Part I - Ordinary Differential Equations
Chapter 1 - First Order Differential Equations
1.1 Preliminary Concepts
1.1.1 General and Particular Solutions
1.1.2 Implicitly Defined Solutions
1.1.3 Integral Curves
1.1.4 The Initial Value Problem
1.1.5 Direction Fields
1.2 Separable Equations
1.2.1 Some Applications of Separable Differential Equations
1.3 Linear Differential Equations
1.4 Exact Differential Equations
1.5 Integrating Factors
1.5.1 Separable Equations and Integrating Factors
1.5.2 Linear Equations and Integrating Factors
1.6 Homogeneous, Bernoulli and Riccati Equations
1.6.1 Homogeneous Differential Equations
1.6.2 The Bernoulli Equation
1.6.3 The Riccati Equation
1.7 Applications to Mechanics, Electrical Circuits and Orthogonal Trajectories
1.7.1 Mechanics
1.7.2 Electrical Circuits
1.7.3 Orthogonal Trajectories
1.8 Existence and Uniqueness for Solutions of Initial Value Problems
Chapter 2 - Linear Second Order Differential Equations
2.1 Preliminary Concepts
2.2 Theory of Solutions of y′′ + p(x)y′ + q(x)y = f(x)
2.2.1 The Homogeneous Equation y′′ + p(x)y′ + q(x) = 0
2.2.2 The Nonhomogeneous Equation y′′ + p(x)y′ + q(x)y = f(x)
2.3 Reduction of Order
2.4 The Constant Coefficient Homogeneous Linear Equation
2.4.1 Case 1 A2 - 4B > 0
2.4.2 Case 2 A2 - 4B = 0
2.4.3 Case 3 A2 - 4B < 0
2.4.4 An Alternative General Solution In the Complex Root Case
2.5 Euler's Equation
2.6 The Nonhomogeneous Equation y′′ + p(x)y′ + q(x)y = f(x)
2.6.1 The Method of Variation of Parameters
2.6.2 The Method of Undetermined Coefficients
2.6.3 The Principle of Superposition
2.6.4 Higher Order Differential equations
2.7 Application of Second Order Differential Equations to a Mechanical System
2.7.1 Unforced Motion
2.7.2 Forced Motion
2.7.3 Resonance
2.7.4 Beats
2.7.5 Analogy With An Electrical Circuit
Chapter 3 - The Laplace Transform
3.1 Definition and Basic Properties
3.2 Solution of Initial Value Problems Using the Laplace Transform
3.3 Shifting Theorems and the Heaviside Function
3.3.1 The First Shifting Theorem
3.3.2 The Heaviside Function and Pulses
3.3.3 The Second Shifting Theorem
3.3.4 Analysis of Electrical Circuits
3.4 Convolution
3.5 Unit Impulses and the Dirac Delta Function
3.6 Laplace Transform Solution of Systems
3.7 Differential Equations With Polynomial Coefficients
Chapter 4 - Series Solutions
4.1 Power Series Solutions of Initial Value Problems
4.2 Power Series Solutions Using Recurrence Relations
4.3 Singular Points and the Method of Frobenius
4.4 Second Solutions and Logarithm Factors
4.5 Appendix on Power Series
4.5.1 Convergence of Power Series
4.5.2 Algebra and Calculus of Power Series
4.5.3 Taylor and Maclaurin Expansions
4.5.4 Shifting Indices
Chapter 5 - Numerical Approximation of Solutions
5.1 Euler's Method
5.1.1 A Problem in Radioactive Waste Disposal
5.2 One-Step Methods
5.2.1 The Second Order Taylor Method
5.2.2 The Modified Euler Method
5.2.3 Runge-Kutta Methods
5.3 Multistep Methods
5.3.1 Multistep Methods
Part II - Vectors and Linear Algebra
Chapter 6 - Vectors and Vector Spaces
6.1 The Algebra and Geometry of Vectors
6.2 The Dot Product
6.3 The Cross Product
6.4 The Vector Space Rⁿ
6.5 Linear Independence, Spanning Sets and Dimension in Rⁿ
6.6 Abstract Vector Spaces
Chapter 7 - Matrices and Systems of Linear Equations
7.1 Matrices
7.1.1 Matrix Algebra
7.1.2 Matrix Notation for Systems of Linear Equations
7.1.3 Some Special Matrices
7.1.4 Another Rationale for the Definition of Matrix Multiplication
7.1.5 Random Walks in Crystals
7.2 Elementary Row Operations and Elementary Matrices
7.3 The Row Echelon Form of a Matrix
7.4 The Row and Column Spaces of a Matrix and Rank of a Matrix
7.5 Solution of Homogeneous Systems of Linear Equations
7.6 The Solution Space of AX = O
7.7 Nonhomogeneous Systems of Linear Equations
7.7.1 The Structure of Solutions of AX = B
7.7.2 Existence and Uniqueness of Solutions of AX = B
7.8 Summary for Linear Systems
7.9 Matrix Inverses
7.9.1 A Method for Finding A-1
Chapter 8 - Determinants
8.1 Permutations
8.2 Definition of the Determinant
8.3 Properties of Determinants
8.4 Evaluation of Determinants by Elementary Row and Column Operations
8.5 Cofactor Expansions
8.6 Determinants of Triangular Matrices
8.7 A Determinant Formula for a Matrix Inverse
8.8 Cramer's Rule
8.9 The Matrix Tree Theorem
Chapter 9 - Eigenvalues, Diagonalization and Special Matrices
9.1 Eigenvalues and Eigenvectors
9.1.1 Gerschgorin's Theorem
9.2 Diagonalization of Matrices
9.3 Orthogonal and Symmetric Matrices
9.4 Quadratic Forms
9.5 Unitary, Hermitian and Skew Hermitian Matrices
Part III - Systems of Differential Equations and Qualitative Methods
Chapter 10 - Systems of Linear Differential Equations
10.1 Theory of Systems of Linear First Order Differential Equations
10.1.1 Theory of the Homogeneous System X′ = AX
10.1.2 General Solution of the Nonhomogeneous System X′ = AX + G
10.2 Solution of X′ = AX When A Is Constant
10.2.1 Solution of X′ = AX When A Has Complex Eigenvalues
10.2.2 Solution of X′ = AX When A Does Not Have n Linearly Independent Eigenvectors
10.2.3 Solution of X′ = AX By Diagonalizing A
10.2.4 Exponential Matrix Solutions of X0 = AX
10.3 Solution of X′ = AX + G
10.3.1 Variation of Parameters
10.3.2 Solution of X′ = AX + G By Diagonalizing A
Chapter 11 - Qualitative Methods and Systems of Nonlinear Differential Equations
11.1 Nonlinear Systems and Existence of Solutions
11.2 The Phase Plane, Phase Portraits and Direction Fields
11.3 Phase Portraits of Linear Systems
11.4 Critical Points and Stability
11.5 Almost Linear Systems
11.6 Predator/Prey Population Models
11.6.1 A Simple Predator/Prey Model
11.6.2 An Extended Predator/Prey Model
11.7 Competing Species Models
11.7.1 A Simple Competing Species Model
11.7.2 An Extended Competing Species Model
11.8 Lyapunov's Stability Criteria
11.9 Limit Cycles and Periodic Solutions
Part IV - Vector Analysis
Chapter 12 - Vector Differential Calculus
12.1 Vector Functions of One Variable
12.2 Velocity, Acceleration, Curvature and Torsion
12.2.1 Tangential and Normal Components of Acceleration
12.2.2 Curvature As a Function of t
12.2.3 The Frenet Formulas
12.3 Vector Fields and Streamlines
12.4 The Gradient Field and Directional Derivatives
12.4.1 Level Surfaces, Tangent Planes and Normal Lines
12.5 Divergence and Curl
12.5.1 A Physical Interpretation of Divergence
12.5.2 A Physical Interpretation of Curl
Chapter 13 - Vector Integral Calculus
13.1 Line Integrals
13.1.1 Line Integral With Respect to Arc Length
13.2 Green's Theorem
13.2.1 An Extension of Green's Theorem
13.3 Independence of Path and Potential Theory In the Plane
13.3.1 A More Critical Look at Theorem 13.5
13.4 Surfaces In 3- Space and Surface Integrals
13.4.1 Normal Vector to a Surface
13.4.2 The Tangent Plane to a Surface
13.4.3 Smooth and Piecewise Smooth Surfaces
13.4.4 Surface Integrals
13.5 Applications of Surface Integrals
13.5.1 Surface Area
13.5.2 Mass and Center of Mass of a Shell
13.5.3 Flux of a Vector Field Across a Surface
13.6 Preparation for the Integral Theorems of Gauss and Stokes
13.7 The Divergence Theorem of Gauss
13.7.1 Archimedes's Principle
13.7.2 The Heat Equation
13.7.3 The Divergence Theorem As A Conservation of Mass Principle
13.7.4 Green's Identities
13.8 The Integral Theorem of Stokes
13.8.1 An Interpretation of Curl
13.8.2 Potential Theory in 3- Space
Part V - Fourier Analysis, Orthogonal Expansions and Wavelets
Chapter 14 - Fourier Series
14.1 Why Fourier Series?
14.2 The Fourier Series of a Function
14.2.1 Even and Odd Functions
14.3 Convergence of Fourier Series
14.3.1 Convergence at the End Points
14.3.2 A Second Convergence Theorem
14.3.3 Partial Sums of Fourier Series
14.3.4 The Gibbs Phenomenon
14.4 Fourier Cosine and Sine Series
14.4.1 The Fourier Cosine Series of a Function
14.4.2 The Fourier Sine Series of a Function
14.5 Integration and Differentiation of Fourier Series
14.6 The Phase Angle Form of a Fourier Series
14.7 Complex Fourier Series and the Frequency Spectrum
14.7.1 Review of Complex Numbers
14.7.2 Complex Fourier Series
Chapter 15 - The Fourier Integral and Fourier Transforms
15.1 The Fourier Integral
15.2 Fourier Cosine and Sine Integrals
15.3 The Complex Fourier Integral and the Fourier Transform
15.4 Additional Properties and Applications of the Fourier Transform
15.4.1 The Fourier Transform of a Derivative
15.4.2 Frequency Differentiation
15.4.3 The Fourier Transform of an Integral
15.4.4 Convolution
15.4.5 Filtering and the Dirac Delta Function
15.4.6 The Windowed Fourier Transform
15.4.7 The Shannon Sampling Theorem
15.4.8 Lowpass and Bandpass Filters
15.5 The Fourier Cosine and Sine Transforms
15.6 The Finite Fourier Cosine and Sine Transforms
15.7 The Discrete Fourier Transform
15.7.1 Linearity and Periodicity
15.7.2 The Inverse N- Point DFT
15.7.3 DFT Approximation of Fourier Coefficients
15.8 Sampled Fourier Series
15.8.1 Approximation of a Fourier Transform by an N- Point DFT
15.8.2 Filtering
15.9 The Fast Fourier Transform
15.9.1 Computational Efficiency of the FFT
15.9.2 Use of the FFT in Analyzing Power Spectral Densities of Signals
15.9.3 Filtering Noise From a Signal
15.9.4 Analysis of the Tides in Morro Bay
Chapter 16 - Special Functions, Orthogonal Expansions and Wavelets
16.1 Legendre Polynomials
16.1.1 A Generating Function for the Legendre Polynomials
16.1.2 A Recurrence Relation for the Legendre Polynomials
16.1.3 Orthogonality of the Legendre Polynomials
16.1.4 Fourier-Legendre Series
16.1.5 Computation of Fourier-Legendre Coefficients
16.1.6 Zeros of the Legendre Polynomials
16.1.7 Derivative and Integral Formulas for Pn(x)
16.2 Bessel Functions
16.2.1 The Gamma Function
16.2.2 Bessel Functions of the First Kind and Solutions of Bessel's Equation
16.2.3 Bessel Functions of the Second Kind
16.2.4 Modified Bessel Functions
16.2.5 Some Applications of Bessel Functions
16.2.6 A Generating Function for Jn(x)
16.2.7 An Integral Formula for Jn(x)
16.2.8 A Recurrence Relation for Jv(x)
16.2.9 Zeros of Jv(x)
16.2.10Fourier-Bessel Expansions
16.2.11Fourier-Bessel Coefficients
16.3 Sturm-Liouville Theory and Eigenfunction Expansions
16.3.1 The Sturm-Liouville Problem
16.3.2 The Sturm-Liouville Theorem
16.3.3 Eigenfunction Expansions
16.3.4 Approximation In the Mean and Bessel's Inequality
16.3.5 Convergence in the Mean and Parseval's Theorem
16.3.6 Completeness of the Eigenfunctions
16.4 Orthogonal Polynomials
16.4.1 Chebyshev Polynomials
16.4.2 Laguerre Polynomials
16.4.3 Hermite Polynomials
16.5 Wavelets
16.5.1 The Idea Behind Wavelets
16.5.2 The Haar Wavelets
16.5.3 A Wavelet Expansion
16.5.4 Multiresolution Analysis With Haar Wavelets
16.5.5 General Construction of Wavelets and Multiresolution Analysis
16.5.6 Shannon Wavelets
Part VI - Partial Differential Equations
Chapter 17 - The Wave Equation
17.1 The Wave Equation and Initial and Boundary Conditions
17.2 Fourier Series Solutions of the Wave Equation
17.2.1 Vibrating String With Zero Initial Velocity
17.2.2 Vibrating String with Given Initial Velocity and Zero Initial Displacement
17.2.3 Vibrating String With Initial Displacement and Velocity
17.2.4 Verification of Solutions
17.2.5 Transformation of Boundary Value Problems Involving the Wave Equation
17.2.6 Effects of Initial Conditions and Constants on the Motion
17.2.7 Numerical Solution of the Wave Equation
17.3 Wave Motion Along Infinite and Semi-Infinite Strings
17.3.1 Wave Motion Along an Infinite String
17.3.2 Wave Motion Along a Semi-Infinite String
17.3.3 Fourier Transform Solution of Problems on Unbounded Domains
17.4 Characteristics and d'Alembert's Solution
17.4.1 A Nonhomogeneous Wave Equation
17.4.2 Forward and Backward Waves
17.5 Normal Modes of Vibration of a Circular Elastic Membrane
17.6 Vibrations of a Circular Elastic Membrane, Revisited
17.7 Vibrations of a Rectangular Membrane
Chapter 18 - The Heat Equation
18.1 The Heat Equation and Initial and Boundary Conditions
18.2 Fourier Series Solutions of the Heat Equation
18.2.1 Ends of the Bar Kept at Temperature Zero
18.2.2 Temperature in a Bar With Insulated Ends
18.2.3 Temperature Distribution in a Bar With Radiating End
18.2.4 Transformations of Boundary Value Problems Involving the Heat Equation
18.2.5 A Nonhomogeneous Heat Equation
18.2.6 Effects of Boundary Conditions and Constants on Heat Conduction
18.2.7 Numerical Approximation of Solutions
18.3 Heat Conduction in Infinite Media
18.3.1 Heat Conduction in an Infinite Bar
18.3.2 Heat Conduction in a Semi-Infinite Bar
18.3.3 Integral Transform Methods for the Heat Equation in an Infinite Medium
18.4 Heat Conduction in an Infinite Cylinder
18.5 Heat Conduction in a Rectangular Plate
Chapter 19 - The Potential Equation
19.1 Harmonic Functions and the Dirichlet Problem
19.2 Dirichlet Problem for a Rectangle
19.3 Dirichlet Problem for a Disk
19.4 Poisson's Integral Formula for the Disk
19.5 Dirichlet Problems in Unbounded Regions
19.5.1 Dirichlet Problem for the Upper Half Plane
19.5.2 Dirichlet Problem for the Right Quarter Plane
19.5.3 An Electrostatic Potential Problem
19.6 A Dirichlet Problem for a Cube
19.7 The Steady-State Heat Equation for a Solid Sphere
19.8 The Neumann Problem
19.8.1 A Neumann Problem for a Rectangle
19.8.2 A Neumann Problem for a Disk
19.8.3 A Neumann Problem for the Upper Half Plane
Part VII - Complex Analysis
Chapter 20 - Geometry and Arithmetic of Complex Numbers
20.1 Complex Numbers
20.1.1 The Complex Plane
20.1.2 Magnitude and Conjugate
20.1.3 Complex Division
20.1.4 Inequalities
20.1.5 Argument and Polar Form of a Complex Number
20.1.6 Ordering
20.1.7 Binomial Expansion of (z + w)n
20.2 Loci and Sets of Points in the Complex Plane
20.2.1 Distance
20.2.2 Circles and Disks
20.2.3 The Equation |z - a| = |z - b|
20.2.4 Other Loci
20.2.5 Interior Points, Boundary Points, and Open and Closed Sets
20.2.6 Limit Points
20.2.7 Complex Sequences
20.2.8 Subsequences
20.2.9 Compactness and the Bolzano-Weierstrass Theorem
Chapter 21 - Complex Functions
21.1 Limits, Continuity and Derivatives
21.1.1 Limits
21.1.2 Continuity
21.1.3 The Derivative of a Complex Function
21.1.4 The Cauchy-Riemann Equations
21.2 Power Series
21.2.1 Series of Complex Numbers
21.2.2 Power Series
21.3 The Exponential and Trigonometric Functions
21.4 The Complex Logarithm
21.5 Powers
21.5.1 Integer Powers
21.5.2 z1/n for Positive Integer n
21.5.3 Rational Powers
21.5.4 Powers zw
Chapter 22 - Complex Integration
22.1 Curves in the Plane
22.2 The Integral of a Complex Function
22.2.1 The Complex Integral in Terms of Real Integrals
22.2.2 Properties of Complex Integrals
22.2.3 Integrals of Series of Functions
22.3 Cauchy's Theorem
22.3.1 Proof of Cauchy's Theorem for a Special Case
22.3.2 Proof of Cauchy's Theorem for a Rectangle
22.4 Consequences of Cauchy's Theorem
22.4.1 Independence of Path
22.4.2 The Deformation Theorem
22.4.3 Cauchy's Integral Formula
22.4.4 Cauchy's Integral Formula for Higher Derivatives
22.4.5 Bounds on Derivatives and Liouville's Theorem
22.4.6 An Extended Deformation Theorem
Chapter 23 - Series Representations of Functions
23.1 Power Series Representations
23.1.1 Isolated Zeros and the Identity Theorem
23.1.2 The Maximum Modulus Theorem
23.2 The Laurent Expansion
Chapter 24 - Singularities and the Residue Theorem
24.1 Singularities
24.2 The Residue Theorem
24.3 Some Applications of the Residue Theorem
24.3.1 The Argument Principle
24.3.2 Rouché's Theorem
24.3.3 Summation of Real Series
24.3.4 An Inversion Formula for the Laplace Transform
24.3.5 Evaluation of Real Integrals
Chapter 25 - Conformal Mappings
25.1 Functions as Mappings
25.2 Conformal Mappings
25.2.1 Linear Fractional Transformations
25.3 Construction of Conformal Mappings Between Domains
25.3.1 Schwarz-Christoffel Transformation
25.4 Harmonic Functions and the Dirichlet Problem
25.4.1 Solution of Dirichlet Problems by Conformal Mapping
25.5 Complex Function Models of Plane Fluid Flow
Part VIII - Probability and Statistics
Chapter 26 - Counting and Probability
26.1 The Multiplication Principle
26.2 Permutations
26.3 Choosing r Objects From n Objects
26.3.1 r Objects From n Objects, With Order
26.3.2 r Objects From n Objects, Without Order
26.3.3 Tree Diagrams
26.4 Events and Sample Spaces
26.5 The Probability of an Event
26.6 Complementary Events
26.7 Conditional Probability
26.8 Independent Events
26.8.1 The Product Rule
26.9 Tree Diagrams In Computing Probabilities
26.10 Bayes' Theorem
26.11 Expected Value
Chapter 27 - Statistics
27.1 Measures of Center and Variation
27.1.1 Measures of Center
27.1.2 Measures of Variation
27.2 Random Variables and Probability Distributions
27.3 The Binomial and Poisson Distributions
27.3.1 The Binomial Distribution
27.3.2 The Poisson Distribution
27.4 A Coin Tossing Experiment, Normally Distributed Data, and the Bell Curve
27.4.1 The Standard Bell Curve
27.4.2 The 68, 95, 99.7 Rule
27.5 Sampling Distributions and the Central Limit Theorem
27.6 Confidence Intervals and Estimating Population Proportion
27.7 Estimating Population Mean and the Student t Distribution
27.8 Correlation and Regression
