Introduction to Analysis

by Wade, William R.

• ISBN13: 9780131453333
• ISBN10: 0131453335
• Edition: 3rd
• Format: Hardcover
• Copyright: 2010-01-01
• Publisher: Pearson College Div

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	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

Excerpts

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