Introduction to Analysis, An, 4th edition
Title overview
For 1- or 2-semester junior or senior level courses in Advanced Calculus, Analysis I, or Real Analysis.
A modern classic
An Introduction to Analysis 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. The 4th Edition 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.
This title is part of the Pearson Modern Classics series. Pearson Modern Classics are acclaimed titles at a value price.
Hallmark features of this title
- 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.
New and updated features of this title
- 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.
- 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.
Table of contents
Preface
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
Author bios
About our author
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 40 years. During that time, he has received multiple awards including 2 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 4th 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.