A Friendly Introduction to Analysis

Cover

S Title

A Friendly Introduction to Analysis

Copyright

     © 2004, 1999 Pearson Education

     ISBN 0-13-045796-5

Dedication

Contents

Preface

     1 Purpose and Background

     2 Design and Organization

     3 Supplements

     4 How the 2nd Edition Differs From the 1st Edition

     5 Acknowledgments

1  Introduction

     1.1 * Algebra of Sets

          Exercises 1.1

     1.2* Relations and Functions

          Exercises 1.2

     1.3* Mathematical Induction

          Exercises 1.3

     1.4* Proof Techniques

          Exercises 1.4

     1.5* Inverse Functions

          Exercises 1.5

     1.6* Finite and Infinite Sets

          Exercises 1.6

     1.7* Ordered Field and a Real Number System

          Exercises 1.7

     1.8* Some Properties of Real Numbers

          Exercises 1.8

     1,9* Review

     1.10* Projects

          Part 1. Fibonacci Numbers

          Part 2. Lucas Numbers

          Part 3. Mean of Real Numbers

2  Sequences

     2.1 Convergence

          Exercises 2.1

     2.2 Limit Theorems

          Exercises 2.2

     2.3 Infinite Limits

          Exercises 2.3

     2.4 Monotone Sequences

          Exercises 2.4

     2.5 Cauchy Sequences

          Exercises 2.5

     2.6 Subsequences

          Exercises 2.6

     2.7* Review

     2.8* Projects

          Part 1. The Transcendental Number e

          Part 2. Summable Sequences

3  Limits of Functions

     3.1 Limit at Infinity

          Exercises 3.1

     3.2 Limit at a Real Number

          Exercises 3.2

     3.3 Sided Limits

          Exercises 3.3

     3.4* Review

     3.5* Projects

          Part 1. Monotone Functions

          Part 2. Continued Fractions

4  Continuity

     4.1 Continuity of a Function

          Exercises 4.1

     4.2* Discontinuity of a Function

          Exercises 4.2

     4.3 Properties of Continuous Functions

          Exercises 4.3

     4.4 Uniform Continuity

          Exercises 4.4

     4.5* Review

     4.6* Projects

          Part 1. Compact Sets

          Part 2. Multiplicative, Subadditive, and Additive Functions

5  Differentiation

     5.1 Derivative of a Function

          Exercises 5.1

     5.2 Properties of Differentiable Functions

          Exercises 5.2

     5.3 Mean Value Theorems

          Exercises 5.3

     5.4 Higher-Order Derivatives

          Exercises 5.4

     5.5 * L'Hopital's Rule

          Exercises 5.5

     5.6 * Review

     5.7 * Projects

          Part 1. Approximation of Derivatives

          Part 2. Lipschitz Condition

          Part 3. Functions of Bounded Variation

          Part 4. Absolutely Continuous Functions

          Part 5. Convex Functions

6  Integration

     6.1 Riemann Integral

          Exercises 6.1

     6.2 Integrable Functions

          Exercises 6.2

     6.3 Properties of the Riemann Integral

          Exercises 6.3

     6.4 Integration in Relation to Differentiation

          Exercises 6.4

     6.5 Improper Integral

          Exercises 6.5

     6.6 * Special Functions

          Exercises 6.6

     6.7 * Review

     6.8 * Projects

          Part 1. Wallis 's Formula

          Part 2. Euler's Summation Formula

          Part 3. Laplace Transforms

          Part 4. Inverse Laplace Transforms

7  Infinite Series

     7.1 Convergence

          Exercises 7.1

     7.2 Tests for Convergence

          Exercises 7.2

     7.3 Ratio and Root Tests

          Exercises 7.3

     7.4 Absolute and Conditional Convergence

          Exercises 7.4

     7.5* Review

     7.6 * Projects

          Part 1. Summation by Parts

          Part 2. Multiplication of Series

          Part 3. Infinite Products

          Part 4. Cantor Set

8  Sequences and Series of Functions

     8.1 Pointwise Convergence

          Exercises 8.1

     8.2 Uniform Convergence

          Exercises 8.2

     8.3 Properties of Uniform Convergence

          Exercises 8.3

     8.4 Pointwise and Uniform Convergence of Series

          Exercises 8.4

     8.5 Power Series

          Exercises 8.5

     8.6 Taylor Series

          Exercises 8.6

     8.7 * Review

     8.8 * Projects

          Part 1. Limit Superior

          Part 2. Irrationality of e

          Part 3. An Everywhere Continuous but Nowhere Differentiable Function

          Part 4. Equicontinuity

9  Vector Calculus

     9.1* Cartesian Coordinates in R^3

          Exercises 9.1

     9.2* Vectors in R^3

          Exercises 9.2

     9.3* Dot Product and Cross Product

          Exercises 9.3

     9.4 Parametric Equations

          Exercises 9.4

     9.5* Lines and Planes in R^3

          Exercises 9.5

     9.6 Vector-Valued Functions

          Exercises 9.6

     9.7 Arc Length

          Exercises 9.7

     9.8 * Review

     9.9 * Projects

          Part 1. Inner Product

          Part 2. Polar Coordinates

          Part 3. Cantor Function

10  Functions of two Variables

     10.1 Basic Topology

          Exercises 10.1

     10.2 Limits and Continuity

          Exercises 10.2

     10.3 Partial Derivatives

          Exercises 10.3

     10.4 Differentiation

          Exercises 10.4

     10.5 Directional Derivative

          Exercises 10.5

     10.6 Chain Rule

          Exercises 10.6

     10.7* Review

     10.8* Projects

          Part 1. Operator Method for Solving Differential Equations

          Part 2. Separable and Homogeneous First-Order Differential Equations

11  Multiple Integration

     11.1 Double Integral

          Exercises 11.1

     11.2 Iterated Integrals

          Exercises 11.2

     11.3 Integrals Over General Regions

          Exercises 11.3

     11.4 Line Integrals

          Exercises 11.4

     11.5 Vector Fields and Work Integrals

          Exercises 11.5

     11.6 Gradient Vector Field

          Exercises 11.6

     11.7 Green's Theorem

          Exercises 11.7

     11.8 * Stokes's and Gauss's Theorems

          Exercises 11.8

     11.9* Review

     11.10* Projects

          Part 1. Change of Variables for Double Integrals

          Part 2. Exact Equations

Hints and Solutions to Selected Exercises

Greek Alphabet

Index of Symbols

Index