Advanced Mathematics for Engineers and Physicists

Preface
Contents
Basic Notations
	Sets
	Hyperbolic Functions
	Euler Integrals
1 First-Order Differential Equations
	1.1 Introduction to Ordinary Differential Equations
	1.2 Separable Equations
	1.3 Homogeneous Equations
	1.4 First-Order Linear Differential Equations
	1.5 Bernoulli Equations
	1.6 Riccati Equations
	1.7 Exact Differential Equations
	1.8 Lagrange Equations and Clairaut Equations
	1.9 Existence and Uniqueness of Solution of the Cauchy Problem
	1.10 Exercises
2 Higher-Order Differential Equations
	2.1 Introduction
	2.2 Homogeneous Linear Differential Equations of Order n
	2.3 Non-Homogeneous Linear Differential Equations of Order n
	2.4 Homogeneous Linear Equations with Constant Coefficients
	2.5 Nonhomogeneous Linear Equations with Constant Coefficients
	2.6 Euler Equations
	2.7 Exercises
3 Systems of Differential Equations
	3.1 Introduction
	3.2 First-Order Systems and Differential Equations of Order n
	3.3 Linear Systems of Differential Equations
	3.4 Linear Systems with Constant Coefficients
		3.4.1 The Homogeneous Case (the Algebraic Method)
		3.4.2 The Non-Homogeneous Case (the Method of Undetermined Coefficients)
			3.4.2.1 The Diagonalizable Case
			3.4.2.2 The Non-Diagonalizable Case
		3.4.3 Matrix Exponential and Linear Systems with Constant Coefficients
			3.4.3.1 Fundamental Matrix
			3.4.3.2 Matrix Exponential
			3.4.3.3 The Exponential of a Diagonalizable Matrix
			3.4.3.4 The Exponential of a Nondiagonalizable Matrix
		3.4.4 Elimination Method for Linear Systems with Constant Coefficients
	3.5 Autonomous Systems of Differential Equations
	3.6 First-Order Partial Differential Equations
		3.6.1 Linear Homogeneous First-Order PDE
		3.6.2 Quasilinear First-Order Partial Differential Equations
	3.7 Exercises
4 Fourier Series
	4.1 Introduction: Periodic, Piecewise Smooth Functions
		4.1.1 Periodic Functions
		4.1.2 Piecewise Continuous and Piecewise Smooth Functions
	4.2 Fourier Series Expansions
		4.2.1 Series of Functions
		4.2.2 A Basic Trigonometric System
		4.2.3 Fourier Coefficients
	4.3 Orthogonal Systems of Functions
		4.3.1 Inner Product
		4.3.2 Best Approximation in the Mean: Bessel's Inequality
	4.4 The Convergence of Fourier Series
	4.5 Differentiation and Integration of the Fourier Series
	4.6 The Convergence in the Mean: Complete Systems
	4.7 Examples of Fourier Expansions
	4.8 The Complex form of the Fourier Series
	4.9 Exercises
5 Fourier Transform
	5.1 Improper Integrals
	5.2 The Fourier Integral Formula
	5.3 The Fourier Transform
	5.4 Solving Linear Differential Equations
	5.5 Moments Theorems
	5.6 Sampling Theorem
	5.7 Discrete Fourier Transform
	5.8 Exercises
6 Laplace Transform
	6.1 Introduction
	6.2 Properties of the Laplace Transform
	6.3 Inverse Laplace Transform
	6.4 Solving Linear Differential Equations
	6.5 The Dirac Delta Function
	6.6 Exercises
7 Second-Order Partial Differential Equations
	7.1 Classification: Canonical Form
	7.2 The Wave Equation
		7.2.1 Infinite Vibrating String: D'Alembert Formula
		7.2.2 Finite Vibrating String: Fourier Method
		7.2.3 Laplace Transform Method for the Vibrating String
		7.2.4 Vibrations of a Rectangular Membrane: Two-Dimensional Wave Equation
	7.3 Vibrations of a Simply Supported Beam: Fourier Method
	7.4 The Heat Equation
		7.4.1 Modeling the Heat Flow from a Body in Space
		7.4.2 Heat Flow in a Finite Rod: Fourier Method
		7.4.3 Heat Flow in an Infinite Rod
		7.4.4 Heat Flow in a Rectangular Plate
	7.5 The Laplace's Equation
		7.5.1 Dirichlet Problem for a Rectangle
		7.5.2 Dirichlet Problem for a Disk
	7.6 Exercises
8 Introduction to the Calculus of Variations
	8.1 Classical Variational Problems
	8.2 General Frame of Calculus of Variations
	8.3 The Case F[y]=abF(x,y,y) dx
	8.4 The Case F[y]=ab F(x, y, y,…,y(n)) dx
	8.5 The Case F[y1,…,yn]=abF(x,y1,…,yn,y1,…,yn) dx
	8.6 The Case F[z]=@汥瑀瑯步渠D F (x,y,z,∂z∂x, ∂z∂y)dxdy
	8.7 Isoperimetric Problems and Geodesic Problems
		8.7.1 Isoperimetric Problems
		8.7.2 Geodesic Problems
	8.8 Exercises
9 Elements of Probability Theory
	9.1 Sample Space: Event Space
	9.2 Probability Space
	9.3 Conditional Probability: Bayes Formula
	9.4 Discrete Random Variables
		9.4.1 Random Variables
		9.4.2 Expected Value; Moments
		9.4.3 Variance
		9.4.4 Discrete Uniform Distribution
		9.4.5 Bernoulli Distribution
		9.4.6 Binomial Distribution
		9.4.7 Poisson Distribution
		9.4.8 Geometric Distribution
	9.5 Continuous Random Variables
		9.5.1 The Probability Density Function; The Distribution Function
		9.5.2 Expected Value, Moments and Variance for Continuous Random Variables
		9.5.3 Characteristic Function
		9.5.4 The Uniform Distribution
		9.5.5 The Exponential Distribution
		9.5.6 The Normal Distribution
		9.5.7 Gamma Distribution
		9.5.8 Chi-Squared Distribution
		9.5.9 Student t-Distribution
	9.6 Limit Theorems
	9.7 Exercises
10 Answers and Solutions to Exercises
	10.1 Chapter 1
	10.2 Chapter 2
	10.3 Chapter 3
	10.4 Chapter 4
	10.5 Chapter 5
	10.6 Chapter 6
	10.7 Chapter 7
	10.8 Chapter 8
	10.9 Chapter 9
11 Supplementary Materials
	11.1 Normed, Metric and Hilbert Spaces
		11.1.1 Normed Vector Spaces
		11.1.2 Sequences and Series of Functions
		11.1.3 Metric Spaces. Some Density Theorems
		11.1.4 The Fields Q, R and C
		11.1.5 Hilbert Spaces
		11.1.6 Continuous Functions and Step Functions
		11.1.7 Orthonormal Systems in a Hilbert Space
	11.2 Complex Function Theory
		11.2.1 Differentiability of Complex Functions
		11.2.2 Integration of Complex Functions
		11.2.3 Power Series Representation
		11.2.4 Residue Theorem and Applications
Bibliography
Index