Introductory Analysis - 2nd Edition - ISBN: 9780122676550, 9780080918273

Introductory Analysis

2nd Edition

The Theory of Calculus

Authors: John Fridy
eBook ISBN: 9780080918273
Hardcover ISBN: 9780122676550
Imprint: Academic Press
Published Date: 10th February 2000
Page Count: 335
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Introductory Analysis, Second Edition, is intended for the standard course on calculus limit theories that is taken after a problem solving first course in calculus (most often by junior/senior mathematics majors). Topics studied include sequences, function limits, derivatives, integrals, series, metric spaces, and calculus in n-dimensional Euclidean space

Key Features

  • Bases most of the various limit concepts on sequential limits, which is done first
  • Defines function limits by first developing the notion of continuity (with a sequential limit characterization)
  • Contains a thorough development of the Riemann integral, improper integrals (including sections on the gamma function and the Laplace transform), and the Stieltjes integral
  • Presents general metric space topology in juxtaposition with Euclidean spaces to ease the transition from the concrete setting to the abstract

New to This Edition

  • Contains new Exercises throughout
  • Provides a simple definition of subsequence
  • Contains more information on function limits and L'Hospital's Rule
  • Provides clearer proofs about rational numbers and the integrals of Riemann and Stieltjes
  • Presents an appendix lists all mathematicians named in the text
  • Gives a glossary of symbols


Researchers, professionals, the general public, and librarians who want to expand or enhance their knowledge of calculus limit theories.

Table of Contents

Introduction: Mathematical Statements and Proofs Types of Mathematical Statements The Structure of Proofs Ordering of the Real Numbers The Order Axiom Least Upper Bounds The Density of the Rational Numbers Sequence Limits Convergent Sequences Algebraic Combinations of Sequences Infinite Limits Subsequences and Limit Points Monotonic Sequences Completeness of the Real Numbers The Bolzano--Weierstrass Theorem Cauchy Sequences The Nested Intervals Theorem The Heine--Borel Covering Theorem Continuous Functions Continuity The Sequential Criterion for Continuity Combinations of Continuous Functions One-Sided Continuity Function Limits The Sequential Criterion for Function Limits Variations of Function Limits Consequences of Continuity The Range of a Continuous Function The Intermediate Value Property Uniform Continuity The Sequential Criterion for Uniform Continuity The Derivative Difference Quotients The Chain Rule The Law of the Mean Cauchy Law of the Mean Taylor's Formula with Remainder L'Hopital's Rule The Riemann Integral Riemann Sums and Integrable Functions Basic Properties The Darboux Criterion for Integrability Integrability of Continuous Functions Products of Integrable Functions The Fundamental Theorem of Calculus Improper Integrals Types of Improper Integrals Integrals over Unbounded Domains Integrals of Unbounded Functions The Gamma Function The Laplace Transform Infinite Series Convergent and Divergent Series Comparison Tests The Cauchy Condensation Test Elementary Tests Delicate Tests Absolute and Conditional Convergence Regrouping and Rearranging Series Multiplication of Series The Riemann--Stieltjes Integral Functions of Bounded Variation The Total Variation Function Riemann--Stieltjes Sums and Integrals Integration by Parts Integrability of Continuous Functions Function Sequences Pointwise Convergence Uniform Convergence Sequences of Continuous Functions Sequences of Integrable Functions Sequences of Differentiable Functions The Weierstrass Approximation Theorem Function Series Power Series Convergence of Power Series Integration and Differentiation of Power Series Taylor Series The Remainder Term Taylor Series of Some Elementary Functions Metric Spaces and Euclidean Spaces Metric Spaces Euclidean n-Space Metric Space Topology Connectedness Point Sequences Completeness of En Dense Subsets of En Continuous Transformations Transformations and Functions Criteria for Continuity The Range of a Continuous Transformation Continuity in En Linear Transformations Differential Calculus in Euclidean Spaces Partrial Derivatives and Directional Derivatives Differentials and the Approximation Property The Chain Rule The Law of the Mean Mixed Partial Derivatives The Implicit Function Theorem Area and Integration in E2 Integration on a Bounded Set Inner and Outer Area Properties of the Double Integral Line Integrals Independence of Path and Exact Differentials Green's Theorem Analogs of Green's Theorem Appendix A Mathematical Induction Appendix B Countable and Uncountable Sets Appendix C Infinite Products Appendix D List of Mathematicians Appendix E Glossary of Symbols Index


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© Academic Press 2000
Academic Press
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About the Author

John Fridy

Affiliations and Expertise

Kent State University, Ohio, U.S.A.

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