An Introduction to Analysis (Global Edition)

✍ Scribed by William R. Wade

• Publisher: Pearson
• Year: 2021
• Tongue: English
• Leaves: 696
• Edition: 4
• Category: Library

✦ Synopsis

For one- or two-semester junior orsenior level courses in Advanced Calculus, Analysis I, or Real Analysis. This title is part of the Pearson Modern Classicsseries. This text prepares students for future coursesthat use analytic ideas, such as real and complex analysis, partial andordinary differential equations, numerical analysis, fluid mechanics, anddifferential geometry. This book is designed to challenge advanced studentswhile encouraging and helping weaker students. Offering readability,practicality and flexibility, Wade presents fundamental theorems and ideas froma practical viewpoint, showing students the motivation behind the mathematicsand enabling them to construct their own proofs.

✦ Table of Contents

Cover
Half Title
Title Page
Copyright
Dedication
Contents
Preface
Acknowledgments for the Global Edition
Chapter 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
Chapter 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
Chapter 3. Functions on R
3.1 Two-Sided Limits
3.2 One-Sided Limits and Limits at Infinity
3.3 Continuity
3.4 Uniform Continuity
Chapter 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
Chapter 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
Chapter 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
Chapter 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
Chapter 8. Euclidean Spaces
8.1 Algebraic Structure
8.2 Planes and Linear Transformations
Chapter 9. Convergence in Rn
9.1 Topology of Rn
9.2 Interior, Closure, and Boundary
∗9.3 Compact Sets
9.4 Heine–Borel Theorem
9.5 Limits of Sequences
9.6 Limits of Functions
9.7 Continuous Functions
∗9.8 Applications
Chapter 10. Metric Spaces
10.1 Introduction
10.2 Interior, Closure, and Boundary
10.3 Compact Sets
10.4 Connected Sets
10.5 Limits of Functions
10.6 Continuous Functions
∗10.7 Stone–Weierstrass Theorem
Chapter 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
Chapter 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
Chapter 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
Chapter 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
Appendix A. Algebraic Laws
Appendix B. Trigonometry
Appendix C. Matrices and Determinants
Appendix D. Quadric Surfaces
Appendix E. Vector Calculus and Physics
Appendix F. Equivalence Relations
References
Answers and Hints to Selected Exercises
Subject Index
Notation Index