Mathematical Background Foundations of Infinitesimal Calculus

Numbers and Functions......Page 7 1.1 Field Axioms......Page 9 1.2 Order Axioms......Page 12 1.3 The Completeness Axiom......Page 13 bers......Page 15 2.1 Speci c Functional Identities......Page 23 2.2 General Functional Identities......Page 24 2.3 The Function Extension Axiom......Page 27 2.4 Additive Functions......Page 30 2.5 The Motion of a Pendulum......Page 32 Limits......Page 35 The Theory of Limits......Page 37 3.1 Plain Limits......Page 38 3.2 Function Limits......Page 40 3.3 Computation of Limits......Page 43 4.1 Uniform Continuity......Page 49 4.2 The Extreme Value Theorem......Page 50 orem......Page 52 1 Variable Differentiation......Page 53 1......Page 55 5.2 Derivatives, Epsilons and Deltas......Page 59 tion and Derivative......Page 60 5.4 Rules Smoothness......Page 62 5.5 The Increment and Increasing......Page 63 5.6 Inverse Functions and Derivatives......Page 64 6.1 Pointwise Limits......Page 75 6.2 Pointwise Derivatives......Page 78 for Inverses......Page 82 7.1 The Mean Value Theorem......Page 85 7.2 Darboux's Theorem......Page 89 are Uniform......Page 91 8.1 Taylor's Formula and Bending......Page 93 lor's Formula......Page 95 tives......Page 97 mula......Page 98 der Derivatives......Page 104 Integration......Page 113 Basic Theory of the Definite Integral......Page 115 9.1 Existence of the Integral......Page 116 continuous Functions......Page 120 9.3 Fundamental Theorem: Part 2......Page 122 9.4 Improper Integrals......Page 125 Multivariable Di erentiation......Page 131 Derivatives of Multivariable Functions......Page 133 Differential Equations......Page 135 lutions......Page 137 Systems......Page 141 11.3 Attraction and Repulsion......Page 147 11.4 Stable Limit Cycles......Page 149 Infinite Series......Page 151 The Theory of Power Series......Page 153 12.1 Uniformly Convergent Series......Page 155 12.2 Robinson's Sequential Lemma......Page 157 12.3 Integration of Series......Page 158 12.4 Radius of Convergence......Page 160 12.5 Calculus of Power Series......Page 162 The Theory of Fourier Series......Page 165 13.1 Computation of Fourier Series......Page 166 Functions......Page 173 tions......Page 179 13.4 Integration of Fourier Series......Page 181 Numbers and Functions 7 Numbers 9 1.1 Field Axioms 9 1.2 Order Axioms 12 1.3 The Completeness Axiom 13 1.4 Small, Medium and Large Num- 15 bers 15 Functional Identities 23 2.1 Speci c Functional Identities 23 2.2 General Functional Identities 24 2.3 The Function Extension Axiom 27 2.4 Additive Functions 30 2.5 The Motion of a Pendulum 32 Limits 35 The Theory of Limits 37 3.1 Plain Limits 38 3.2 Function Limits 40 3.3 Computation of Limits 43 Continuous Functions 49 4.1 Uniform Continuity 49 4.2 The Extreme Value Theorem 50 4.3 Bolzano's Intermediate Value The- 52 orem 52 1 Variable Differentiation 53 The Theory of Derivatives 55 5.1 The Fundamental Theorem: Part 55 1 55 5.2 Derivatives, Epsilons and Deltas 59 5.3 Smoothness Continuity of Func- 60 tion and Derivative 60 5.4 Rules Smoothness 62 5.5 The Increment and Increasing 63 5.6 Inverse Functions and Derivatives 64 Pointwise Derivatives 75 6.1 Pointwise Limits 75 6.2 Pointwise Derivatives 78 6.3 Pointwise Derivatives Aren't Enough 82 for Inverses 82 The Mean Value Theorem 85 7.1 The Mean Value Theorem 85 7.2 Darboux's Theorem 89 7.3 Continuous Pointwise Derivatives 91 are Uniform 91 Higher Order Derivatives 93 8.1 Taylor's Formula and Bending 93 8.2 Symmetric Di erences and Tay- 95 lor's Formula 95 8.3 Approximation of Second Deriva- 97 tives 97 8.4 The General Taylor Small Oh For- 98 mula 98 8.5 Direct Interpretation of Higher Or- 104 der Derivatives 104 Integration 113 Basic Theory of the Definite Integral 115 9.1 Existence of the Integral 116 9.2 You Can't Always Integrate Dis- 120 continuous Functions 120 9.3 Fundamental Theorem: Part 2 122 9.4 Improper Integrals 125 Multivariable Di erentiation 131 Derivatives of Multivariable Functions 133 Differential Equations 135 Theory of Initial Value Problems 137 11.1 Existence and Uniqueness of So- 137 lutions 137 11.2 Local Linearization of Dynamical 141 Systems 141 11.3 Attraction and Repulsion 147 11.4 Stable Limit Cycles 149 Infinite Series 151 The Theory of Power Series 153 12.1 Uniformly Convergent Series 155 12.2 Robinson's Sequential Lemma 157 12.3 Integration of Series 158 12.4 Radius of Convergence 160 12.5 Calculus of Power Series 162 The Theory of Fourier Series 165 13.1 Computation of Fourier Series 166 13.2 Convergence for Piecewise Smooth 173 Functions 173 13.3 Uniform Convergence for Con- 179 tinuous Piecewise Smooth Func- 179 tions 179 13.4 Integration of Fourier Series 181