 | Hrbacek K. Analysis with ultrasmall numbers / K.Hrbacek, O.Lessmann, R.O'Donovan. - Boca Raton: CRC Press/Taylor & Francis, 2015. - xxvii, 294 p.: ill. - (Textbooks in mathematics). - Bibliogr.: p.287-289. - Ind.: p.291-294. - ISBN 978-1-4987-0265-2
Шифр: (И/В16-H86) 02
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Preface ........................................................ xi
Preface for Students .......................................... xix
Acknowledgments ............................................... xxv
Authors ..................................................... xxvii
I Elementary Analysis .......................................... 1
1 Basic Concepts ............................................... 3
1.1 Introduction ............................................ 3
1.2 Observability ........................................... 7
1.3 First Principles ........................................ 9
1.4 Closure ................................................ 15
1.5 Relativization and Stability ........................... 22
1.6 Sets and Induction ..................................... 30
1.7 Summary ................................................ 32
1.8 Additional Exercises ................................... 33
2 Continuity and Limits ....................................... 37
2.1 Continuity ............................................. 37
2.2 Properties of Continuous Functions ..................... 42
2.3 Limits ................................................. 48
2.4 Exponential and Logarithmic Functions .................. 55
2.5 Additional Exercises ................................... 61
3 Differentiability ........................................... 67
3.1 Derivative ............................................. 67
3.2 Rules of Differentiation ............................... 72
3.3 Basic Theorems about Derivatives ....................... 78
3.4 Smooth Functions ....................................... 82
3.5 Derivatives of Trigonometric Functions ................. 83
3.6 Second Order Derivatives ............................... 88
3.7 Additional Exercises ................................... 94
4 Integration of Continuous Functions ......................... 99
4.1 Fundamental Theorem of Calculus ........................ 99
4.2 Antiderivatives ....................................... 107
4.3 Rules of Integration .................................. 109
4.4 Geometric Interpretation of Integrals ................. 112
4.5 Applications of the Integral .......................... 115
4.6 Natural Logarithm and Exponential ..................... 121
4.7 Numerical Integration ................................. 127
4.8 Improper Integrals .................................... 132
4.9 Additional Exercises .................................. 137
II Higher Analysis ............................................ 141
5 Basic Concepts Revisited ................................... 143
5.1 Real and Natural Numbers .............................. 143
5.2 Epsilon-Delta Method .................................. 147
5.3 Alternative Characterization of Limits ................ 150
5.4 Additional Exercises .................................. 152
6 L'Hôpital's Rule and Higher Order Derivatives .............. 155
6.1 L'Hopital's Rule ...................................... 155
6.2 Higher Order Derivatives .............................. 158
6.3 Additional Exercises .................................. 161
7 Sequences and Series ....................................... 163
7.1 Sequences ............................................. 163
7.2 Series ................................................ 168
7.3 Taylor Series ......................................... 177
7.4 Uniform Convergence ................................... 183
7.5 Power Series .......................................... 185
7.6 Additional Exercises .................................. 188
8 First Order Differential Equations ......................... 193
8.1 Solutions of Some Differential Equations .............. 193
8.2 Existence and Uniqueness .............................. 196
8.3 Additional Exercises .................................. 200
9 Integration ................................................ 205
9.1 Riemann Integral ...................................... 205
9.2 Darboux Integral ...................................... 214
9.3 Additional Exercises .................................. 217
10 Topology of Real Numbers ................................... 221
10.1 Open and Closed Sets .................................. 221
10.2 Dense Sets ............................................ 229
10.3 Compact Sets .......................................... 231
10.4 Additional Exercises .................................. 235
Answers to Exercises .......................................... 239
Appendix: Foundations and Relative Set Theory ................. 267
Bibliography .................................................. 287
Index ......................................................... 291
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