Introduction to calculus:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English Polish |
| Published: |
Oxford [u.a.]
Pergamon Press
1969
|
| Edition: | 2. ed. |
| Series: | International series of monographs in pure and applied mathematics
17 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Einheitssacht.: Wyklady rachunku rozniczkowego i calkowego <engl.>. - Aus dem Poln. übers. |
| Physical Description: | 336 S. |
Staff View
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| 245 | 1 | 0 | |a Introduction to calculus |c by Kazimierz Kuratowski |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Oxford [u.a.] |b Pergamon Press |c 1969 | |
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Record in the Search Index
| _version_ | 1804135986273189888 |
|---|---|
| adam_text | CONTENTS
Preface to the English edition 9
Preface to the Polish edition 10
I. SEQUENCES AND SERIES
§ 1. Introduction
1.1. Various kinds of numbers 11
1.2. The principle of mathematical induction 12
1.3. The Newton binomial formula 15
1.4*. Schwarz inequality 17
1.5. The principle of continuity (Dedekind) 18
1.6. The absolute value of a number 19
1.7. Bounded sets. The upper bound and the lower bound
of a set 19
1.8*. The axiomatic treatment of real numbers 21
1.9*. Real numbers as sets of rational numbers 23
Exercises on § 1 25
§ 2. Infinite sequences
2.1. Definition and examples 26
2.2. The notion of limit 28
2.3. Bounded sequences 32
2.4. Operations on sequences 32
. 2.5. Further properties of the limit 36
2.6. Subsequences 38
2.7. Cauchy theorem 42
2.8. Divergence to oo 44
2.9. Examples 46
2.10. The number « 47
2.11*. The sequences of the arithmetic means and of the
geometric means of a given sequence 49
Exercises on § 2 52
§ 3. Infinite series
3.1. Definitions and examples 55
3.2. General properties of series 66
3.3. Alternating series. Abel theorem 59
3.4. Series with positive terms. D Alembert and Cauchy
convergence criterions 61
6 CONTENTS
3.5. Applications and examples 64
3.6. Other convergence criteria 66
3.7. Absolutely convergent series 68
3.8. Multiplication of series 71
3.9*. Infinite products 74
Exercises on § 3 79
II. FUNCTIONS
§ 4. Functions and their limits
4.1. Definitions 81
4.2. Monotone functions 83
4.3. One to one functions. Inverse functions 85
4.4. Elementary functions 86
4.5. The limit of a function / at a point a 89
4.6. Operations on the limit 93
4.7. Conditions for the existence of the limit 97
Exercises on § 4 101
§ 5. Continuous functions
5.1. Definition 102
5.2. Cauchy characterization of continuity. Geometrical
interpretation 104
5.3. Continuity of elementary functions 105
5.4. General properties of continuous functions 109
5.5. Continuity of inverse functions 114
Exercises on § 5 116
§ 6. Sequences and series of functions
6.1. Uniform convergence 118
6.2. Uniformly convergent series 121
6.3. Power series 123
6.4. Approximation of continuous functions by polygonal
functions 127
6.5*. The symbolism of mathematical logic 129
Exercises on § 6 137
III. DIFFERENTIAL CALCULUS
§ 7. Derivatives of the first order
7.1. Definitions 138
7.2. Differentiation of elementary functions 142
7.3. Differentiation of inverse functions 147
CONTENTS 7
7.4. Extrema of functions. Rolle theorem 149
7.5. Lagrange and Cauchy theorems 152
7.6. Differentiation of composite functions 156
7.7. Geometrical interpretation of the sign of a derivative 161
7.8. Indeterminate expressions 163
7.9. The derivative of a limit 168
7.10. The derivative of a power series 169
7.11. The expansion of the functions log(l+x) and
arc tan x in power series 172
7.12*. Asymptotes 174
7.13*. The concept of a differential 175
Exercises on § 7 178
§ 8. Derivatives of higher orders
8.1. Definition and examples 180
8.2*. Differentials of higher order 182
8.3. Arithmetical operations 184
8.4. Taylor formula 185
8.5. Expansions in power series 190
8.6. A criterion for extrema 194
8.7. Geometrical interpretation of the second derivative.
Points of inflexion 196
Exercises on § 8 199
IV. INTEGRAL CALCULUS
§ 9. Indefinite integrals
9.1. Definition 201
9.2. The integral of the limit. Integrability of continuous
functions 204
9.3. General formulae for integration 205
9.4. Integration of rational functions 211
9.5. Integration of irrational functions of the second degree 215
9.6. Integration of trigonometric functions 219
Exercises on § 9 223
§ 10. Definite integrals
10.1. Definition and examples 224
10.2. Calculation formulae 227
10.3. Definite integral as a limit of sums 234
10.4. The integral as an area 237
10.5. The length of an arc 241
10.6. The volume and surf ace area of a solid of revolution 248
10.7. Two mean value theorems 253
8 CONTENTS
10.8. Methods of approximate integrations. La grange
interpolation 256
10.9. Wallis formula 259
10.10. Stirling formula 261
10.11*. Riemann integral. Upper and lower Darboux
integrals 262
Exercises on § 10 270
§ 11. Improper integrals and their connection with infinite series
11.1. Integrals with an unbounded interval of integration 273
11.2. Integrals of functions not defined in one point 276
11.3. Calculation formulae 280
11.4. Examples 282
11.5. The Gamma function 292
11.6. The relation between the convergence of an integral
and the convergence of an infinite series 294
11.7. Fourier series 299
11.8. Applications and examples 304
Exercises on § 11 309
Supplement. Additional exercises 311
Index 328
Other titleB in the Series in Pure and Applied Mathematics 333
|
| adam_txt |
CONTENTS
Preface to the English edition 9
Preface to the Polish edition 10
I. SEQUENCES AND SERIES
§ 1. Introduction
1.1. Various kinds of numbers 11
1.2. The principle of mathematical induction 12
1.3. The Newton binomial formula 15
1.4*. Schwarz inequality 17
1.5. The principle of continuity (Dedekind) 18
1.6. The absolute value of a number 19
1.7. Bounded sets. The upper bound and the lower bound
of a set 19
1.8*. The axiomatic treatment of real numbers 21
1.9*. Real numbers as sets of rational numbers 23
Exercises on § 1 25
§ 2. Infinite sequences
2.1. Definition and examples 26
2.2. The notion of limit 28
2.3. Bounded sequences 32
2.4. Operations on sequences 32
. 2.5. Further properties of the limit 36
2.6. Subsequences 38
2.7. Cauchy theorem 42
2.8. Divergence to oo 44
2.9. Examples 46
2.10. The number « 47
2.11*. The sequences of the arithmetic means and of the
geometric means of a given sequence 49
Exercises on § 2 52
§ 3. Infinite series
3.1. Definitions and examples 55
3.2. General properties of series 66
3.3. Alternating series. Abel theorem 59
3.4. Series with positive terms. D'Alembert and Cauchy
convergence criterions 61
6 CONTENTS
3.5. Applications and examples 64
3.6. Other convergence criteria 66
3.7. Absolutely convergent series 68
3.8. Multiplication of series 71
3.9*. Infinite products 74
Exercises on § 3 79
II. FUNCTIONS
§ 4. Functions and their limits
4.1. Definitions 81
4.2. Monotone functions 83
4.3. One to one functions. Inverse functions 85
4.4. Elementary functions 86
4.5. The limit of a function / at a point a 89
4.6. Operations on the limit 93
4.7. Conditions for the existence of the limit 97
Exercises on § 4 101
§ 5. Continuous functions
5.1. Definition 102
5.2. Cauchy characterization of continuity. Geometrical
interpretation 104
5.3. Continuity of elementary functions 105
5.4. General properties of continuous functions 109
5.5. Continuity of inverse functions 114
Exercises on § 5 116
§ 6. Sequences and series of functions
6.1. Uniform convergence 118
6.2. Uniformly convergent series 121
6.3. Power series 123
6.4. Approximation of continuous functions by polygonal
functions 127
6.5*. The symbolism of mathematical logic 129
Exercises on § 6 137
III. DIFFERENTIAL CALCULUS
§ 7. Derivatives of the first order
7.1. Definitions 138
7.2. Differentiation of elementary functions 142
7.3. Differentiation of inverse functions 147
CONTENTS 7
7.4. Extrema of functions. Rolle theorem 149
7.5. Lagrange and Cauchy theorems 152
7.6. Differentiation of composite functions 156
7.7. Geometrical interpretation of the sign of a derivative 161
7.8. Indeterminate expressions 163
7.9. The derivative of a limit 168
7.10. The derivative of a power series 169
7.11. The expansion of the functions log(l+x) and
arc tan x in power series 172
7.12*. Asymptotes 174
7.13*. The concept of a differential 175
Exercises on § 7 178
§ 8. Derivatives of higher orders
8.1. Definition and examples 180
8.2*. Differentials of higher order 182
8.3. Arithmetical operations 184
8.4. Taylor formula 185
8.5. Expansions in power series 190
8.6. A criterion for extrema 194
8.7. Geometrical interpretation of the second derivative.
Points of inflexion 196
Exercises on § 8 199
IV. INTEGRAL CALCULUS
§ 9. Indefinite integrals
9.1. Definition 201
9.2. The integral of the limit. Integrability of continuous
functions 204
9.3. General formulae for integration 205
9.4. Integration of rational functions 211
9.5. Integration of irrational functions of the second degree 215
9.6. Integration of trigonometric functions 219
Exercises on § 9 223
§ 10. Definite integrals
10.1. Definition and examples 224
10.2. Calculation formulae 227
10.3. Definite integral as a limit of sums 234
10.4. The integral as an area 237
10.5. The length of an arc 241
10.6. The volume and surf ace area of a solid of revolution 248
10.7. Two mean value theorems 253
8 CONTENTS
10.8. Methods of approximate integrations. La grange
interpolation 256
10.9. Wallis formula 259
10.10. Stirling formula 261
10.11*. Riemann integral. Upper and lower Darboux
integrals 262
Exercises on § 10 270
§ 11. Improper integrals and their connection with infinite series
11.1. Integrals with an unbounded interval of integration 273
11.2. Integrals of functions not defined in one point 276
11.3. Calculation formulae 280
11.4. Examples 282
11.5. The Gamma function 292
11.6. The relation between the convergence of an integral
and the convergence of an infinite series 294
11.7. Fourier series 299
11.8. Applications and examples 304
Exercises on § 11 309
Supplement. Additional exercises 311
Index 328
Other titleB in the Series in Pure and Applied Mathematics 333 |
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| spellingShingle | Kuratowski, Kazimierz 1896-1980 Introduction to calculus International series of monographs in pure and applied mathematics Calculo (matematica) - elementar larpcal Calculus Zahlentheorie (DE-588)4067277-3 gnd Reelle Analysis (DE-588)4627581-2 gnd Analysis (DE-588)4001865-9 gnd Infinitesimalrechnung (DE-588)4072798-1 gnd |
| subject_GND | (DE-588)4067277-3 (DE-588)4627581-2 (DE-588)4001865-9 (DE-588)4072798-1 (DE-588)4151278-9 |
| title | Introduction to calculus |
| title_auth | Introduction to calculus |
| title_exact_search | Introduction to calculus |
| title_exact_search_txtP | Introduction to calculus |
| title_full | Introduction to calculus by Kazimierz Kuratowski |
| title_fullStr | Introduction to calculus by Kazimierz Kuratowski |
| title_full_unstemmed | Introduction to calculus by Kazimierz Kuratowski |
| title_short | Introduction to calculus |
| title_sort | introduction to calculus |
| topic | Calculo (matematica) - elementar larpcal Calculus Zahlentheorie (DE-588)4067277-3 gnd Reelle Analysis (DE-588)4627581-2 gnd Analysis (DE-588)4001865-9 gnd Infinitesimalrechnung (DE-588)4072798-1 gnd |
| topic_facet | Calculo (matematica) - elementar Calculus Zahlentheorie Reelle Analysis Analysis Infinitesimalrechnung Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015226382&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV001888024 |
| work_keys_str_mv | AT kuratowskikazimierz introductiontocalculus |