Table of Contents 

 

Unit One: Ordinary Differential Equations - Part One 

Introduction - Unit One 

Chapter 1:                First-Order Differential Equations 

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 

Chapter 2:                Models Containing ODEs 

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 

Chapter 3:                Methods for Solving First-Order ODEs 

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 

Chapter 4:                Numeric Methods for Solving First-Order ODEs 

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 

Chapter 5:                Second-Order Differential Equations 

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 

Chapter 6:                The Laplace Transform 

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 

Unit Two: Infinite Series 

Introduction - Unit Two 

Chapter 7:                Sequences and Series of Numbers 

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 

Chapter 8:                Sequences and Series of Functions 

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 

Chapter 9:                Power Series 

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 

Chapter 10:              Fourier Series 

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 

Chapter 11:             Asymptotic Series 

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 

Unit Three: Ordinary Differential Equations - Part Two 

Introduction - Unit Three 

Chapter 12:              Systems of First-Order ODEs 

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 

Chapter 13:              Numerical Techniques: First-Order Systems and Second-Order ODEs 

Introduction - Chapter 13 

Section 13.1 

Runge-Kutta-Nystrom 

Section 13.2 

rk4 for First-Order Systems 

Review Exercises - Chapter 13 

Chapter 14:              Series Solutions 

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 

Chapter 15:              Boundary Value Problems 

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 

Chapter 16:              The Eigenvalue Problem 

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 

Unit Four: Vector Calculus 

Introduction - Unit Four 

Chapter 17:              Space Curves 

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 

Chapter 18:              The Gradient Vector 

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 

Chapter 19:              Line Integrals in the Plane 

Introduction - Chapter 19 

Section 19.1 

Work and Circulation 

Section 19.2  

Flux through a Plane Curve 

Review Exercises - Chapter 19 

Chapter 20:              Additional Vector Differential Operators 

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 

Chapter 21:              Integration 

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 

Chapter 22:              NonCartesian Coordinates 

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 

Chapter 23:              Miscellaneous Results 

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 

Unit Five: Boundary Value Problems for PDEs 

Introduction - Unit Five 

Chapter 24:              Wave Equation 

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 

Chapter 25:              Heat Equation 

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 

Chapter 26:              Laplace's Equation in a Rectangle 

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 

Chapter 27:              Nonhomogeneous Boundary Value Problems 

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 

Chapter 28:              Time-Dependent Problems in Two Spatial Dimensions 

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 

Chapter 29:              Separation of Variables in NonCartesian Coordinates 

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 

Chapter 30:              Transform Techniques 

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 

Unit Six: Matrix Algebra 

Introduction - Unit Six 

Chapter 31:              Vectors as Arrows 

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 

Chapter 32:              Change of Coordinates 

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 

Chapter 33:              Matrix Computations 

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 

Chapter 34:              Matrix Factorization 

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 

Unit Seven: Complex Variables 

Introduction - Unit Seven 

Chapter 35:              Fundamentals 

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 

Chapter 36:              Applications 

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 

Unit Eight: Numerical Methods 

Introduction - Unit Eight 

Chapter 37:             Equations in One Variable - Preliminaries 

Introduction - Chapter 37 

Section 37.1 

Accuracy and Errors 

Section 37.2 

Rate of Convergence 

Review Exercises - Chapter 37 

Chapter 38:             Equations in One Variable - Methods 

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 

Chapter 39:             Systems of Equations 

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 

Chapter 40:             Interpolation 

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 

Chapter 41:             Approximation of Continuous Functions 

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 

Chapter 42:             Numeric Differentiation 

Introduction - Chapter 42 

Section 42.1 

Basic Formulas 

Section 42.2 

Richardson Extrapolation 

Review Exercises - Chapter 42 

Chapter 43:             Numeric Integration 

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 

Chapter 44:             Approximation of Discrete Data 

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 

Chapter 45:             Numerical Calculation of Eigenvalues 

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 

Unit Nine: Calculus of Variations 

Introduction - Unit Nine 

Chapter 46:             Basic Formalisms 

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 

Chapter 47:             Constrained Optimization 

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 

Chapter 48:             Variational Mechanics 

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