An Interactive Introduction to Mathematical Analysis Paperback with CD-ROMThis book provides a rigorous course in the calculus of functions of a real variable. Its gentle approach, particularly in its early chapters, makes it especially suitable for students who are not headed for graduate school but, for those who are, this book also provides the opportunity to engage in a penetrating study of real analysis.The companion onscreen version of this text contains hundreds of links to alternative approaches, more complete explanations and solutions to exercises; links that make it more friendly than any printed book could be. In addition, there are links to a wealth of optional material that an instructor can select for a more advanced course, and that students can use as a reference long after their first course has ended. The on-screen version also provides exercises that can be worked interactively with the help of the computer algebra systems that are bundled with Scientific Notebook. |
Contents
A Note to the Reader | 1 |
Background Material | 3 |
The Emergence of Rigorous Calculus | 5 |
12 The Pythagorean Crisis | 6 |
13 The Zeno Crisis | 7 |
14 The Set Theory Crisis | 10 |
Mathematical Grammar | 12 |
22 Negating a Mathematical Sentence | 17 |
Differentiation | 220 |
92 Derivatives and Differentiability | 222 |
93 Some Elementary Properties of Derivatives | 227 |
94 The Mean Value Theorem | 233 |
95 Taylor Polynomials | 239 |
96 Indeterminate Forms | 246 |
The Exponential and Logarithmic Functions | 253 |
102 Integer Exponents | 254 |
23 Combining Two or More Statements | 19 |
Strategies for Writing Proofs | 26 |
32 Statements that Contain the Word and | 27 |
33 Statements that Contain the Word or | 29 |
34 Statements of the Form P Q | 32 |
35 Statements of the Form 𝑥P𝑥 | 33 |
36 Statements of the Form 𝑥P𝑥 | 37 |
37 Proof by Contradiction | 41 |
38 Some Further Examples | 44 |
Elements of Set Theory | 50 |
42 Sets and Subsets | 52 |
43 Functions | 59 |
Elementary Concepts of Analysis | 69 |
The Real Number System | 71 |
52 Axioms for the Real Number System | 79 |
53 Arithmetical Properties of R | 81 |
55 Integers and Rationals | 84 |
57 The Axiom of Completeness | 88 |
58 Some Consequences of the Completeness Axiom | 91 |
59 The Archimedean Property of the System R | 93 |
510 Boundedness of Functions | 97 |
511 Sequences Finite Sets and Infinite Sets | 98 |
512 Sequences of Sets | 100 |
513 Mathematical Induction | 103 |
515 The Complex Number System Optional | 106 |
Elementary Topology of the Real Line | 107 |
62 Interior Points and Neighborhoods | 108 |
63 Open Sets and Closed Sets | 111 |
64 Some Properties of Open Sets and Closed Sets | 113 |
65 The Closure of a Set | 117 |
66 Limit Points | 122 |
67 Neighborhoods of Infinity | 125 |
Limits of Sequences | 127 |
72 Subsequences | 128 |
73 Limits and Partial Limits of Sequences | 129 |
74 Some Elementary Facts About Limits and Partial Limits | 137 |
75 The Algebraic Rules for Limits | 143 |
76 The Relationship Between Sequences and the Topology of R | 148 |
77 Limits of Monotone Sequences | 151 |
78 The Cantor Intersection Theorem | 158 |
79 The Existence of Partial Limits | 161 |
710 Upper and Lower Limits | 164 |
Limits and Continuity of Functions | 165 |
82 Limits at Infinity and Infinite Limits | 176 |
83 OneSided Limits | 181 |
84 The Relationship Between Limits of Functions and Limits of Sequences | 183 |
85 Some Facts About Limits of Functions | 186 |
86 The Composition Theorem for Limits | 188 |
87 Continuity | 192 |
88 The Distance Function of a Set | 198 |
810 The Behavior of Continuous Functions on Intervals | 200 |
811 Inverse Function Theorems for Continuity | 207 |
812 Uniform Continuity | 210 |
103 Rational Exponents | 255 |
104 Real Exponents | 258 |
Intuitive Approach | 260 |
Rigorous Approach | 263 |
The Riemann Integral | 270 |
112 Partitions and Step Functions | 275 |
113 Integration of Step Functions | 279 |
114 Elementary Sets | 287 |
115 Riemann Integrability and the Riemann Integral | 294 |
116 Some Examples of Integrable and Nonintegrable Functions | 299 |
117 Some Properties of the Riemann Integral | 304 |
118 Upper Lower and Oscillation Functions | 307 |
119 Riemann Sums and Darbouxs Theorem Optional | 314 |
1110 The Role of Continuity in Riemann Integration | 317 |
1111 The Composition Theorem for Riemann Integrability | 320 |
1112 The Fundamental Theorem of Calculus | 324 |
1113 The Change of Variable Theorem | 327 |
1114 Integration of Complex Functions Optional | 334 |
Infinite Series | 335 |
122 Elementary Properties of Series | 343 |
123 Some Elementary Facts About Convergence | 345 |
124 Convergence of Series with Nonnegative Terms | 346 |
125 Decimals | 353 |
127 Convergence of Series Whose Terms May Change Sign | 365 |
128 Rearrangements of Series | 375 |
1210 Multiplication of Series | 377 |
1211 The Cantor Set | 382 |
Improper Integrals | 383 |
132 Elementary Properties of Improper Integrals | 387 |
133 Convergence of Integrals of Nonnegative Functions | 389 |
134 Absolute and Conditional Convergence | 392 |
Sequences and Series of Functions | 399 |
141 The Three Types of Convergence | 400 |
142 The Important Properties of Uniform Convergence | 412 |
143 The Important Property of Bounded Convergence | 414 |
144 Power Series | 426 |
145 Power Series Expansion of the Exponential Function | 440 |
146 Binomial Series | 442 |
147 The Trigonometric Functions | 448 |
148 Analytic Functions of a Real Variable | 456 |
Calculus of a Complex Variable Optional | 458 |
Integration of Functions of Two Variables | 459 |
162 Functions of Two Variables | 460 |
163 Continuity of a Partial Integral | 464 |
164 Differentiation of a Partial Integral | 466 |
165 Some applications of Partial Integrals | 468 |
166 Interchanging Iterated Riemann Integrals | 470 |
Sets of Measure Zero Optional | 478 |
Calculus of Several Variables Optional | 479 |
| 480 | |
Index of Symbols and General Index | 482 |
Other editions - View all
An Interactive Introduction to Mathematical Analysis Hardback with CD-ROM Jonathan Lewin Limited preview - 2003 |
An Interactive Introduction to Mathematical Analysis Paperback with CD-ROM Jonathan Lewin Limited preview - 2003 |
Common terms and phrases
absolutely convergent An+1 assertion bounded function bounded set Cauchy product chapter choose a number closed conditions are equivalent continuous function converges uniformly deduce derivative differentiable Dirichlet's test diverges elementary set equation exercise exists a number f is continuous f is integrable f(xn fact finite function defined function f ƒ and g Given any number given number Given that ƒ graph holds icon Improper Integrals infinite interval L'Hôpital's rule lim f(x limit point limits of sequences mathematical neighborhood nonempty number ɛ number x on-screen version partial limit partition polynomial positive integer positive number Proof rational numbers real number system real numbers Riemann integrable satisfies the inequality Scientific Notebook sequence fn sequence of real set of real statement step function Subsection Suppose that ƒ true uniformly continuous upper bound