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An interactive introduction to mathematical analysis

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Cambridge University Press, 2003 - Mathematics - 492 pages
This 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. For those who are, this book gives an 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 CD provides exercises that can be worked interactively with the help of the computer algebra systems that are bundled with Scientific Notebook.
  

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Great book. Mathematicians should use a language that is simple enough to comprehend the complexity in this field, usually in math books we see the opposite, i.e. writers use complicated language to present simplicity and leave the readers lost. This books grabs your interest, keeps you wanting for more. The smartness lies in making people understand what you know or atleast what they should know. Then, we have a great teacher. Finding an analysis book that speaks to me mind to mind really makes me happy. I unfortunately, cannot speak for all the chapters but certain topics that I looked up on this online book were the boundedness of sets, and I am looking forward for the rest.  

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Selected pages

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
Bibliography
480
Index of Symbols and General Index
482
Copyright

Other editions - View all

An Interactive Introduction to Mathematical Analysis Hardback with CD-ROM
An Interactive Introduction to Mathematical Analysis Hardback with CD-ROM
Jonathan Lewin
Limited preview - 2003

Common terms and phrases

absolutely convergent assertion bounded function bounded set Cauchy product chapter choose a number choose an integer closed conditions are equivalent continuous function converges uniformly deduce derivative differentiable Dirichlet's test diverges elementary set equation exercise exists a number f(xn fact finite function defined function f Given any number given number graph holds icon illustrated in Figure Improper Integrals infinite interior point interval 0,1 largest member limit point limits of sequences Maclaurin series mathematical mean value theorem need to show neighborhood nonempty number x G on-screen version oo oo partial limit partition positive integer positive number power series Proof prove rational numbers real number system Riemann integrable satisfies the inequality Scientific Notebook sequence of real sequence xn set H set of real statement step function Subsection Suppose that f true Uniform Convergence uniformly continuous upper bound

References to this book

From Google Scholar

Desarrollo De Los Contenidos.
Regla de L’Hopital, Polinomio de Taylor

About the author (2003)

Lewin of Kennesaw State University, Georgia

Bibliographic information

QR code for An interactive introduction to mathematical analysis
TitleAn interactive introduction to mathematical analysis
AuthorJonathan Lewin
Editionillustrated
PublisherCambridge University Press, 2003
ISBN0521017181, 9780521017183
Length492 pages
Subjects › 

Mathematics / Discrete Mathematics
  
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