# Introductory Analysis

## The Theory of Calculus

**By**

- John Fridy

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

**Audience**

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

Hardbound, 335 Pages

Published: January 2000

Imprint: Academic Press

ISBN: 978-0-12-267655-0

## 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 *