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.
|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.
Background Number Systems.- §1. Counting: The Natural Numbers.- §2. Measurement: The Rational Numbers.- The Axioms of Ordered Fields.- §3. Decimal Representation. Irrationals.- I ANALYSIS.- 1 Approximation: The Real Numbers.- §1. Least Upper Bound.- §2. Completeness. Nested Intervals.- §3. Bounded Monotonic Sequences.- §4. Cauchy Sequences.- §5. The Real Number System.- §6. Countability.- Appendix. The Fundamental Theorem of Algebra. Complex Numbers.- 2 The Extreme-Value Problem.- §1. Continuity, Compactness, and the Extreme-Value Theorem.- §2. Continuity of Rational Functions. Limits of Sequences.- Appendix. Completion of the Proof of the Fundamental Theorem of Algebra.- §3. Sequences and Series of Reals. The Number e.- §4. Sets of Reals. Limits of Functions.- 3 Continuous Functions.- §1. Implicit Functions,$$sqrt[n]{x}$$The Intermediate-Value Theorem.- §2. Inverse Functions. xr for r ? ?.- §3. Continuous Extension. Uniform Continuity. The Exponential and Logarithm.- §4. The Elementary Functions.- §5. Uniformity. The Heine-Borel Theorem.- §6. Uniform Convergence. A Nowhere Differentiable Continuous function.- §7. The Weierstrass Approximation Theorem.- Summary of the Main Properties of Continuous Functions.- Appendix. A Space-Filling Continuous Curve.- II FOUNDATIONS OF CALCULUS.- 4 Differentiation.- §1. Differential and Derivative. Tangent Line.- §2. The Foundations of Differentiation.- §3. Curve Sketching. The Mean-Value Theorem.- §4. Taylor´s Theorem.- §5. Functions Defined Implicitly.- 5 Integration.- §1. Definitions. Darboux Theorem.- §2. Foundations of Integral Calculus. The Fundamental Theorem of Calculus.- §3. The Nature of Integrability. Lebesgue´s Theorem.- §4. Improper Integral.- §5. Arclength. Bounded Variation.- A Word About the Stieltjes Integral and Measure Theory.- 6 Infinite Series.- §1. The Vibrating String.- §2. Convergence: General Considerations.- §3. Convergence: Series of Positive Terms.- §4. Computation with Series.- §5. Power Series.- §6. Fourier Series.
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.
Background Number Systems.-
1. Counting: The Natural Numbers.-
2. Measurement: The Rational Numbers.- The Axioms of Ordered Fields.-
3. Decimal Representation. Irrationals.- I ANALYSIS.- 1 Approximation: The Real Numbers.- 2 The Extreme-Value Problem.- 3 Continuous Functions.- II FOUNDATIONS OF CALCULUS.- 4 Differentiation.- 5 Integration.- 6 Infinite Series.
Über den Autor
Inhaltsverzeichnis
Background Number Systems.- §1. Counting: The Natural Numbers.- §2. Measurement: The Rational Numbers.- The Axioms of Ordered Fields.- §3. Decimal Representation. Irrationals.- I ANALYSIS.- 1 Approximation: The Real Numbers.- §1. Least Upper Bound.- §2. Completeness. Nested Intervals.- §3. Bounded Monotonic Sequences.- §4. Cauchy Sequences.- §5. The Real Number System.- §6. Countability.- Appendix. The Fundamental Theorem of Algebra. Complex Numbers.- 2 The Extreme-Value Problem.- §1. Continuity, Compactness, and the Extreme-Value Theorem.- §2. Continuity of Rational Functions. Limits of Sequences.- Appendix. Completion of the Proof of the Fundamental Theorem of Algebra.- §3. Sequences and Series of Reals. The Number e.- §4. Sets of Reals. Limits of Functions.- 3 Continuous Functions.- §1. Implicit Functions,$$sqrt[n]{x}$$The Intermediate-Value Theorem.- §2. Inverse Functions. xr for r ? ?.- §3. Continuous Extension. Uniform Continuity. The Exponential and Logarithm.- §4. The Elementary Functions.- §5. Uniformity. The Heine-Borel Theorem.- §6. Uniform Convergence. A Nowhere Differentiable Continuous function.- §7. The Weierstrass Approximation Theorem.- Summary of the Main Properties of Continuous Functions.- Appendix. A Space-Filling Continuous Curve.- II FOUNDATIONS OF CALCULUS.- 4 Differentiation.- §1. Differential and Derivative. Tangent Line.- §2. The Foundations of Differentiation.- §3. Curve Sketching. The Mean-Value Theorem.- §4. Taylor¿s Theorem.- §5. Functions Defined Implicitly.- 5 Integration.- §1. Definitions. Darboux Theorem.- §2. Foundations of Integral Calculus. The Fundamental Theorem of Calculus.- §3. The Nature of Integrability. Lebesgue¿s Theorem.- §4. Improper Integral.- §5. Arclength. Bounded Variation.- A Word About the Stieltjes Integral and Measure Theory.- 6 Infinite Series.- §1. The Vibrating String.- §2. Convergence: General Considerations.- §3. Convergence: Series of Positive Terms.- §4. Computation with Series.- §5. Power Series.- §6. Fourier Series.
Klappentext
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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