Foundations of Infinitesimal Calculus

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 Chapter 1: Numbers
    1.1 Field Axioms
    1.2 Order Axioms
    1.3 The Completeness Axiom
    1.4 Small, Medium and Large Numbers

 Chapter 2: Functional Identities
    2.1 Specific Functional Identities
    2.2 General Functional Identities
    2.3 The Function Extension Axiom
    2.4 Additive Functions
    2.5 The Motion of a Pendulum

 Chapter 3: The Theory of Limits
    3.1 Plain Limits
    3.2 Function Limits
    3.3 Computation of Limits

 Chapter 4: Continuous Functions
    4.1 Uniform Continuity
    4.2 The Extreme Value Theorem
    4.3 Bolzano's Intermediate Value Theorem

 Chapter 5: The Theory of Derivatives
    5.1 The Fundamental Theorem: Part 1
    5.2 Derivatives, Epsilons and Deltas
    5.3 Smoothness -> Continuity of Function and Derivative
    5.4 Rules -> Smoothness
    5.5 The Increment and Increasing
    5.6 Inverse Functions and Derivatives

 Chapter 6: Pointwise Derivatives
    6.1 Pointwise Limits
    6.2 Pointwise Derivatives
    6.3 Pointwise Derivatives Aren't Enough for Inverses

 Chapter 7: The Mean Value Theorem
    7.1 The Mean Value Theorem
    7.2 Darboux's Theorem
    7.3 Continuous Pointwise Derivatives are Uniform

 Chapter 8: Higher Order Derivatives
    8.1 Taylor's Formula and Bending
    8.2 Symmetric Differences and Taylor's Formula
    8.3 Approximation of Second Derivatives
    8.4 The General Taylor Small Oh Formula
    8.5 Direct Interpretation of Higher Order Derivatives

 Chapter 9: Basic Theory of the Definite Integral
    9.1 Existence of the Integral
    9.2 You Can't Always Integrate Discontinuous Functions
    9.3 Fundamental Theorem: Part 2
    9.4 Improper Integrals

 Chapter 10: Derivatives of Multivariable Functions

 Chapter 11: Theory of Initial Value Problems
    11.1 Existence and Uniqueness of Solutions
    11.2 Local Linearization of Dynamical Systems
    11.3 Attraction and Repulsion
    11.4 Stable Limit Cycles

 Chapter 12: The Theory of Power Series
    12.1 Uniformly Convergent Series
    12.2 Robinson's Sequential Lemma
    12.3 Integration of Series
    12.4 Radius of Convergence
    12.5 Calculus of Power Series

 Chapter 13: The Theory of Fourier Series
    13.1 Computation of Fourier Series
    13.2 Convergence for Piecewise Smooth Functions
    13.3 Uniform Convergence for Continuous Piecewise Smooth Functions
    13.4 Integration of Fourier Series

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