9780130144096

Introduction to Analysis

by
  • ISBN13:

    9780130144096

  • ISBN10:

    0130144096

  • Edition: 2nd
  • Format: Hardcover
  • Copyright: 1/1/2000
  • Publisher: Pearson College Div
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Summary

Offering readability, practicality and flexibility, Wade presents Fundamental Theorems from a practical viewpoint. Introduces central ideas of analysis in a one-dimensional setting, then covers multidimensional theory. Offers separate coverage of topology and analysis. Numbers theorems, definitions and remarks consecutively. Uniform writing style and notation. Practical focus on analysis. For those interested in learning more about analysis.

Table of Contents

Preface ix
Part I. ONE-DIMENSIONAL THEORY
The Real Number System
1(33)
Ordered field axioms
1(12)
The Well-Ordering Principle
13(5)
The Completeness Axiom
18(6)
Functions, countability, and the algebra of sets
24(10)
Sequences in R
34(23)
Limits of sequences
34(4)
Limit theorems
38(6)
The Bolzano--Weierstrass Theorem
44(4)
Cauchy sequences
48(3)
Limits supremum and infimum
51(6)
Continuity on R
57(27)
Two-sided limits
57(8)
One-sided limits and limits at infinity
65(5)
Continuity
70(9)
Uniform continuity
79(5)
Differentiability on R
84(22)
The derivative
84(7)
Differentiability theorems
91(2)
The Mean Value Theorem
93(8)
Monotone functions and the Inverse Function Theorem
101(5)
Integrability on R
106(46)
The Riemann integral
106(9)
Riemann sums
115(10)
The Fundamental Theorem of Calculus
125(9)
Improper Riemann integration
134(6)
Functions of bounded variation
140(5)
Convex functions
145(7)
Infinite Series of Real Numbers
152(30)
Introduction
152(6)
Series with nonnegative terms
158(5)
Absolute convergence
163(8)
Alternating series
171(4)
Estimation of series
175(4)
Additional tests
179(3)
Infinite Series of Functions
182(41)
Uniform convergence of sequences
182(8)
Uniform convergence of series
190(5)
Power series
195(10)
Analytic functions
205(12)
Applications
217(6)
Part II. MULTIDIMENSIONAL THEORY
Euclidean Spaces
223(30)
Algebraic structure
223(10)
Limits of sequences
233(3)
Limits of functions
236(9)
The total derivative
245(8)
Topology of Euclidean Spaces
253(31)
Interior, closure, boundary
253(8)
Compact sets
261(4)
Connected sets
265(3)
Continuous functions
268(6)
Applications
274(10)
Metric Spaces
284(31)
Introduction
284(6)
Limits of functions
290(5)
Interior, closure, boundary
295(5)
Compact sets
300(6)
Connected sets
306(4)
Continuous functions
310(5)
Differentiability on Rn
315(57)
Partial derivatives and partial integrals
315(10)
The definition of differentiability
325(10)
Differentiability theorems
335(6)
The Mean Value Theorem and Taylor's Formula
341(9)
The Inverse Function Theorem
350(10)
Optimization
360(12)
Integration on Rn
372(65)
Jordan regions
372(10)
Riemann integration on Jordan regions
382(11)
Iterated integrals
393(13)
Change of variables
406(13)
Partitions of unity
419(10)
The gamma function and volume
429(8)
Fundamental Theorems of Vector Calculus
437(56)
Curves
437(12)
Oriented curves
449(7)
Surfaces
456(11)
Oriented surfaces
467(8)
Theorems of Green and Gauss
475(9)
Stokes's Theorem
484(9)
Fourier Series
493(32)
Introduction
493(6)
Summability of Fourier series
499(7)
Growth of Fourier coefficients
506(7)
Convergence of Fourier series
513(6)
Uniqueness
519(6)
Differentiable Manifolds
525(32)
Differential forms on Rn
525(12)
Differentiable manifolds
537(11)
Stokes's Theorem on manifolds
548(9)
Appendices 557(22)
A. Algebraic laws
557(3)
B. Trigonometry
560(4)
C. Matrices and determinants
564(6)
D. Quadric surfaces
570(4)
E. Vector calculus and physics
574(3)
F. Equivalence relations
577(2)
References 579(1)
Answers and Hints to Exercises 580(17)
Subject Index 597(13)
Notation Index 610

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