Introduction - Chapter 1
Section 1.1	Introduction
Section 1.2	Terminology
Section 1.3	The Direction Field
Section 1.4	Picard Iteration
Section 1.5	Existence and Uniqueness for the Initial Value Problem
Review Exercises - Chapter 1

Introduction - Chapter 2
Section 2.1	Exponential Growth and Decay
Section 2.2	Logistic Models
Section 2.3	Mixing Tank Problems - Constant and Variable Volumes
Section 2.4	Newton's Law of Cooling
Review Exercises - Chapter 2

Introduction - Chapter 3
Section 3.1	Separation of Variables
Section 3.2	Equations with Homogeneous Coefficients
Section 3.3	Exact Equations
Section 3.4	Integrating Factors and the First-Order Linear Equation
Section 3.5	Variation of Parameters and the First-Order Linear Equation
Section 3.6	The Bernoulli Equation
Review Exercises - Chapter 3

Introduction - Chapter 4
Section 4.1	Fixed-Step Methods - Order and Error
Section 4.2	The Euler Method
Section 4.3	Taylor Series Methods
Section 4.4	Runge-Kutta Methods
Section 4.5	Adams-Bashforth Multistep Methods
Section 4.6	Adams-Moulton Predictor-Corrector Methods
Section 4.7	Milne's Method
Section 4.8	rkf45, the Runge-Kutta-Fehlberg Method
Review Exercises - Chapter 4

Introduction - Chapter 5
Section 5.1	Springs 'n' Things
Section 5.2	The Initial Value Problem
Section 5.3	Overview of the Solution Process
Section 5.4	Linear Dependence and Independence
Section 5.5	Free Undamped Motion
Section 5.6	Free Damped Motion
Section 5.7	Reduction of Order and Higher-Order Equations
Section 5.8	The Bobbing Cylinder
Section 5.9	Forced Motion and Variation of Parameters
Section 5.10	Forced Motion and Undetermined Coefficients
Section 5.11	Resonance
Section 5.12	The Euler Equation
Section 5.13	The Green's Function Technique for IVPs
Review Exercises - Chapter 5

Introduction - Chapter 6
Section 6.1	Definition and Examples
Section 6.2	Transform of Derivatives
Section 6.3	First Shifting Law
Section 6.4	Operational Laws
Section 6.5	Heaviside Functions and the Second Shifting Law
Section 6.6	Pulses and the Third Shifting Law
Section 6.7	Transforms of Periodic Functions
Section 6.8	Convolution and the Convolution Theorem
Section 6.9	Convolution Products by the Convolution Theorem
Section 6.10	The Dirac Delta Function
Section 6.11	Transfer Function, Fundamental Solution, and the Green's Function
Review Exercises - Chapter 6

Introduction - Chapter 7
Section 7.1	Sequences
Section 7.2	Infinite Series
Section 7.3	Series with Positive Terms
Section 7.4	Series with Both Negative and Positive Terms
Review Exercises - Chapter 7

Introduction - Chapter 8
Section 8.1	Sequences of Functions
Section 8.2	Pointwise Convergence
Section 8.3	Uniform Convergence
Section 8.4	Convergence in the Mean
Section 8.5	Series of Functions
Review Exercises - Chapter 8

Introduction - Chapter 9
Section 9.1	Taylor Polynomials
Section 9.2	Taylor Series
Section 9.3	Termwise Operations on Taylor Series
Review Exercises - Chapter 9

Introduction - Chapter 10
Section 10.1	General Formalism
Section 10.2	Termwise Integration and Differentiation
Section 10.3	Odd and Even Functions and Their Fourier Series
Section 10.4	Sine Series and Cosine Series
Section 10.5	Periodically Driven Damped Oscillator
Section 10.6	Optimizing Property of Fourier Series
Section 10.7	Fourier-Legendre Series
Review Exercises - Chapter 10

Introduction - Chapter 11
Section 11.1	Computing with Divergent Series
Section 11.2	Definitions
Section 11.3	Operations with Asymptotic Series
Review Exercises - Chapter 11

Introduction - Chapter 12
Section 12.1	Mixing Tanks - Closed Systems
Section 12.2	Mixing Tanks - Open Systems
Section 12.3	Vector Structure of Solutions
Section 12.4	Determinants and Cramer's Rule
Section 12.5	Solving Linear Algebraic Equations
Section 12.6	Homogeneous Equations and the Null Space
Section 12.7	Inverses
Section 12.8	Vectors and the Laplace Transform
Section 12.9	The Matrix Exponential
Section 12.10	Eigenvalues and Eigenvectors
Section 12.11	Solutions by Eigenvalues and Eigenvectors
Section 12.12	Finding Eigenvalues and Eigenvectors
Section 12.13	System versus Second-Order ODE
Section 12.14	Complex Eigenvalues
Section 12.15	The Deficient Case
Section 12.16	Diagonalization and Uncoupling
Section 12.17	A Coupled Linear Oscillator
Section 12.18	Nonhomogeneous Systems and Variation of Parameters
Section 12.19	Phase Portraits
Section 12.20	Stability
Section 12.21	Nonlinear Systems
Section 12.22	Linearization
Section 12.23	The Nonlinear Pendulum
Review Exercises - Chapter 12

Introduction - Chapter 13
Section 13.1	Runge-Kutta-Nystrom
Section 13.2	rk4 for First-Order Systems
Review Exercises - Chapter 13

Introduction - Chapter 14
Section 14.1	Power Series
Section 14.2	Asymptotic Solutions
Section 14.3	Perturbation Solution of an Algebraic Equation
Section 14.4	Poincare Perturbation Solution for Differential Equations
Section 14.5	The Nonlinear Spring and Lindstedt's Method
Section 14.6	The Method of Krylov and Bogoliubov
Review Exercises - Chapter 14

Introduction - Chapter 15
Section 15.1	Analytic Solutions
Section 15.2	Numeric Solutions
Section 15.3	Least-Squares, Rayleigh-Ritz, Galerkin, and Collocation Techniques
Section 15.4	Finite Elements
Review Exercises - Chapter 15

Introduction - Chapter 16
Section 16.1	Regular Sturm-Liouville Problems
Section 16.2	Bessel's Equation
Section 16.3	Legendre's Equation
Section 16.4	Solution by Finite Differences
Review Exercises - Chapter 16

Introduction - Chapter 17
Section 17.1	Curves and Their Tangent Vectors
Section 17.2	Arc Length
Section 17.3	Curvature
Section 17.4	Principal Normal and Binormal Vectors
Section 17.5	Resolution of R'' into Tanential and Normal Components
Section 17.6	Applications to Dynamics
Review Exercises - Chapter 17

Introduction - Chapter 18
Section 18.1	Visualizing Vector Fields and Their Flows
Section 18.2	The Directional Derivative and Gradient Vector
Section 18.3	Properties of the Gradient Vector
Section 18.4	Lagrange Multipliers
Section 18.5	Conservative Forces and the Scalar Potential
Review Exercises - Chapter 18

Introduction - Chapter 19
Section 19.1	Work and Circulation
Section 19.2	Flux through a Plane Curve
Review Exercises - Chapter 19

Introduction - Chapter 20
Section 20.1	Divergence and Its Meaning
Section 20.2	Curl and Its Meaning
Section 20.3	Products - One f and Two Operands
Section 20.4	Products - Two f's and One Operand
Review Exercises - Chapter 20

Introduction - Chapter 21
Section 21.1	Surface Area
Section 21.2	Surface Integrals and Surface Flux
Section 21.3	The Divergence Theorem and the Theorems of Green and Stokes
Section 21.4	Green's Theorem
Section 21.5	Conservative, Solenoidal, and Irrotational Fields
Section 21.6	Integral Equivalents of div, grad, and curl
Review Exercises - Chapter 21

Introduction - Chapter 22
Section 22.1	Mappings and Changes of Coordinates
Section 22.2	Vector Operators in Polar Coordinates
Section 22.3	Vector Operators in Cylindrical and Spherical Coordinates
Review Exercises - Chapter 22

Introduction - Chapter 23
Section 23.1	Gauss' Theorem
Section 23.2	Surface Area for Parametrically Given Surfaces
Section 23.3	The Equation of Continuity
Section 23.4	Green's Identities
Review Exercises - Chapter 23

Introduction - Chapter 24
Section 24.1	The Plucked String
Section 24.2	The Struck String
Section 24.3	D'Alembert's Solution
Section 24.4	Derivation of the Wave Equation
Section 24.5	Longitudinal Vibrations in an Elastic Rod
Section 24.6	Finite-Difference Solution of the One-Dimensional Wave Equation
Review Exercises - Chapter 24

Introduction - Chapter 25
Section 25.1	One-Dimensional Heat Diffusion
Section 25.2	Derivation of the One-Dimensional Heat Equation
Section 25.3	Heat Flow in a Rod with Insulated Ends
Section 25.4	Finite-Difference Solution of the One-Dimensional Heat Equation
Review Exercises - Chapter 25

Introduction - Chapter 26
Section 26.1	Nonzero Temperature on the Bottom Edge
Section 26.2	Nonzero Temperature on the Top Edge
Section 26.3	Nonzero Temperature on the Left Edge
Section 26.4	Finite-Difference Solution of Laplace's Equation
Review Exercises - Chapter 26

Introduction - Chapter 27
Section 27.1	One-Dimensional Heat Equation with Different Endpoint Temperatures
Section 27.2	One-Dimensional Heat Equation with Time-Varying Endpoint Temperatures
Review Exercises - Chapter 27

Introduction - Chapter 28
Section 28.1	Oscillations of a Rectangular Membrane
Section 28.2	Time-Varying Temperatures in a Rectangular Plate
Review Exercises - Chapter 28

Introduction - Chapter 29
Section 29.1	Laplace's Equation in a Disk
Section 29.2	Laplace's Equation in a Cylinder
Section 29.3	The Circular Drumhead
Section 29.4	Laplace's Equation in a Sphere
Section 29.5	The Spherical Dielectric
Review Exercises - Chapter 29

Introduction - Chapter 30
Section 30.1	Solution by Laplace Transform
Section 30.2	The Fourier Integral Theorem
Section 30.3	The Fourier Transform
Section 30.4	Wave Equation on the Infinite String - Solution by Fourier Transform
Section 30.5	Heat Equation on the Infinite Rod - Solution by Fourier Transform
Section 30.6	Laplace's Equation on the Infinite Strip - Solution by Fourier Transform
Section 30.7	The Fourier Sine Transform
Section 30.8	The Fourier Cosine Transform
Review Exercises - Chapter 30

Introduction - Chapter 31
Section 31.1	The Algebra and Geometry of Vectors
Section 31.2	Inner and Dot Products
Section 31.3	The Cross-Product
Review Exercises - Chapter 31

Introduction - Chapter 32
Section 32.1	Change of Basis
Section 32.2	Rotations and Orthogonal Matrices
Section 32.3	Change of Coordinates
Section 32.4	Reciprocal Bases and Gradient Vectors
Section 32.5	Gradient Vectors and the Covariant Transformation Law
Review Exercises - Chapter 32

Introduction - Chapter 33
Section 33.1	Summary
Section 33.2	Projections
Section 33.3	The Gram-Schmidt Orthogonalization Process
Section 33.4	Quadratic Forms
Section 33.5	Vector and Matrix Norms
Section 33.6	Least Squares
Review Exercises - Chapter 33

Introduction - Chapter 34
Section 34.1	LU Decomposition
Section 34.2	PJP-1 and Jordan Canonical Form
Section 34.3	QR Decomposition
Section 34.4	QR Algorithm for Finding Eigenvalues
Section 34.5	SVD, The Singular Value Decomposition
Section 34.6	Minimum-Length Least-Squares Solution, and the Pseudoinverse
Review Exercises - Chapter 34

Introduction - Chapter 35
Section 35.1	Complex Numbers
Section 35.2	The Function w = f(z) = z2
Section 35.3	The Function w = f(z) = z3
Section 35.4	The Exponential Function
Section 35.5	The Complex Logarithm
Section 35.6	Complex Exponents
Section 35.7	Trigonometric and Hyperbolic Functions
Section 35.8	Inverses of Trigonometric and Hyperbolic Functions
Section 35.9	Differentiation and the Cauchy-Riemann Equations
Section 35.10	Analytic and Harmonic Functions
Section 35.11	Integration
Section 35.12	Series in Powers of z
Section 35.13	The Calculus of Residues
Review Exercises - Chapter 35

Introduction - Chapter 36
Section 36.1	Evaluation of Integrals
Section 36.2	The Laplace Transform
Section 36.3	Fourier Series and the Fourier Transform
Section 36.4	The Root Locus
Section 36.5	The Nyquist Stability Criterion
Section 36.6	Conformal Mapping
Section 36.7	The Joukowski Map
Section 36.8	Solving the Dirichlet Problem by Conformal Mapping
Section 36.9	Planar Fluid Flow
Section 36.10	Conformal Mapping of Elementary Flows
Review Exercises - Chapter 36

Introduction - Chapter 37
Section 37.1	Accuracy and Errors
Section 37.2	Rate of Convergence
Review Exercises - Chapter 37

Introduction - Chapter 38
Section 38.1	Fixed-Point Iteration
Section 38.2	The Bisection Method
Section 38.3	Newton-Raphson Iteration
Section 38.4	The Secant Method
Section 38.5	Muller's Method
Review Exercises - Chapter 38

Introduction - Chapter 39
Section 39.1	Gaussian Arithmetic
Section 39.2	Condition Numbers
Section 39.3	Iterative Improvement
Section 39.4	The Method of Jacobi
Section 39.5	Gauss-Seidel Iteration
Section 39.6	Relaxation and SOR
Section 39.7	Iterative Methods for Nonlinear Systems
Section 39.8	Newton's Iteration for Nonlinear Systems
Review Exercises - Chapter 39

Introduction - Chapter 40
Section 40.1	Lagrange Interpolation
Section 40.2	Divided Differences
Section 40.3	Chebyshev Interpolation
Section 40.4	Spline Interpolation
Section 40.5	Bezier Curves
Review Exercises - Chapter 40

Introduction - Chapter 41
Section 41.1	Least-Squares Approximation
Section 41.2	Pade Approximations
Section 41.3	Chebyshev Approximation
Section 41.4	Chebyshev-Pade and Minimax Approximations
Review Exercises - Chapter 41

Introduction - Chapter 42
Section 42.1	Basic Formulas
Section 42.2	Richardson Extrapolation
Review Exercises - Chapter 42

Introduction - Chapter 43
Section 43.1	Methods from Elementary Calculus
Section 43.2	Recursive Trapezoid Rule and Romberg Integration
Section 43.3	Gauss-Legendre Quadrature
Section 43.4	Adaptive Quadrature
Section 43.5	Iterated Integrals
Review Exercises - Chapter 43

Introduction - Chapter 44
Section 44.1	Least-Squares Regression Line
Section 44.2	The General Linear Model
Section 44.3	The Role of Orthogonality
Section 44.4	Nonlinear Least Squares
Review Exercises - Chapter 44

Introduction - Chapter 45
Section 45.1	Power Methods
Section 45.2	Householder Reflections
Section 45.3	QR Decomposition via Householder Reflections
Section 45.4	Upper Hessenberg Form, Givens Rotations, and the Shifted QR-Algorithm
Section 45.5	The Generalized Eigenvalue Problem
Review Exercises - Chapter 45

Introduction - Chapter 46
Section 46.1	Motivational Examples
Section 46.2	Direct Methods
Section 46.3	The Euler-Lagrange Equation
Section 46.4	First Integrals
Section 46.5	Derivation of the Euler-lagrange Equation
Section 46.6	Transversality Conditions
Section 46.7	Derivation of the Transversality Conditions
Section 46.8	Three Generalizations
Review Exercises - Chapter 46

Introduction - Chapter 47
Section 47.1	Application of Lagrange Multipliers
Section 47.2	Queen Dido's Problem
Section 47.3	Isoperimetric Problems
Section 47.4	The Hanging Chain
Section 47.5	A Variable-Endpoint Problem
Section 47.6	Differential Constraints
Review Exercises - Chapter 47

Introduction - Chapter 48
Section 48.1	Hamilton's Principle
Section 48.2	The Simple Pendulum
Section 48.3	A Compound Pendulum
Section 48.4	The Spherical Pendulum
Section 48.5	Pendulum with Oscillating Support
Section 48.6	Legendre and Extended Legendre Transformations
Section 48.7	Hamilton's Canonical Equations
Review Exercises - Chapter 48