. 154
Functions Defined Implicitly	p. 160
Integration	p. 169
Definitions. Darboux Theorem	p. 171
Foundations of Integral Calculus. The Fundamental Theorem of Calculus	p. 179
The Nature of Integrability. Lebesgue's Theorem	p. 189
Improper Integral	p. 197
Arclength. Bounded Variation	p. 206
A Word About the Stieltjes Integral and Measure Theory	p. 215
Infinite Series	p. 218
The Vibrating String	p. 219
Convergence: General Considerations	p. 224
Convergence: Series of Positive Terms	p. 230
Computation with Series	p. 237
Power Series	p. 244
Fourier Series	p. 254
Bibliography	p. 266
Index	p. 269
Table of Contents provided by Blackwell. All Rights Reserved.

Undergraduate Texts in Mathematics

At a Glance

Hardcover

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