Serious mathematics, written with the reader in mind.
[MsE] Matrix Editions
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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