1 Preliminaries

1.1 The Logic of Quantifiers

1.1.1 Rules of Quantifiers

1.1.2 Examples

1.1.3 Exercises

1.2 Infinite Sets

1.2.1 Countable Sets

1.2.2 Uncountable Sets

1.2.3 Exercises

1.3 Proofs

1.3.1 How to Discover Proofs

1.3.2 How to Understand Proofs

1.4 The Rational Number System

1.5 The Axiom of Choice

2 Construction of the Real Number System

2.1 Cauchy Sequences

2.1.1 Motivation

2.1.2 The Definition

2.1.3 Exercises

2.2 The Reals as an Ordered Field

2.2.1 Defining Arithmetic

2.2.2 The Field Axioms

2.2.3 Order

2.2.4 Exercises

2.3 Limits and Completeness

2.3.1 Proof of Completeness

2.3.2 Square Roots

2.3.3 Exercises

2.4 Other Versions and Visions

2.4.1 Infinite Decimal Expansions

2.4.2 Dedekind Cuts

2.4.3 Non-Standard Analysis

2.4.4 Constructive Analysis

2.4.5 Exercises

2.5 Summary

3 Topology of the Real Line

3.1 The Theory of Limits

3.1.1 Limits,Sups,and Infs

3.1.2 Limit Points

3.1.3 Exercises

3.2 Open Sets and Closed Sets

3.2.1 Open Sets

3.2.2 Closed Sets

3.2.3 Exercises

3.3 Compact Sets

3.3.1 Exercises

3.4 Summary

4 Continuous Functions

4.1 Concepts of Continuity

4.1.1 Definitions

4.1.2 Limits of Functions and Limits of Sequences

4.1.3 Inverse Images of Open Sets

4.1.4 Related Definitions

4.1.5 Exercises

4.2 Properties of Continuous Functions

4.2.1 Basic Properties

4.2.2 Continuous Functions on Compact Domains

4.2.3 Monotone Functions

4.2.4 Exercises

4.3 Summary

5 Differential Calculus

5.1 Concepts of the Derivative

5.1.1 Equivalent Definitions

5.1.2 Continuity and Continuous Differentiability

5.1.3 Exercises

5.2 Properties of the Derivative

5.2.1 Local Properties

5.2.2 Intermediate Value and Mean Value Theorems

5.2.3 Global Properties

5.2.4 Exercises

5.3 The Calculus of Derivatives

5.3.1 Product and Quotient Rules

5.3.2 The Chain Rule

5.3.3 Inverse Function Theorem

5.3.4 Exercises

5.4 Higher Derivatives and Taylor's Theorem

5.4.1 Interpretations of the Second Derivative

5.4.2 Taylor's Theorem

5.4.3 L'H？pital's Rule

5.4.4 Lagrange Remainder Formula

5.4.5 Orders of Zeros

5.4.6 Exercises

5.5 Summary

6 Integral Calculus

6.1 Integrals of Continuous Functions

6.1.1 Existence of the Integral

6.1.2 Fundamental Theorems of Calculus

6.1.3 Useful Integration Formulas

6.1.4 Numerical Integration

6.1.5 Exercises

6.2 The Riemann Integral

6.2.1 Definition of the Integral

6.2.2 Elementary Properties of the Integral

6.2.3 Functions with a Countable Number of Discon-tinuities

6.2.4 Exercises

6.3 Improper Integrals

6.3.1 Definitions and Examples

6.3.2 Exercises

6.4 Summary

7 Sequences and Series of Functions

7.1 Complex Numbers

7.1.1 Basic Properties of C

7.1.2 Complex-Valued Functions

7.1.3 Exercises

7.2 Numerical Series and Sequences

7.2.1 Convergence and Absolute Convergence

7.2.2 Rearrangements

7.2.3 Summation by Parts

7.2.4 Exercises

7.3 Uniform Convergence

7.3.1 Uniform Limits and Continuity

7.3.2 Integration and Differentiation of Limits

7.3.3 Unrestricted Convergence

7.3.4 Exercises

7.4 Power Series

7.4.1 The Radius of Convergence

7.4.2 Analytic Continuation

7.4.3 Analytic Functions on Complex Domains

7.4.4 Closure Properties of Analytic Functions

7.4.5 Exercises

7.5 Approximation by Polynomials

7.5.1 Lagrange Interpolation

7.5.2 Convolutions and Approximate Identities

7.5.3 The Weierstrass Approximation Theorem

7.5.4 Approximating Derivatives

7.5.5 Exercises

7.6 Equicontinuity

7.6.1 The Definition of Equicontinuity

7.6.2 The Arzela-Ascoli Theorem

7.6.3 Exercises

7.7 Summary

8 Transcendental Functions

8.1 The Exponential and Logarithm

8.1.1 Five Equivalent Definitions

8.1.2 Exponential Glue and Blip Functions

8.1.3 Functions with Prescribed Taylor Expansions

8.1.4 Exercises

8.2 Trigonometric Functions

8.2.1 Definition of Sine and Cosine

8.2.2 Relationship Between Sines,Cosines,and Com-plex Exponentials

8.2.3 Exercises

8.3 Summary

9 Euclidean Space and Metric Spaces

9.1 Structures on Euclidean Space

9.1.1 Vector Space and Metric Space

9.1.2 Norm and Inner Product

9.1.3 The Complex Case

9.1.4 Exercises

9.2 Topology of Metric Spaces

9.2.1 Open Sets

9.2.2 Limits and Closed Sets

9.2.3 Completeness

9.2.4 Compactness

9.2.5 Exercises

9.3 Continuous Functions on Metric Spaces

9.3.1 Three Equivalent Definitions

9.3.2 Continuous Functions on Compact Domains

9.3.3 Connectedness

9.3.4 The Contractive Mapping Principle

9.3.5 The Stone-Weierstrass Theorem

9.3.6 Nowhere Differentiable Functions,and Worse

9.3.7 Exercises

9.4 Summary

10 Differential Calculus in Euclidean Space

10.1 The Differehtial

10.1.1 Definition of Differentiability

10.1.2 Partial Derivatives

10.1.3 The Chain Rule

10.1.4 Differentiation of Integrals

10.1.5 Exercises

10.2 Higher Derivatives

10.2.1 Equality of Mixed Partials

10.2.2 Local Extrema

10.2.3 Taylor Expansions

10.2.4 Exercises

10. 3 Summary

11 Ordinary Differential Equations

11.1 Existence and Uniqueness

11.1.1 Motivation

11.1.2 Picard Iteration

11.1.3 Linear Equations

11.1.4 Local Existence and Uniqueness

11.1.5 Higher Order Equations

11.1.6 Exercises

11.2 Other Methods of Solution

11.2.1 Difference Equation Approximation

11.2.2 Peano Existence Theorem

11.2.3 Power-Series Solutions

11.2.4 Exercises

11.3 Vector Fields and Flows

11.3.1 Integral Curves

11.3.2 Hamiltonian Mechanics

11.3.3 First-Order Linear P.D.E.'s

11.3.4 Exercises

11.4 Summary

12 Fourier Series

12.1 Origins of Fourier Series

12.1.1 Fourier Series Solutions of P.D.E.'s

12.1.2 Spectral Theory

12.1.3 Harmonic Analysis

12.1.4 Exercises

12.2 Convergence of Fourier Series

12.2.1 Uniform Convergence for C1 Functions

12.2.2 Summability of Fourier Series

12.2.3 Convergence in the Mean

12.2.4 Divergence and Gibb's Phenomenon

12.2.5 Solution of the Heat Equation

12.2.6 Exercises

12.3 Summary

13 Implicit Functions,Curves,and Surfaces

13.1 The Implicit Function Theorem

13.1.1 Statement of the Theorem

13.1.2 The Proof

13.1.3 Exercises

13.2 Curves and Surfaces

13.2.1 Motivation and Examples

13.2.2 Immersions and Embeddings

13.2.3 Parametric Description of Surfaces

13.2.4 Implicit Description of Surfaces

13.2.5 Exercises

13.3 Maxima and Minima on Surfaces

13.3.1 Lagrange Multipliers

13.3.2 A Second Derivative Test

13.3.3 Exercises

13.4 Arc Length

13.4.1 Rectifiable Curves

13.4.2 The Integral Formula for Arc Length

13.4.3 Arc Length Parameterization

13.4.4 Exercises

13.5 Summary

14 The Lebesgue Integral

14.1 The Concept of Measure

14.1.1 Motivation

14.1.2 Properties of Length

14.1.3 Measurable Sets

14.1.4 Basic Properties of Measures

14.1.5 A Formula for Lebesgue Measure

14.1.6 Other Examples of Measures

14.1.7 Exercises

14.2 Proof of Existence of Measures

14.2.1 Outer Measures

14.2.2 Metric Outer Measure

14.2.3 Hausdorff Measures

14.2.4 Exercises

14.3 The Integral

14.3.1 Non-negative Measurable Functions

14.3.2 The Monotone Convergence Theorem

14.3.3 Integrable Functions

14.3.4 Almost Everywhere

14.3.5 Exercises

14.4 The Lebesgue Spaces L1 and L2

14.4.1 L1 as a Banach Space

14.4.2 L2 as a Hilbert Space

14.4.3 Fourier Series for L2 Functions

14.4.4 Exercises

14.5 Summary

15 Multiple Integrals

15.1 Interchange of Integrals

15.1.1 Integrals of Continuous Functions

15.1.2 Fubini's Theorem

15.1.3 The Monotone Class Lemma

15.1.4 Exercises

15.2 Change of Variable in Multiple Integrals

15.2.1 Determinants and Volume

15.2.2 The Jacobian Factor

15.2.3 Polar Coordinates

15.2.4 Change of Variable for Lebesgue Integrals

15.2.5 Exercises

15.3 Summary

Index