Table of Contents
CLASSIC ENGINEERING MATHEMATICS
First-Order Ordinary Differential Equations
Classification of Differential Equations
Separation of Variables
Homogeneous Equations
Exact Equations
Linear Equations
Graphical Solutions
Numerical Methods
Higher-Order Ordinary Differential Equations
Homogeneous Linear Equations with Constant Coefficients
Simple Harmonic Motion
Damped Harmonic Motion
Method of Undetermined Coefficients
Forced Harmonic Motion
Variation of Parameters
Euler-Cauchy Equation
Phase Diagrams
Numerical Methods
Linear Algebra
Fundamentals of Linear Algebra
Determinants
Cramer’s Rule
Row Echelon Form and Gaussian Elimination
Eigenvalues and Eigenvectors
Systems of Linear Differential Equations
Matrix Exponential
Vector Calculus
Review
Divergence and Curl
Line Integrals
The Potential Function
Surface Integrals
Green’s Lemma
Stokes’ Theorem
Divergence Theorem
Fourier Series
Fourier Series
Properties of Fourier Series
Half-Range Expansions
Fourier Series with Phase Angles
Complex Fourier Series
The Use of Fourier Series in the Solution of Ordinary Differential Equations
Finite Fourier Series
The Sturm-Liouville Problem
Eigenvalues and Eigenfunctions
Orthogonality of Eigenfunctions
Expansion in Series of Eigenfunctions
A Singular Sturm-Liouville Problem: Legendre’s Equation
Another Singular Sturm-Liouville Problem: Bessel’s Equation
Finite Element Method
The Wave Equation
The Vibrating String
Initial Conditions: Cauchy Problem
Separation of Variables
D’Alembert’s Formula
Numerical Solution of the Wave Equation
The Heat Equation
Derivation of the Heat Equation
Initial and Boundary Conditions
Separation of Variables
Numerical Solution of the Heat Equation
Laplace’s Equation
Derivation of Laplace’s Equation
Boundary Conditions
Separation of Variables
Poisson’s Equation on a Rectangle
Numerical Solution of Laplace’s Equation
Finite Element Solution of Laplace’s Equation
TRANSFORM METHODS
Complex Variables
Complex Numbers
Finding Roots
The Derivative in the Complex Plane: The Cauchy-Riemann Equations
Line Integrals
The Cauchy-Goursat Theorem
Cauchy’s Integral Formula
Taylor and Laurent Expansions and Singularities
Theory of Residues
Evaluation of Real Definite Integrals
Cauchy’s Principal Value Integral
Conformal Mapping
The Fourier Transform
Fourier Transforms
Fourier Transforms Containing the Delta Function
Properties of Fourier Transforms
Inversion of Fourier Transforms
Convolution
Solution of Ordinary Differential Equations
The Solution of Laplace’s Equation on the Upper Half-Plane
The Solution of the Heat Equation
The Laplace Transform
Definition and Elementary Properties
The Heaviside Step and Dirac Delta Functions
Some Useful Theorems
The Laplace Transform of a Periodic Function
Inversion by Partial Fractions: Heaviside’s Expansion Theorem
Convolution
Integral Equations
Solution of Linear Differential Equations with Constant Coefficients
Inversion by Contour Integration
The Solution of the Wave Equation
The Solution of the Heat Equation
The Superposition Integral and the Heat Equation
The Solution of Laplace’s Equation
The Z-Transform
The Relationship of the Z-Transform to the Laplace Transform
Some Useful Properties
Inverse Z-Transforms
Solution of Difference Equations
Stability of Discrete-Time Systems
The Hilbert Transform
Definition
Some Useful Properties
Analytic Signals
Causality: The Kramers-Kronig Relationship
Green’s Functions
What Is a Green’s Function?
Ordinary Differential Equations
Joint Transform Method
Wave Equation
Heat Equation
Helmholtz’s Equation
Galerkin Methods
STOCHASTIC PROCESSES
Probability
Review of Set Theory
Classic Probability
Discrete Random Variables
Continuous Random Variables
Mean and Variance
Some Commonly Used Distributions
Joint Distributions
Random Processes
Fundamental Concepts
Power Spectrum
Two-State Markov Chains
Birth and Death Processes
Poisson Processes
Itˆo’s Stochastic Calculus
Random Differential Equations
Random Walk and Brownian Motion
Itˆo’s Stochastic Integral
Itˆo’s Lemma
Stochastic Differential Equations
Numerical Solution of Stochastic Differential Equations
