9780131453333

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.

Preface

xi

Part I. ONE-DIMENSIONAL THEORY

1 The Real Number System

1

(34)

1.1 Ordered field axioms

1

(12)

1.2 Well-Ordering Principle

13

(5)

1.3 Completeness Axiom

18

(6)

1.4 Functions, countability, and the algebra of sets

24

(11)

2 Sequences in R

35

(23)

2.1 Limits of sequences

35

(4)

2.2 Limit theorems

39

(6)

2.3 Bolzano-Weierstrass Theorem

45

(4)

2.4 Cauchy sequences

49

(3)

2.5 Limits supremum and infimum

52

(6)

3 Continuity on R

58

(27)

3.1 Two-sided limits

58

(8)

3.2 One-sided limits and limits at infinity

66

(5)

3.3 Continuity

71

(8)

3.4 Uniform continuity

79

(6)

4 Differentiability on R

85

(22)

4.1 The derivative

85

(7)

4.2 Differentiability theorems

92

(2)

4.3 Mean Value Theorem

94

(8)

4.4 Monotone functions and Inverse Function Theorem

102

(5)

5 Integrability on R

107

(47)

5.1 Riemann integral

107

(10)

5.2 Riemann sums

117

(10)

5.3 Fundamental Theorem of Calculus

127

(9)

5.4 Improper Riemann integration

136

(6)

5.5 Functions of bounded variation

142

(5)

5.6 Convex functions

147

(7)

6 Infinite Series of Real Numbers

154

(30)

6.1 Introduction

154

(6)

6.2 Series with nonnegative terms

160

(5)

6.3 Absolute convergence

165

(8)

6.4 Alternating series

173

(4)

6.5 Estimation of series

177

(4)

6.6 Additional tests

181

(3)

7 Infinite Series of Functions

184

(41)

7.1 Uniform convergence of sequences

184

(8)

7.2 Uniform convergence of series

192

(5)

7.3 Power series

197

(10)

7.4 Analytic functions

207

(12)

7.5 Applications

219

(6)

Part II. MULTIDIMENSIONAL THEORY

8 Euclidean Spaces

225

(31)

8.1 Algebraic structure

225

(9)

8.2 Planes and linear transformations

234

(8)

8.3 Topology of Rn

242

(7)

8.4 Interior, closure and boundary

249

(7)

9 Convergence in Rn

256

(34)

9.1 Limits of sequences

256

(7)

9.2 Limits of functions

263

(7)

9.3 Continuous functions

270

(7)

9.4 Compact sets

277

(3)

9.5 Applications

280

(10)

10 Metric Spaces

290

(31)

10.1 Introduction

290

(6)

10.2 Limits of functions

296

(5)

10.3 Interior, closure and boundary

301

(5)

10.4 Compact sets

306

(6)

10.5 Connected sets

312

(4)

10.6 Continuous functions

316

(5)

11 Differentiability on Rn

321

(60)

11.1 Partial derivatives and partial integrals

321

(11)

11.2 Definition of differentiability

332

(7)

11.3 Derivatives, differentials, and tangent planes

339

(9)

11.4 Chain Rule

348

(4)

11.5 Mean Value Theorem and Taylor's Formula

352

(6)

11.6 Inverse Function Theorem

358

(11)

11.7 Optimization

369

(12)

12 Integration on Rn

381

(68)

12.1 Jordan regions

381

(13)

12.2 Riemann integration on Jordan regions

394

(13)

12.3 Iterated integrals

407

(13)

12.4 Change of variables

420

(12)

12.5 Partitions of unity

432

(9)

12.6 Gamma function and volume

441

(8)

13 Fundamental Theorems of Vector Calculus

449

(57)

13.1 Curves

449

(12)

13.2 Oriented curves

461

(7)

13.3 Surfaces

468

(11)

13.4 Oriented surfaces

479

(9)

13.5 Theorems of Green and Gauss

488

(8)

13.6 Stokes's Theorem

496

(10)

14 Fourier Series

506

(32)

14.1 Introduction

506

(6)

14.2 Summability of Fourier series

512

(7)

14.3 Growth of Fourier coefficients

519

(7)

14.4 Convergence of Fourier series

526

(6)

14.5 Uniqueness

532

(6)

15 Differentiable Manifolds

538

(32)

15.1 Differential forms on Rn

538

(12)

15.2 Differentiable manifolds

550

(11)

15.3 Stokes's Theorem on manifolds

561

Appendices

A. Algebraic laws

570

(3)

B. Trigonometry

573

(4)

C. Matrices and determinants

577

(6)

D. Quadric surfaces

583

(4)

E. Vector calculus and physics

587

(3)

F. Equivalence relations

590

(2)

References

592

(1)

Answers and Hints to Exercises

593

(18)

Subject Index

611

(13)

Notation Index

624

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

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This text provides a bridge from "sophomore" calculus to graduate courses that use analytic ideas, e.g., real and complex analysis, partial and ordinary differential equations, numerical analysis, fluid mechanics, and differential geometry. For a two-semester course, the first semester should end with Chapter 8. For a three-quarter course, the second quarter should begin in Chapter 6 and end somewhere in the middle of Chapter 11. Our presentation is divided into two parts. The first half, Chapters 1 through 7 together with Appendices A and B, gradually introduces the central ideas of analysis in a one-dimensional setting. The second half, Chapters 8 through 15 together with Appendices C through F, covers multidimensional theory. More specifically, Chapter 1 introduces the real number system as a complete, ordered field, Chapters 2 through 5 cover calculus on the real line; and Chapters 6 and 7 discuss infinite series, including uniform and absolute convergence. The first two sections of Chapter 8 give a short introduction to the algebraic, structure ofR n , including the connection between linear functions and matrices. At that point instructors have two options. They can continue covering Chapters 8 and 9 to explore topology and convergence in the concrete Euclidean space setting, or they can cover these same concepts in the abstract metric space setting (Chapter 10). Since either of these options provides the necessary foundation for the rest of the book, instructors are free to choose the approach that they feel best suits their aims. With this background material out of the way, Chapters 11 through 13 develop the, machinery and theory of vector calculus. Chapter 14 gives a short introduction to Fourier series, including summability and convergence bf Fourier series, growth of Fourier coefficients, and uniqueness of trigonometric series. Chapter 15 gives a short introduction to differentiable manifolds which culminates in a proof of Stokes's Theorem on differentiable manifolds. Separating the one-dimensional from then-dimensional material is not the most efficient way to present the material, but it does have two advantages. The more abstract, geometric concepts can be postponed until students have been given a thorough introduction to analysis on the real line. Students have two chances to master some of the deeper ideas of analysis (e.g., convergence of sequences, limits of functions, and uniform continuity). We have made this text flexible in another way by including core material and enrichment material. The core material, occupying fewer than 384 pages, can be covered easily in a one-year course. The enrichment material is included for two reasons: Curious students can use it to delve deeper into the core material or as a jumping off place to pursue more general topics, and instructors can use it to supplement their course or to add variety from year to year. Enrichment material appears in enrichment sections, marked with a superscripte, or in core sections, where it is marked with an asterisk. Exercises that use enrichment material are also marked with an asterisk, and the material needed to solve them is cited in the Answers and Hints section. To make course planning easier, each enrichment section begins with a statement which indicates whether that section uses material from any other enrichment section. Since no core material depends on enrichment material, any of the latter can be skipped without loss in the integrity of the course. Most enrichment sections (5.5, 5.6, 6.5, 6.6, 7.5, 9.4, 11.6, 12.6, 14.1, 15.1) are independent and can be covered in any order after the core material that precedes them has been dealt with. Sections 9.5, 12.5, and 15.2 require 9.4, Section 15.3 requires 12.5, and Section 14.3 requires 5.5 only to establish Lemma 14.25. This result can easily be proved for continuously differentiable functions, th

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Introduction to Analysis

Introduction to Analysis

0131453335

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