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Concrete Introduction to Real Analysis 2nd edition [Kõva köide]

(University of Colorado, Colorado Springs, USA)
  • Formaat: Hardback, 314 pages, kõrgus x laius: 234x156 mm, kaal: 580 g, 1 Tables, black and white; 28 Illustrations, black and white
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 06-Dec-2017
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 1498778135
  • ISBN-13: 9781498778138
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  • Formaat: Hardback, 314 pages, kõrgus x laius: 234x156 mm, kaal: 580 g, 1 Tables, black and white; 28 Illustrations, black and white
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 06-Dec-2017
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 1498778135
  • ISBN-13: 9781498778138
Teised raamatud teemal:
A Concrete Introduction to Analysis, Second Edition offers a major reorganization of the previous edition with the goal of making it a much more comprehensive and accessible for students.

The standard, austere approach to teaching modern mathematics with its emphasis on formal proofs can be challenging and discouraging for many students. To remedy this situation, the new edition is more rewarding and inviting. Students benefit from the text by gaining a solid foundational knowledge of analysis, which they can use in their fields of study and chosen professions.

The new edition capitalizes on the trend to combine topics from a traditional transition to proofs course with a first course on analysis. Like the first edition, the text is appropriate for a one- or two-semester introductory analysis or real analysis course. The choice of topics and level of coverage is suitable for mathematics majors, future teachers, and students studying engineering or other fields requiring a solid, working knowledge of undergraduate mathematics.

Key highlights:











Offers integration of transition topics to assist with the necessary background for analysis





Can be used for either a one- or a two-semester course





Explores how ideas of analysis appear in a broader context





Provides as major reorganization of the first edition





Includes solutions at the end of the book

Arvustused

This is the second, revised edition of a textbook for a course that would typically follow an introductory calculus course in the standard American curriculum. The intended student readers would be those planning to continue with more advanced mathematics, or those in areas that require an understanding of mathematics from a conceptual perspective. While there are many texts with this focus, and libraries may already have several, this one is a bit different. It approaches the subject in an easy-to-read and intuitive manner, rather than in the more rigidly traditional "definition, theorem, proof" approach. It doesnt cover as much material as some other texts, since it approaches the topics through a historical, contextual lens. Even so, a library may wish to consider this book, even if it has an existing real analysis collection. It will appeal to the student who is struggling with the structured formalism of more traditional texts. Enhancing its utility for self-study, the book includes complete solutions to half of the exercises. It contains a short bibliography, mostly of classic mathematical analysis books. --D. Z. Spicer, University System of Maryland

List of Figures
xiii
Preface xv
1 Real numbers and mathematical proofs
1(28)
1.1 Real number axioms
3(6)
1.1.1 Field axioms
3(3)
1.1.2 Order axioms
6(3)
1.2 Proofs
9(15)
1.2.1 Proof by induction
10(4)
1.2.2 Irrational real numbers
14(1)
1.2.3 Prepositional logic
15(1)
1.2.3.1 Truth tables
16(2)
1.2.3.2 Valid consequences
18(2)
1.2.4 Rules of inference
20(1)
1.2.5 Predicates and quantifiers
20(4)
1.3 Problems
24(5)
2 Infinite sequences
29(28)
2.1 Limits of infinite sequences
30(8)
2.1.1 Basic ideas
30(4)
2.1.2 Properties of limits
34(4)
2.2 Completeness axioms
38(5)
2.3 Subsequences and compact intervals
43(3)
2.4 Cauchy sequences
46(1)
2.5 Continued fractions
47(6)
2.6 Problems
53(4)
3 Infinite series
57(16)
3.1 Basics
57(2)
3.2 Positive series
59(3)
3.3 General series
62(4)
3.3.1 Absolute convergence
63(1)
3.3.2 Alternating series
64(2)
3.4 Power series
66(3)
3.5 Problems
69(4)
4 More sums
73(20)
4.1 Grouping and rearrangement
73(6)
4.2 A calculus of sums and differences
79(6)
4.3 Computing the sums of powers
85(5)
4.4 Problems
90(3)
5 Functions
93(38)
5.1 Basics
94(2)
5.2 Limits and continuity
96(12)
5.2.1 Limits
96(1)
5.2.1.1 Limit as x → ∞
96(1)
5.2.1.2 Limit as x → x0
97(2)
5.2.1.3 Limit rules
99(2)
5.2.2 Continuity
101(3)
5.2.2.1 Rootfinding 1
104(1)
5.2.3 Uniform continuity
105(3)
5.3 Derivatives
108(16)
5.3.1 Computation of derivatives
109(5)
5.3.2 The Mean Value Theorem
114(3)
5.3.3 Contractions
117(2)
5.3.3.1 Rootfinding 2: Newton's Method
119(1)
5.3.4 Convexity
120(4)
5.4 Problems
124(7)
6 Integrals
131(40)
6.1 Areas under power function graphs
134(5)
6.2 Integrable functions
139(8)
6.3 Properties of integrals
147(5)
6.4 Arc length and trigonometric functions
152(2)
6.5 Improper integrals
154(11)
6.5.1 Integration of positive functions
156(4)
6.5.2 Integrals and sums
160(1)
6.5.3 Absolutely convergent integrals
161(1)
6.5.4 Conditionally convergent integrals
162(3)
6.6 Problems
165(6)
7 The Natural logarithm
171(22)
7.1 Introduction
171(4)
7.2 The natural exponential function
175(3)
7.3 Infinite products
178(4)
7.4 Stirling's formula
182(7)
7.5 Problems
189(4)
8 Taylor polynomials and series
193(24)
8.1 Taylor polynomials
195(5)
8.2 Taylor's Theorem
200(3)
8.3 The remainder
203(4)
8.3.1 Calculating e
205(1)
8.3.2 Calculating π
206(1)
8.4 Additional results
207(6)
8.4.1 Taylor series by algebraic manipulations
207(2)
8.4.2 The binomial series
209(4)
8.5 Problems
213(4)
9 Uniform convergence
217(26)
9.1 Introduction
217(2)
9.2 Uniform Convergence
219(5)
9.3 Convergence of power series
224(2)
9.4 The Weierstrass Approximation Theorem
226(3)
9.5 Trigonometric approximation
229(10)
9.5.1 Solving a heat equation
229(1)
9.5.2 Approximation by trigonometric functions
230(4)
9.5.3 Fourier series
234(5)
9.6 Problems
239(4)
A Solutions to odd numbered problems
243(50)
A.1
Chapter 1 Solutions
243(7)
A.2
Chapter 2 Solutions
250(4)
A.3
Chapter 3 Solutions
254(5)
A.4
Chapter 4 Solutions
259(5)
A.5
Chapter 5 Solutions
264(9)
A.6
Chapter 6 Solutions
273(6)
A.7
Chapter 7 Solutions
279(3)
A.8
Chapter 8 Solutions
282(4)
A.9
Chapter 9 Solutions
286(7)
Bibliography 293(2)
Index 295
Robert Carlson is professor of mathematics at the University of Colorado, Colorado Springs. He holds a Ph.D. in Mathematics from UCLA and has written extensively for several noted journals about graphs, differential equations, eigenvalue problems, and other mathematical topics. This is his third book.