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袁萌 陈启清 6月20日
附件:
Mathematical Background: Foundations of Infinitesimal Calculus second edition
Contents（章节的目录）
Part 1 Numbers and Functions
Chapter 1. Numbers 3
1.1 Field Axioms 3
1.2 Order Axioms 6
1.3 The Completeness Axiom 7
1.4 Small, Medium and Large Numbers 9
Chapter 2. Functional Identities 17
2.1 Specific Functional Identities 17
2.2 General Functional Identities 18
2.3 The Function Extension Axiom 21
2.4 Additive Functions 24 2.5 The Motion of a Pendulum 26
Part 2 Limits
Chapter 3. The Theory of Limits 31
3.1 Plain Limits 32
3.2 Function Limits 34
3.3 Computation of Limits 37
Chapter 4. Continuous Functions 43
4.1 Uniform Continuity 43 4.2 The Extreme Value Theorem 44
iii
iv Contents
4.3 Bolzano’s Intermediate Value Theorem 46
Part 3 1 Variable Differentiation
Chapter 5. The Theory of Derivatives 49
5.1 The Fundamental Theorem:
Part 1 49
5.1.1 Rigorous Infinitesimal Justification 52
5.1.2 Rigorous Limit Justification 53
5.2 Derivatives, Epsilons and Deltas 53
5.3 Smoothness ⇒ Continuity of Function and Derivative 54
5.4 Rules ⇒ Smoothness 56 5.5 The Increment and Increasing 57
5.6 Inverse Functions and Derivatives 58
Chapter 6. Pointwise Derivatives 69
6.1 Pointwise Limits 69
6.2 Pointwise Derivatives 72 6.3 Pointwise Derivatives Aren’t Enough for Inverses 76
Chapter 7. The Mean Value Theorem 79
7.1 The Mean Value Theorem 79
7.2 Darboux’s Theorem 83 7.3 Continuous Pointwise Derivatives are Uniform 85
Chapter 8. Higher Order Derivatives 87
8.1 Taylor’s Formula and Bending 87
8.2 Symmetric Differences and Taylor’s Formula 89
8.3 Approximation of Second Derivatives 91
8.4 The General Taylor Small Oh Formula 92
8.4.1 The Converse of Taylor’s Theorem 95
8.5 Direct Interpretation of Higher Order Derivatives 98
8.5.1 Basic Theory of Interpolation 99
8.5.2 Interpolation where f is Smooth 101
8.5.3 Smoothness From Differences 102
Part 4 Integration
Chapter 9. Basic Theory of the Definite Integral 109
9.1 Existence of the Integral 110
Contents v
9.2 You Can’t Always Integrate Discontinuous Functions 114
9.3 Fundamental Theorem: Part 2 116 9.4
Improper Integrals 119
9.4.1 Comparison of Improper Integrals 121
9.4.2 A Finite Funnel with Infinite Area? 123
Part 5 Multivariable Differentiation
Chapter 10. Derivatives of Multivariable Functions 127 Part 6 Differential Equations Chapter 11. Theory of Initial Value Problems 131
11.1 Existence and Uniqueness of Solutions 131 11.2 Local Linearization of Dynamical Systems 135
11.3 Attraction and Repulsion 141
11.4 Stable Limit Cycles 143 Part 7 Infinite Series Chapter 12. The Theory of Power Series 147
12.1 Uniformly Convergent Series 149
12.2 Robinson’s Sequential Lemma 151
12.3 Integration of Series 152
12.4 Radius of Convergence 154
12.5 Calculus of Power Series 156
Chapter 13. The Theory of Fourier Series 159
13.1 Computation of Fourier Series 160
13.2 Convergence for Piecewise Smooth Functions 167
13.3 Uniform Convergence for Continuous Piecewise Smooth Functions 173
13.4 Integration of Fourier Series 175
袁萌 陈启清 6月20日
附件:
Mathematical Background: Foundations of Infinitesimal Calculus second edition
Contents（章节的目录）
Part 1 Numbers and Functions
Chapter 1. Numbers 3
1.1 Field Axioms 3
1.2 Order Axioms 6
1.3 The Completeness Axiom 7
1.4 Small, Medium and Large Numbers 9
Chapter 2. Functional Identities 17
2.1 Specific Functional Identities 17
2.2 General Functional Identities 18
2.3 The Function Extension Axiom 21
2.4 Additive Functions 24 2.5 The Motion of a Pendulum 26
Part 2 Limits
Chapter 3. The Theory of Limits 31
3.1 Plain Limits 32
3.2 Function Limits 34
3.3 Computation of Limits 37
Chapter 4. Continuous Functions 43
4.1 Uniform Continuity 43 4.2 The Extreme Value Theorem 44
iii
iv Contents
4.3 Bolzano’s Intermediate Value Theorem 46
Part 3 1 Variable Differentiation
Chapter 5. The Theory of Derivatives 49
5.1 The Fundamental Theorem:
Part 1 49
5.1.1 Rigorous Infinitesimal Justification 52
5.1.2 Rigorous Limit Justification 53
5.2 Derivatives, Epsilons and Deltas 53
5.3 Smoothness ⇒ Continuity of Function and Derivative 54
5.4 Rules ⇒ Smoothness 56 5.5 The Increment and Increasing 57
5.6 Inverse Functions and Derivatives 58
Chapter 6. Pointwise Derivatives 69
6.1 Pointwise Limits 69
6.2 Pointwise Derivatives 72 6.3 Pointwise Derivatives Aren’t Enough for Inverses 76
Chapter 7. The Mean Value Theorem 79
7.1 The Mean Value Theorem 79
7.2 Darboux’s Theorem 83 7.3 Continuous Pointwise Derivatives are Uniform 85
Chapter 8. Higher Order Derivatives 87
8.1 Taylor’s Formula and Bending 87
8.2 Symmetric Differences and Taylor’s Formula 89
8.3 Approximation of Second Derivatives 91
8.4 The General Taylor Small Oh Formula 92
8.4.1 The Converse of Taylor’s Theorem 95
8.5 Direct Interpretation of Higher Order Derivatives 98
8.5.1 Basic Theory of Interpolation 99
8.5.2 Interpolation where f is Smooth 101
8.5.3 Smoothness From Differences 102
Part 4 Integration
Chapter 9. Basic Theory of the Definite Integral 109
9.1 Existence of the Integral 110
Contents v
9.2 You Can’t Always Integrate Discontinuous Functions 114
9.3 Fundamental Theorem: Part 2 116 9.4
Improper Integrals 119
9.4.1 Comparison of Improper Integrals 121
9.4.2 A Finite Funnel with Infinite Area? 123
Part 5 Multivariable Differentiation
Chapter 10. Derivatives of Multivariable Functions 127 Part 6 Differential Equations Chapter 11. Theory of Initial Value Problems 131
11.1 Existence and Uniqueness of Solutions 131 11.2 Local Linearization of Dynamical Systems 135
11.3 Attraction and Repulsion 141
11.4 Stable Limit Cycles 143 Part 7 Infinite Series Chapter 12. The Theory of Power Series 147
12.1 Uniformly Convergent Series 149
12.2 Robinson’s Sequential Lemma 151
12.3 Integration of Series 152
12.4 Radius of Convergence 154
12.5 Calculus of Power Series 156
Chapter 13. The Theory of Fourier Series 159
13.1 Computation of Fourier Series 160
13.2 Convergence for Piecewise Smooth Functions 167
13.3 Uniform Convergence for Continuous Piecewise Smooth Functions 173
13.4 Integration of Fourier Series 175
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