Serious mathematics, written with the reader in mind.
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Tentative Table of Contents

Advanced Topics in Calculus

(book in preparation)

John H. Hubbard and Barbara Burke Hubbard


Chapter 7: Differential Equations

7.0 Introduction

7.1 Differential Equations as Models

7.2 Qualitative Methods

7.3 Analytic Methods

7.4 Numerical Methods

7.5 Higher-order and Higher-dimensional Equations

7.6 The Fundamental Inequality

7.7 Existence and Uniqueness

7.8 Review Exercises

Chapter 8: Inner Products and Signal Processing

8.0 Introduction

8.1 Fields and Vector Spaces

8.2 Change of Basis

8.3 Real Inner products and Schwarz

8.4 Orthonormal bases and Gram-Schmidt

8.5 Orthogonal matrices and Gram-Schmidt

8.6 Complex Inner Products

8.7 Fourier Series

8.8 Fourier Convergence

8.9 Fast Fourier Transform

8.10 Orthogonal Polynomials and Gaussian Integration

8.11 Fourier Transforms

8.12 Fourier Transforms and Differential Equations

8.13 Signal Processing, Heisenberg Uncertainty Principle and the Sampling Theorem

8.14 Wavelets

8.15 Review Exercises

Chapter 9: Eigenvalues, Eigenvectors, and Linear Differential Equations

9.0 Introduction

9.1 Eigenvalues and Eigenvectors

9.2 Canonical Forms

9.3 Linear Differential Equations with Constant Coefficients

9.4 The Bifurcation Diagram for 2 x 2 Linear Systems

9.5 Nonhomogeneous Systems

9.6 Planar Vector Fields

9.7 Review Exercises

Chapter 10: Advanced Topics in Differential Forms

10.0 Introduction

10.1 Pullbacks byLinear Transformations

10.2 Pullbacks by Nonlinear Mappings and the Exterior Derivative

10.3 The Poincaré Lemma

10.4 The Brouwer Fixed Point Theorem

10.5 Review Exercises

Chapter 11: Inner Products, Forms, and Electromagnetism

11.0 Introduction

11.1 Maxwell's Equations in Terms of Forms

11.2 Inner Products on Forms

11.3 The Laplacian

11.4 Minkowski Space and the Lorentz Metric

11.5 The Star Operator in Minkowski Space

11.6 Some Electromagnetic Fields

11.7 The Conservation of Energy and Momentum

11.8 Review Exercises

Chapter 12: Introduction to Partial Differential Equations

12.0 Introduction

12.1 The Wave Equation

12.2 Solution of the Wave Equation

12.3 The Heat Equation

12.3 Review Exercises

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