The principal aim of this book is to introduce university level mathematics - both algebra and calculus. The text is suitable for first and second year students. It treats the material in depth, and thus can also be of interest to beginning graduate students. New concepts are motivated before being introduced through rigorous definitions. All theorems are proved and great care is taken over the logical structure of the material presented. To facilitate understanding, a large number of diagrams are included. Most of the material is presented in the traditional way, but an innovative approach is taken with emphasis on the use of Maple and in presenting a modern theory of integration. To help readers with their own use of this software, a list of Maple commands employed in the book is provided. The book advocates the use of computers in mathematics in general, and in pure mathematics in particular. It makes the point that results need not be correct just because they come from the computer. A careful and critical approach to using computer algebra systems persists throughout the text.
Author(s): P. Adams, K. Smith, R Výborný
Publisher: WSPC
Year: 2004
Language: English
Pages: 544
Cover Page......Page 1
Title Page......Page 4
ISBN 9812389318......Page 5
Preface......Page 8
Outline of the book......Page 9
Notes on notation......Page 11
Exercises......Page 13
Acknowledgments......Page 14
Contents......Page 16
1.1 Our aims......Page 22
1.2.2 Starting Maple......Page 25
1.2.3 Worksheets in Maple......Page 26
1.2.4 Entering commands into Maple......Page 27
1.2.6 Using previous results......Page 29
1.3.1 Help......Page 30
1.3.2 Error messages......Page 31
1.4.1 Basic mathematical operators......Page 32
1.4.3 Performing calculations......Page 33
1.4.4 Exact versus floating point numbers......Page 34
1.5.1 Assigning variables and giving names......Page 39
1.5.2 Useful inbuilt functions......Page 41
1.5.2.1 expand()......Page 42
1.5.2.3 simplify()......Page 43
1.5.2.4 normal()......Page 44
1.5.2.6 sort()......Page 45
1.5.2.7 combine()......Page 46
1.6 Examples of the use of Maple......Page 47
2.1 Sets......Page 54
2.1.1 Union, intersection and difference of sets......Page 56
2.1.2 Sets in Maple......Page 59
2.1.2.1 Expression sequences, lists......Page 60
2.1.3 Families of sets......Page 63
2.1.5 Some common sets......Page 64
2.2 Correct and incorrect reasoning......Page 66
2.3 Propositions and their combinations......Page 68
2.4 Indirect proof......Page 72
2.5 Comments and supplements......Page 74
2.5.1 Divisibility: An example of an axiomatic theory......Page 77
3.1 Relations......Page 84
3.2 Functions......Page 89
3.3.1 Library of functions......Page 96
3.3.2 Defining functions in Maple......Page 98
3.3.3 Boolean functions......Page 101
3.3.4 Graphs of functions in Maple......Page 102
3.4 Composition of functions......Page 110
3.5 Bijections......Page 111
3.6 Inverse functions......Page 112
3.7 Comments......Page 116
4.1 Fields......Page 118
4.2 Order axioms......Page 121
4.3 Absolute value......Page 126
4.4 Using Maple for solving inequalities......Page 129
4.5 Inductive sets......Page 133
4.6 The least upper bound axiom......Page 135
4.7 Operation with real valued functions......Page 141
4.8 Supplement. Peano axioms. Dedekind cuts......Page 142
Order......Page 145
Addition......Page 146
When is a cut a rational number?......Page 147
Summary of Peano's axioms......Page 148
5.1 Inductive reasoning......Page 150
5.2 Aim high!......Page 156
5.3 Notation for sums and products......Page 157
5.3.1 Sums in Maple......Page 162
5.3.2 Products in Maple......Page 164
5.4 Sequences......Page 166
5.5 Inductive definitions......Page 167
5.6 The binomial theorem......Page 170
5.7 Roots and powers with rational exponents......Page 173
5.8 Some important inequalities......Page 178
5.9 Complete induction......Page 182
5.10 Proof of the recursion theorem......Page 185
5.11 Comments......Page 187
6.1 Polynomial functions......Page 188
6.2 Algebraic viewpoint......Page 190
6.3 Long division algorithm......Page 196
6.4 Roots of polynomials......Page 199
6.5 The Taylor polynomial......Page 201
6.6 Factorization......Page 205
7.1 Field extensions......Page 212
7.2 Complex numbers......Page 215
7.2.1 Absolute value of a complex number......Page 217
7.2.2 Square root of a complex number......Page 218
7.2.4 Geometric representation of complex numbers. Trigonometric form of a complex number......Page 220
7.2.5 The binomial equation......Page 223
8.1 General remarks......Page 228
8.2 Maple commands solve and fsolve......Page 230
8.3 Algebraic equations......Page 234
Rational roots......Page 235
Multiple roots......Page 238
Real roots......Page 239
The concept of an algebraic solution......Page 240
Quadratic equations......Page 241
Cubic equations......Page 242
8.3.1 Equations of higher orders and fsolve......Page 246
8.4 Linear equations in several unknowns......Page 249
9.1 Equivalent sets......Page 252
10.1 The concept of a limit......Page 260
10.2 Basic theorems......Page 270
10.3 Limits of sequences in Maple......Page 276
10.4 Monotonic sequences......Page 278
10.5 Infinite limits......Page 284
10.6 Subsequences......Page 288
10.7 Existence theorems......Page 289
10.8 Comments and supplements......Page 296
11.1 Definition of convergence......Page 300
11.2 Basic theorems......Page 306
11.3 Maple and infinite series......Page 310
11.4 Absolute and conditional convergence......Page 311
11.5 Rearrangements......Page 316
11.6 Convergence tests......Page 318
11.7 Power series......Page 321
11.8.1 More convergence tests......Page 324
11.8.2 Rearrangements revisited......Page 327
11.8.3 Multiplication of series......Page 328
11.8.4 Concluding comments......Page 330
12.1 Limits......Page 334
12.1.1 Limits of functions in Maple......Page 340
12.2 The Cauchy definition......Page 343
12.3 Infinite limits......Page 350
12.4 Continuity at a point......Page 353
12.5 Continuity of functions on closed bounded intervals......Page 359
12.6 Comments and supplements......Page 374
13.1 Introduction......Page 378
13.2 Basic theorems on derivatives......Page 383
The chain rule......Page 385
Derivative of the inverse function......Page 387
13.3 Significance of the sign of derivative......Page 390
13.4 Higher derivatives......Page 401
13.4.1 Higher derivatives in Maple......Page 402
13.4.2 Significance of the second derivative......Page 403
13.5 Mean value theorems......Page 409
13.6 The Bernoulli-I'Hospital rule......Page 412
13.7 Taylor's formula......Page 415
13.8 Differentiation of power series......Page 419
13.9 Comments and supplements......Page 422
14.1 Introduction......Page 428
14.2 The exponential function......Page 429
14.3 The logarithm......Page 432
14.4 The general power......Page 436
14.5 Trigonometric functions......Page 439
The school definition......Page 444
14.6 Inverses to trigonometric functions.......Page 446
14.7 Hyperbolic functions......Page 451
15.1 Intuitive description of the integral......Page 452
A tentative attempt at definition of the integral......Page 454
15.2 The definition of the integral......Page 459
15.2.1 Integration in Maple......Page 464
15.3 Basic theorems......Page 466
15.4 Bolzano-Cauchy principle......Page 471
15.5 Antiderivates and areas......Page 476
15.6 Introduction to the fundamental theorem of calculus......Page 478
15.7 The fundamental theorem of calculus......Page 479
Direct integration......Page 482
15.8 Consequences of the fundamental theorem......Page 490
15.9 Remainder in the Taylor formula......Page 498
15.10 The indefinite integral......Page 502
15.11 Integrals over unbounded intervals......Page 509
15.12 Interchange of limit and integration......Page 513
15.13 Comments and supplements......Page 519
A.1.1 Introduction......Page 522
A.1.2 The conditional statement......Page 523
A.1.3 The while statement......Page 525
A.2 Examples......Page 527
References......Page 532
Index of Maple commands used in this book......Page 534
Index......Page 540
