数理解析入門(CD-ROM付)<br>An Interactive Introduction to Mathematical Analysis (PAP/CDR)

数理解析入門(CD-ROM付)
An Interactive Introduction to Mathematical Analysis (PAP/CDR)

  • ただいまウェブストアではご注文を受け付けておりません。 ⇒古書を探す
  • 製本 Paperback:紙装版/ペーパーバック版/ページ数 525 p.
  • 言語 ENG,ENG
  • 商品コード 9780521017183
  • DDC分類 515

基本説明

A rigorous course in the calculus of functions of a real variable. The CD requires Windows 95 or later and features hundreds of links to video presentations, solutions to exercises for lecturers, and more.

Table of Contents

Preface                                            xi
The Purpose of This Book xi
Global Structure of the Book xii
The On-Screen Version of This Book xv
Instructor's Manual xx
Preparation of This Book xxi
Reading This Book On-Screen xxiii
What do I Need to Read This Book On-Screen? xxiii
What Is Scientific Notebook? xxiii
Getting Started xxiv
Using the Movie and Installation CD xxiv
Setting Your Screen View xxv
Navigating in the On-Screen Book xxvii
Reading and Writing in Scientific Notebook xxx
Interactive Reading with Scientific Notebook xxx
Updating Your Copy of the Book xxxi
Summing Up xxxii
A Note to the Reader 1 (2)
PART I Background Material 3 (66)
The Emergence of Rigorous Calculus 5 (7)
What Is Mathematical Analysis? 5 (1)
The Pythagorean Crisis 6 (1)
The Zeno Crisis 7 (3)
The Set Theory Crisis 10 (2)
Mathematical Grammar 12 (14)
The Quantifiers For Every and There Exists 12 (5)
Negating a Mathematical Sentence 17 (2)
Combining Two or More Statements 19 (7)
Strategies for Writing Proofs 26 (24)
Introduction 26 (1)
Statements that Contain the Word and 27 (2)
Statements that Contain the Word or 29 (3)
Statements of the Form P → Q 32 (1)
Statements of the Form Ex (P(x)) 33 (4)
Statements of the Form Ax (P(x)) 37 (4)
Proof by Contradiction 41 (3)
Some Further Examples 44 (6)
Elements of Set Theory 50 (19)
Introduction 50 (2)
Sets and Subsets 52 (7)
Functions 59 (10)
PART II Elementary Concepts of Analysis 69 (411)
The Real Number System 71 (36)
Introduction to the System R 71 (8)
Axioms for the Real Number System 79 (2)
Arithmetical Properties of R 81 (1)
Order Properties of R 81 (3)
Integers and Rationals 84 (1)
Upper and Lower Bounds 84 (4)
The Axiom of Completeness 88 (3)
Some Consequences of the Completeness 91 (2)
Axiom
The Archimedean Property of the System R 93 (4)
Boundedness of Functions 97 (1)
Sequences, Finite Sets, and Infinite Sets 98 (2)
Sequences of Sets 100 (3)
Mathematical Induction 103 (1)
The Extended Real Number System 103 (3)
The Complex Number System (Optional) 106 (1)
Elementary Topology of the Real Line 107 (20)
The Role of Topology 107 (1)
Interior Points and Neighborhoods 108 (3)
Open Sets and Closed Sets 111 (2)
Some Properties of Open Sets and Closed 113 (4)
Sets
The Closure of a Set 117 (5)
Limit Points 122 (3)
Neighborhoods of Infinity 125 (2)
Limits of Sequences 127 (38)
The Concepts ``Eventually'' and 127 (1)
``Frequently''
Subsequences 128 (1)
Limits and Partial Limits of Sequences 129 (8)
Some Elementary Facts About Limits and 137 (6)
Partial Limits
The Algebraic Rules for Limits 143 (5)
The Relationship Between Sequences and 148 (3)
the Topology of R
Limits of Monotone Sequences 151 (7)
The Cantor Intersection Theorem 158 (3)
The Existence of Partial Limits 161 (3)
Upper and Lower Limits 164 (1)
Limits and Continuity of Functions 165 (55)
Limits of Functions 165 (11)
Limits at Infinity and Infinite Limits 176 (5)
One-Sided Limits 181 (2)
The Relationship Between Limits of 183 (3)
Functions and Limits of Sequences
Some Facts About Limits of Functions 186 (2)
The Composition Theorem for Limits 188 (4)
Continuity 192 (6)
The Distance Function of a Set 198 (1)
The Behavior of Continuous Functions on 198 (2)
Closed Bounded Sets
The Behavior of Continuous Functions on 200 (7)
Intervals
Inverse Function Theorems for Continuity 207 (3)
Uniform Continuity 210 (10)
Differentiation 220 (33)
Introduction to the Concept of a 220 (2)
Derivative
Derivatives and Differentiability 222 (5)
Some Elementary Properties of Derivatives 227 (6)
The Mean Value Theorem 233 (6)
Taylor Polynomials 239 (7)
Indeterminate Forms 246 (7)
The Exponential and Logarithmic Functions 253 (17)
The Purpose of This Chapter 253 (1)
Integer Exponents 254 (1)
Rational Exponents 255 (3)
Real Exponents 258 (2)
Differentiating the Exponential Function: 260 (3)
Intuitive Approach
Differentiating the Exponential Function: 263 (7)
Rigorous Approach
The Riemann Integral 270 (65)
Introduction to the Concept of an Integral 270 (5)
Partitions and Step Functions 275 (4)
Integration of Step Functions 279 (8)
Elementary Sets 287 (7)
Riemann Integrability and the Riemann 294 (5)
Integral
Some Examples of Integrable and 299 (5)
Nonintegrable Functions
Some Properties of the Riemann Integral 304 (3)
Upper, Lower, and Oscillation Functions 307 (7)
Riemann Sums and Darboux's Theorem 314 (3)
(Optional)
The Role of Continuity in Riemann 317 (3)
Integration
The Composition Theorem for Riemann 320 (4)
Integrability
The Fundamental Theorem of Calculus 324 (3)
The Change of Variable Theorem 327 (7)
Integration of Complex Functions 334 (1)
(Optional)
Infinite Series 335 (48)
Introduction to Infinite Series 335 (8)
Elementary Properties of Series 343 (2)
Some Elementary Facts About Convergence 345 (1)
Convergence of Series with Nonnegative 346 (7)
Terms
Decimals 353 (1)
The Ratio Tests 353 (12)
Convergence of Series Whose Terms May 365 (10)
Change Sign
Rearrangements of Series 375 (1)
Iterated Series 375 (2)
Multiplication of Series 377 (5)
The Cantor Set 382 (1)
Improper Integrals 383 (16)
Introduction to Improper Integrals 383 (4)
Elementary Properties of Improper 387 (2)
Integrals
Convergence of Integrals of Nonnegative 389 (3)
Functions
Absolute and Conditional Convergence 392 (7)
Sequences and Series of Functions 399 (59)
The Three Types of Convergence 400 (12)
The Important Properties of Uniform 412 (2)
Convergence
The Important Property of Bounded 414 (12)
Convergence
Power Series 426 (14)
Power Series Expansion of the Exponential 440 (2)
Function
Binomial Series 442 (6)
The Trigonometric Functions 448 (8)
Analytic Functions of a Real Variable 456 (1)
The Inadequacy of Riemann Integration 456 (2)
Calculus of a Complex Variable (Optional) 458 (1)
Integration of Functions of Two Variables 459 (19)
The Purpose of This Chapter 459 (1)
Functions of Two Variables 460 (4)
Continuity of a Partial Integral 464 (2)
Differentiation of a Partial Integral 466 (2)
Some Applications of Partial Integrals 468 (2)
Interchanging Iterated Riemann Integrals 470 (8)
Sets of Measure Zero (Optional) 478 (1)
Calculus of Several Variables (Optional) 479 (1)
Bibliography 480 (2)
Index of Symbols and General Index 482