Synopses & Reviews
This text prepares readers for fluency with 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 readers while encouraging and helping readers with weaker skills. Offering readability, practicality and flexibility, Wade presents fundamental theorems and ideas from a practical viewpoint, showing readers the motivation behind the mathematics and enabling them to construct their own proofs.
ONE-DIMENSIONAL THEORY; The Real Number System; Sequences in R; Continuity on R; Differentiability on R; Integrability on R; Infinite Series of Real Numbers; Infinite Series of Functions; MULTIDIMENSIONAL THEORY; Euclidean Spaces; Convergence in R^{n}; Metric Spaces; Differentiability on R^{n}; Integration on R^{n}; Fundamental Theorems of Vector Calculus; Fourier Series
For all readers interested in analysis.
Synopsis
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.
About the 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 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.
Table of Contents
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 R^{n}
8.4 Interior, closure, boundary
9. Convergence in R^{n}
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 R^{n}
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 R^{n}
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