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An Introduction to Analysis William Wade 4th GE 9781292357874
TITLE : An Introduction to Analysis William Wade
ISBN13 : 9781292357874
PUBLISHER : PEARSON (2021)
EDITION : 4TH GLOBAL EDITION PAPERBACK
PAGES : 696 PAGES
For one- or two-semester junior or senior level courses in Advanced Calculus, Analysis I, or Real Analysis.
This title is part of the Pearson Modern Classics series.
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.
Table of contents
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. Functions on R
3.1 Two-sided limits
3.2 One-sided limits and limits atinfinity
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'Hopital'sRule
4.5 Inverse function theorems
5 Integrability on R
5.1 The Riemann integral
5.2 Riemann sums
5.3 The Fundamental Theorem ofCalculus
5.4 Improper Riemann integration
*5.5 Functions of boundedvariation
*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 ofsequences
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 lineartransformations
8.3 Topology of Rn
8.4 Interior, closure, and 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 andpartial integrals
11.2 The definition ofdifferentiability
11.3 Derivatives, differentials, andtangent planes
11.4 The Chain Rule
11.5 The Mean Value Theorem andTaylor's Formula
11.6 The Inverse Function Theorem
*11.7 Optimization
12. Integration on Rn
12.1 Jordan regions
12.2 Riemann integration on Jordanregions
12.3 Iterated integrals
12.4 Change of variables
*12.5 Partitions of unity
*12.6 The gamma function andvolume
13. Fundamental Theorems of VectorCalculus
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 Fourierseries
*14.3 Growth of Fouriercoefficients
*14.4 Convergence of Fourierseries
*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 Selected Exercises
Subject Index
Notation Index