Introduction to Analysis

by Wade, William R.

• ISBN13: 9780130144096
• ISBN10: 0130144096
• Edition: 2nd
• Format: Hardcover
• Copyright: 2000-01-01
• 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	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