An Introduction to Analysis,9780130930897

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An Introduction to Analysis

by Wade, William R.

ISBN13:

9780130930897

ISBN10:

013093089X

Format:

Hardcover

Pub. Date:

1/1/1995

Publisher(s):

Prentice Hall

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Summary

Junior level course for math majorsgenerally requiredusually for 2 terms. Chapters 1-5 are for 1st semester, chapters 6-10 for 2nd semester. Text offers strategies of proof for major theorems. This is a "friendly, baby Rudin." Covers both single and multivariable analysis.

Preface

vii

Part I. One-dimensional theory

The Real Number System

Ordered field axioms

(11)

The Well-Ordering Principle

(5)

The Completeness Axiom

(6)

Sequences

(7)

The Bolzano-Weierstrass Theorem

(7)

The extended real number system

(7)

Functions, countability, and the algebra of sets

(11)

Continuity and Differentiability on R

Limits

(11)

Continuity

(8)

Uniform continuity

(4)

Differentiability

(8)

The Mean Value Theorem

(7)

Monotone functions and the Inverse Function Theorem

(6)

Integrability on R

The Riemann integral

(9)

Riemann sums

108

(10)

The Fundamental Theorem of Calculus

118

(9)

Improper Riemann integration

127

(6)

Functions of bounded variation

133

(4)

Convex functions

137

(8)

Infinite Series

Introduction

145

(6)

Tests for convergence

151

(10)

Absolute convergence

161

(10)

Uniform convergence of sequences

171

(7)

Uniform convergence of series

178

(6)

Power series

184

(9)

Analytic functions

193

(12)

Applications

205

(6)

Part II. Multidimensional theory

Euclidean Spaces

Algebraic structure of Rn

211

(14)

Open sets and closed sets in Rn

225

(7)

Sequences and compact sets in Rn

232

(8)

Convex sets and connected sets in Rn

240

(4)

Limits of functions on Rn

244

(8)

Continuous functions on Rn

252

(5)

Applications

257

(9)

Metric spaces

266

(11)

Differentiability on Rn

Partial derivatives and partial integrals

277

(10)

The definition of differentiability

287

(9)

Differentiability theorems

296

(6)

The Mean Value Theorem and Taylor's Formula

302

(8)

The Inverse Function Theorem

310

(9)

Extrema

319

(11)

Differentiability and tangent planes

330

(6)

Integration on Rn

Jordan regions

336

(11)

Riemann integrability on Jordan regions

347

(10)

Iterated integrals

357

(13)

Change of variables

370

(13)

Partitions of unity

383

(9)

The gamma function and volume

392

(8)

Fundamental Theorems of Multivariable Calculus

Curves

400

(11)

Oriented curves

411

(7)

Surfaces

418

(11)

Oriented surfaces

429

(8)

Theorems of Green and Gauss

437

(9)

Stokes's Theorem

446

(9)

Fourier Series

Introduction

455

(6)

Summability of Fourier series

461

(7)

Growth of Fourier coefficients

468

(7)

Convergence of Fourier series

475

(6)

Uniqueness

481

(6)

Stokes's Theorem on Manifolds

Differential forms on Rn

487

(12)

Differentiable manifolds

499

(11)

Stokes's Theorem on manifolds

510

(9)

APPENDICES

A. Algebraic laws

519

(1)

B. Trigonometry

520

(4)

C. Matrices and determinants

524

(6)

D. Quadric surfaces

530

(4)

E. Vector calculus and physics

534

(3)

F. Equivalence relations

537

(2)

References

539

(1)

Answers and hints to exercises

540

(13)

Subject index

553

(11)

Symbol Index

564

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An Introduction to Analysis

9780130930897

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Table of Contents