BOOK INTRODUCTION TO REAL ANALYSIS


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BOOK INTRODUCTION TO REAL ANALYSIS

BOOK INTRODUCTION TO REAL ANALYSIS : 586 pages BY :William F. Trench 

This book has been judged to meet the evaluation criteria set by the Editorial Board of the American Institute of Mathematics in connection with the Institute’s Open Textbook Initiative.

Contents

Chapter 1 : The Real Numbers

.1 The Real Number System .2 Mathematical Induction  .3 The Real Line

Chapter 2 Differential Calculus of Functions of One Variable

.1 Functions and Limits  .2 Continuity  .3 Differentiable Functions of One Variable

.4 L’Hospital’s Rule  .5 Taylor’s Theorem

Chapter 3 Integral Calculus of Functions of One Variable

.1 Definition of the Integral  .2 Existence of the Integral  .3 Properties of the Integral

.4 Improper Integrals  .5 A More Advanced Look at the Existence of the Proper Riemann                                    Integral

Chapter 4 Infinite Sequences and Series

.1 Sequences of Real Numbers  .2 Earlier Topics Revisited With Sequences .3 Infinite Series of                         Constants  .4 Sequences and Series of Functions  .5 Power Series 257

Chapter 5 Real-Valued Functions of Several Variables

.1 Structure of Rn  .2 Continuous Real-Valued Function of n Variables  .3 Partial Derivatives                        the Differential  .4 The Chain Rule and Taylor’s Theorem

Chapter 6 Vector-Valued Functions of Several Variables

.1 Linear Transformations and Matrices .2 Continuity and Differentiability of Transformations                   .3 The Inverse Function Theorem  .4. The Implicit Function Theorem

Chapter 7 Integrals of Functions of Several Variables

.1 Definition and Existence of the Multiple Integral .2 Iterated Integrals and Multiple Integrals                    .3 Change of Variables in Multiple Integrals

Chapter 8 Metric Spaces

.1 Introduction to Metric Spaces  .2 Compact Sets in a Metric Space .3 Continuous Functions on                  Metric Spaces

 Answers to Selected Exercises

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