A University of Iowa professor is offering his textbook, *Mathematical Background: Foundations of Infinitesimal Calculus* from the university website. Keith Duncan Stroyan wrote this textbook for advanced undergraduate and beginning graduate students.

## Table of Contents

- 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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Infinitesimal Calculus