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

 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

Unit Two: Infinite Series

 Introduction - Unit Two

 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

Unit Three: Ordinary Differential Equations - Part Two

 Introduction - Unit Three

 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

Unit Four: Vector Calculus

 Introduction - Unit Four

 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

Unit Five: Boundary Value Problems for PDEs

 Introduction - Unit Five

 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

Unit Six: Matrix Algebra

 Introduction - Unit Six

 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

Unit Seven: Complex Variables

 Introduction - Unit Seven

 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

Unit Eight: Numerical Methods

 Introduction - Unit Eight

 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

Unit Nine: Calculus of Variations

 Introduction - Unit Nine

 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