Introduction to Analysis (4e)

For one- or two-semester junior or senior level courses in Advanced Calculus, Analysis I, or Real Analysis.

 

This text prepares students for future courses that use analytic ideas, such as real and complex analysis, partial and ordinary differential equations, numerical analysis, fluid mechanics, and differential geometry. This book is designed to challenge advanced students while encouraging and helping weaker students. Offering readability, practicality and flexibility, Wade presents fundamental theorems and ideas from a practical viewpoint, showing students the motivation behind the mathematics and enabling them to construct their own proofs.

Preface

Part I. ONE-DIMENSIONAL THEORY

 

1. The Real Number System

1.1 Introduction

1.2 Ordered field axioms

1.3 Completeness Axiom

1.4 Mathematical Induction

1.5 Inverse functions and images

1.6 Countable and uncountable sets

 

2. Sequences in R

2.1 Limits of sequences

2.2 Limit theorems

2.3 Bolzano-Weierstrass Theorem

2.4 Cauchy sequences

*2.5 Limits supremum and infimum

 

3. Continuity on R

3.1 Two-sided limits

3.2 One-sided limits and limits at infinity

3.3 Continuity

3.4 Uniform continuity

 

4. Differentiability on R

4.1 The derivative

4.2 Differentiability theorems

4.3 The Mean Value Theorem

4.4 Taylor's Theorem and l'Hôpital's Rule

4.5 Inverse function theorems

 

5 Integrability on R

5.1 The Riemann integral

5.2 Riemann sums

5.3 The Fundamental Theorem of Calculus

5.4 Improper Riemann integration

*5.5 Functions of bounded variation

*5.6 Convex functions

 

6. Infinite Series of Real Numbers

6.1 Introduction

6.2 Series with nonnegative terms

6.3 Absolute convergence

6.4 Alternating series

*6.5 Estimation of series

*6.6 Additional tests

 

7. Infinite Series of Functions

7.1 Uniform convergence of sequences

7.2 Uniform convergence of series

7.3 Power series

7.4 Analytic functions

*7.5 Applications

 

Part II. MULTIDIMENSIONAL THEORY

 

8. Euclidean Spaces

8.1 Algebraic structure

8.2 Planes and linear transformations

8.3 Topology of Rⁿ

8.4 Interior, closure, boundary

 

9. Convergence in Rⁿ

9.1 Limits of sequences

9.2 Heine-Borel Theorem

9.3 Limits of functions

9.4 Continuous functions

*9.5 Compact sets

*9.6 Applications

 

10. Metric Spaces

10.1 Introduction

10.2 Limits of functions

10.3 Interior, closure, boundary

10.4 Compact sets

10.5 Connected sets

10.6 Continuous functions

10.7 Stone-Weierstrass Theorem

 

11. Differentiability on Rⁿ

11.1 Partial derivatives and partial integrals

11.2 The definition of differentiability

11.3 Derivatives, differentials, and tangent planes

11.4 The Chain Rule

11.5 The Mean Value Theorem and Taylor's Formula

11.6 The Inverse Function Theorem

*11.7 Optimization

 

12. Integration on Rⁿ

12.1 Jordan regions

12.2 Riemann integration on Jordan regions

12.3 Iterated integrals

12.4 Change of variables

*12.5 Partitions of unity

*12.6 The gamma function and volume

 

13. Fundamental Theorems of Vector Calculus

13.1 Curves

13.2 Oriented curves

13.3 Surfaces

13.4 Oriented surfaces

13.5 Theorems of Green and Gauss

13.6 Stokes's Theorem

 

*14. Fourier Series

*14.1 Introduction

*14.2 Summability of Fourier series

*14.3 Growth of Fourier coefficients

*14.4 Convergence of Fourier series

*14.5 Uniqueness

 

Appendices

A. Algebraic laws

B. Trigonometry

C. Matrices and determinants

D. Quadric surfaces

E. Vector calculus and physics

F. Equivalence relations

 

References

Answers and Hints to Exercises

Subject Index

Symbol Index

 

*Enrichment section

Changes to the Exercises

• Computational exercises have been rewritten so that answers are simpler and easier to obtain.
• Calculus-style exercises at the beginning of the book have been revised to be more conceptual, emphasizing the same ideas, but at a higher level.
• Theoretical exercises of medium difficulty have been added throughout the book.
• New True/False questions in the first six chapters confront common misconceptions that students sometimes acquire at this level.

 Content Updates

• A new section 1.1, Introduction, combines introductory material that was previously scattered over several sections. This section includes two accessible examples about why proof is necessary and why we cannot always trust what we see.
• The number of axioms has been reduced from four to three by introducing the Completeness Axiom first, and using it to prove the Well Ordering Principle and the Principle of Mathematical Induction.
• The material on countable sets and inverse images of sets has been postponed to Chapter 3, making it possible to begin discussing limits of sequences even earlier than before.
• Coverage of Taylor's Formula has been moved from Chapter 7 to Chapter 4 to offer another example of the utility of the Mean Value Theorem.
• The Heine-Borel Theorem now has its own section and includes several exercises designed to give students practice in making a local condition on a compact set into a global one.
• Section 12.1, Jordan regions, has been organized to simplify the presentation and make it easier to teach.

• Flexible presentation, with uniform writing style and notation, covers the material in small sections, allowing instructors to adapt this book to their syllabus.
• The practical focus explains assumptions so that students learn the motivation behind the mathematics and are able to construct their own proofs.
• Early introduction of the fundamental goals of analysis Refers and examines how a limit operation interacts with algebraic operation.
• Optional appendices and enrichment sections enables students to understand the material and allows instructors to tailor their courses.
• An alternate chapter on metric spaces allows instructors to cover either chapter independently without mentioning the other.
• More than 200 worked examples and 600 exercises encourage students to test comprehension of concepts, while using techniques in other contexts.
• Separate coverage of topology and analysis presents purely computational material first, followed by topological material in alternate chapters.
• Rigorous presentation of integers provides shorter presentations while focusing on analysis.
• Reorganized coverage of series separates series of constants and series of functions into separate chapters.
• Consecutive numbering of theorems, definitions and remarks allows students and instructors to find citations easily.

William Wade received his PhD in harmonic analysis from the University of California—Riverside. He has been a professor of the Department of Mathematics at the University of Tennessee for more than forty years. During that time, he has received multiple awards including two Fulbright Scholarships, the Chancellor's Award for Research and Creative Achievements, the Dean's Award for Extraordinary Service, and the National Alumni Association Outstanding Teaching Award.

 

Wade’s research interests include problems of uniqueness, growth and dyadic harmonic analysis, on which he has published numerous papers, two books and given multiple presentations on three continents. His current publication, An Introduction to Analysis,is now in its fourth edition.

 

In his spare time, Wade loves to travel and take photographs to document his trips. He is also musically inclined, and enjoys playing classical music, mainly baroque on the trumpet, recorder, and piano.