product iamge

A Friendly Introduction to Analysis

This document was uploaded by one of our users. The uploader already confirmed that they had the permission to publish it. If you are author/publisher or own the copyright of this documents, please report to us by using this DMCA report form.

Download
Search on Amazon

Simply click on the Download Book button.

Yes, Book downloads on Ebookily are 100% Free.

Sometimes the book is free on Amazon As well, so go ahead and hit "Search on Amazon"

For two-semester/three-quarter, first undergraduate courses in Advanced Calculus or Real Analysis. This book is an easy, readable, intimidation-free analysis textbook. Ideas and methods of proof build upon each other and are explained thoroughly. This is the first text to cover both single and multivariable analysis in such a student friendly setting. Features NEW - Revised and reorganized content—Hundreds of small improvements enhance the presentation of material throughout the text. Provides students with more thorough treatments of existing material in a clearer, more readable, and student-friendly format. NEW - Added examples and explanations. NEW - Reworded exercises. Further enhances the precision of the instructions, making it easier for students to follow. NEW - Expanded use of geometry and illustrations. Enhances the visual appeal of the text and students' understanding and visualization. NEW - Author website with additional topics. New To This Edition Revised and reorganized content—Hundreds of small improvements enhance the presentation of material throughout the text. Provides students with more thorough treatments of existing material in a clearer, more readable, and student-friendly format. Added examples and explanations. Reworded exercises. Further enhances the precision of the instructions, making it easier for students to follow. Expanded use of geometry and illustrations. Enhances the visual appeal of the text and students' understanding and visualization. Author website with additional topics. The chapter on Fourier Analysis is, for example, available now in this form.

Author(s): Witold A.J. Kosmala
Edition: 2nd
Publisher: Pearson
Year: 2004

Language: English
Pages: C, XV, 574

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