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E-raamat: Mathematical Analysis and Proof

  • Formaat: PDF+DRM
  • Ilmumisaeg: 30-Apr-2009
  • Kirjastus: Horwood Publishing Ltd
  • Keel: eng
  • ISBN-13: 9780857099341
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  • Formaat: PDF+DRM
  • Ilmumisaeg: 30-Apr-2009
  • Kirjastus: Horwood Publishing Ltd
  • Keel: eng
  • ISBN-13: 9780857099341
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"This fundamental and straightforward text addresses a weakness observed among present-day students, namely a lack of familiarity with formal proof. Beginning with the idea of mathematical proof and the need for it, associated technical and logical skills are developed with care and then brought to bear on the core material of analysis in such a lucid presentation that the development reads naturally and in a straightforward progression. Retaining the core text, the second edition has additional worked examples which users have indicated a need for, in addition to more emphasis on how analysis can be used to tell the accuracy of the approximations to the quantities of interest which arise in analytical limits."--Publisher's description.



This fundamental and straightforward text addresses a weakness observed among present-day students, namely a lack of familiarity with formal proof. Beginning with the idea of mathematical proof and the need for it, associated technical and logical skills are developed with care and then brought to bear on the core material of analysis in such a lucid presentation that the development reads naturally and in a straightforward progression. Retaining the core text, the second edition has additional worked examples which users have indicated a need for, in addition to more emphasis on how analysis can be used to tell the accuracy of the approximations to the quantities of interest which arise in analytical limits.

Arvustused

"Carefully treads the fine line between accuracy and exactitude. A comprehensive introduction, very much in the classical mould, chatty and written with common student misunderstandings in mind. Should be in your undergraduate reference library." --The Mathematical Gazette

"Self-contained and one of the better books. I will definitely and without hesitation recommend and encourage other lecturers to give it serious consideration as a teaching aid." --Mathematics Today

Author's Preface iii
Setting the Scene
1(6)
Introduction
1(1)
The Common Number Systems
2(5)
Logic and Deduction
7(9)
Introduction
7(2)
Implication
9(1)
Is all this Necessary - or Worthwhile?
10(2)
Using the Right Words
12(4)
Problems
15(1)
Mathematical Induction
16(18)
Introduction
16(1)
Arithmetic Progressions
16(2)
The Principle of Mathematical Induction
18(2)
Why All the Fuss About Induction?
20(1)
Examples of Induction
20(8)
The Binomial Theorem
28(6)
Problems
31(3)
Sets and Numbers
34(13)
Sets
34(1)
Standard Sets
35(1)
Proof by Contradiction
36(6)
Sets Again
42(1)
Where we have Got To - and the Way Ahead
43(1)
A Digression
43(4)
Problems
45(2)
Order and Inequalities
47(21)
Basic Properties
47(2)
Consequences of the Basic Properties
49(6)
Bernoulli's Inequality
55(2)
The Modulus (or Absolute Value)
57(11)
Problems
62(6)
Decimals
68(12)
Decimal Notation
68(2)
Decimals of Real Numbers
70(7)
Some Interesting Consequences
77(3)
Problems
78(2)
Limits
80(17)
The Idea of a Limit
80(3)
Manipulating Limits
83(5)
Developments
88(9)
Problems
94(3)
Infinite Series
97(14)
Introduction
97(1)
Convergence Tests
98(9)
Power Series
107(1)
Decimals Again
108(3)
Problems
108(3)
The Structure of the Real Number System
111(10)
Problems
120(1)
Continuity
121(18)
Introduction
121(1)
The Limit of a Function of a Real Variable
122(6)
Continuity
128(6)
Inverse Functions
134(2)
Some Discontinuous Functions
136(3)
Problems
138(1)
Differentiation
139(20)
Basic Results
139(4)
The Mean Value Theorem and its Friends
143(10)
Approximating the Value of a Limit
153(6)
Problems
156(3)
Functions Defined by Power Series
159(17)
Introduction
159(1)
Functions Defined by Power Series
160(3)
Some Standard Functions of Mathematics
163(9)
Further Examples
172(4)
Problems
174(2)
Integration
176(26)
The Integral
176(15)
Approximating the Value of an Integral
191(2)
Improper Integrals
193(9)
Problems
198(4)
Functions of Several Variables
202(24)
Continuity
202(7)
Differentiation
209(5)
Results Involving Interchange of Limits
214(4)
Solving Differential Equations
218(8)
Problems
222(4)
Appendix The Expression of an Integer as a Decimal 226(1)
References and Suggestions for Further Reading 227(2)
Hints and Solutions to Selected Problems 229(18)
Notation Index 247(2)
Subject Index 249
David Stirling, formerly, University of Reading, UK