Advanced Engineering Mathematics

Front Cover
Copyright Page
CONTENTS
Preface
PART 1 - Ordinary Differential Equations
	CHAPTER 1  Introduction to Differential Equations
		Outline & Intro
		1.1 Definitions and Terminology
		1.2 Initial-Value Problems
		1.3 Differential Equations as Mathematical Models
		Review
	CHAPTER 2  First-Order Differential Equations
		Outline & Intro
		2.1 Solution Curves Without a Solution
		2.2 Separable Equations
		2.3 Linear Equations
		2.4 Exact Equations
		2.5 Solutions by Substitutions
		2.6 A Numerical Method
		2.7 Linear Models
		2.8 Nonlinear Models
		2.9 Modeling with Systems of First-Order DEs
		Review
		Contributed Problems
	CHAPTER 3  Higher-Order Differential Equations
		Outline & Intro
		3.1 Theory of Linear Equations
		3.2 Reduction of Order
		3.3 Homogeneous Linear Equations with Constant Coefficients
		3.4 Undetermined Coefficients
		3.5 Variation of Parameters
		3.6 Cauchy-Euler Equations
		3.7 Nonlinear Equations
		3.8 Linear Models: Initial-Value Problems
		3.9 Linear Models: Boundary-Value Problems
		3.10 Green's Functions
		3.11 Nonlinear Models
		3.12 Solving Systems of Linear Equations
		Review
		Contributed Problems
	CHAPTER 4  The Laplace Transform
		Outline & Intro
		4.1 Definition of the Laplace Transform
		4.2 The Inverse Transform and Transforms of Derivatives
		4.3 Translation Theorems
		4.4 Additional Operational Properties
		4.5 The Dirac Delta Function
		4.6 Systems of Linear Differential Equations
		Review
	CHAPTER 5  Series Solutions of Linear Differential Equations
		Outline & Intro
		5.1 Solutions about Ordinary Points
		5.2 Solutions about Singular Points
		5.3 Special Functions
		Review
	CHAPTER 6  Numerical Solutions of Ordinary Differential Equations
		Outline & Intro
		6.1 Euler Methods and Error Analysis
		6.2 Runge-Kutta Methods
		6.3 Multistep Methods
		6.4 Higher-Order Equations and Systems
		6.5 Second-Order Boundary-Value Problems
		Review
PART 2 - Vectors, Matrices, and Vector Calculus
	CHAPTER 7  Vectors
		Outline & Intro
		7 .1 Vectors in 2-Space
		7 .2 Vectors in 3-Space
		7 .3 Dot Product
		7 .4 Cross Product
		7.5 Lines and Planes in 3-Space
		7 .6 Vector Spaces
		7.7 Gram-Schmidt Orthogonalization Process
		Review
	CHAPTER 8  Matrices
		Outline & Intro
		8.1 Matrix Algebra
		8.2 Systems of Linear Algebraic Equations
		8.3 Rank of a Matrix
		8.4 Determinants
		8.5 Properties of Determinants
		8.6 Inverse of a Matrix
		8.7 Cramer's Rule
		8.8 The Eigenvalue Problem
		8.9 Powers of Matrices
		8.10 Orthogonal Matrices
		8.11 Approximation of Eigenvalues
		8.12 Diagonalization
		8.13 LU-Factorization
		8.14 Cryptography
		8.15 An Error-Correcting Code
		8.16 Method of Least Squares
		8.17 Discrete Compartmental Models
		Review
	CHAPTER 9  Vector Calculus
		Outline & Intro
		9.1 Vector Functions
		9.2 Motion on a Curve
		9.3 Curvature and Components of Acceleration
		9.4 Partial Derivatives
		9.5 Directional Derivative
		9.6 Tangent Planes and Normal Lines
		9.7 Curl and Divergence
		9.8 Line Integrals
		9.9 Independence of the Path
		9.10 Double Integrals
		9.11 Double Integrals in Polar Coordinates
		9.12 Green's Theorem
		9.13 Surface Integrals
		9.14 Stokes' Theorem
		9.15 Triple Integrals
		9.16 Divergence Theorem
		9.17 Change of Variables in Multiple Integrals
		Review
PART 3 - Systems of Differential Equations
	CHAPTER 10  Systems of Linear Differential Equations
		Outline & Intro
		10.1 Theory of Linear Systems
		10.2 Homogeneous Linear Systems
		10.3 Solution by Diagonalization
		10.4 Nonhomogeneous Linear Systems
		10.5 Matrix Exponential
		Review
	CHAPTER 11  Systems of Nonlinear Differential Equations
		Outline & Intro
		11.1 Autonomous Systems
		11.2 Stability of Linear Systems
		11.3 Linearization and Local Stability
		11.4 Autonomous Systems as Mathematical Models
		11.5 Periodic Solutions, Limit Cycles, and Global Stability
		Review
PART 4 - Partial Differential Equations
	CHAPTER 12  Orthogonal Functions and Fourier Series
		Outline & Intro
		12.1 Orthogonal Functions
		12.2 Fourier Series
		12.3 Fourier Cosine and Sine Series
		12.4 Complex Fourier Series
		12.5 Sturm-Liouville Problem
		12.6 Bessel and Legendre Series
		Review
	CHAPTER 13  Boundary-Value Problems in Rectangular Coordinates
		Outline & Intro
		13.1 Separable Partial Differential Equations
		13.2 Classical PDEs and Boundary-Value Problems
		13.3 Heat Equation
		13.4 Wave Equation
		13.5 Laplace's Equation
		13.6 Nonhomogeneous BVPs
		13. 7 Orthogonal Series Expansions
		13.8 Fourier Series in Two Variables
		Review
	CHAPTER 14  Boundary-Value Problems in Other Coordinate Systems
		Outline & Intro
		14.1 Problems in Polar Coordinates
		14.2 Problems in Cylindrical Coordinates
		14.3 Problems in Spherical Coordinates
		Review
	CHAPTER 15  Integral Transform Method
		Outline & Intro
		15.1 Error Function
		15.2 Applications of the Laplace Transform
		15.3 Fourier Integral
		15.4 Fourier Transforms
		15.5 Fast Fourier Transform
		Review
	CHAPTER 16  Numerical Solutions of Partial Differential Equations
		Outline & Intro
		16.1 Laplace's Equation
		16.2 The Heat Equation
		16.3 The Wave Equation
		Review
PART 5 - Complex Analysis
	CHAPTER 17  Functions of a Complex Variable
		Outline & Intro
		17.1 Complex Numbers
		17.2 Powers and Roots
		17.3 Sets in the Complex Plane
		17.4 Functions of a Complex Variable
		17.5 Cauchy-Riemann Equations
		17.6 Exponential and Logarithmic Functions
		17. 7 Trigonometric and Hyperbolic Functions
		17.8 Inverse Trigonometric and Hyperbolic Functions
		Review
	CHAPTER 18  Integration in the Complex Plane
		Outline & Intro
		18.1 Contour Integrals
		18.2 Cauchy-Goursat Theorem
		18.3 Independence of the Path
		18.4 Cauchy's Integral Formulas
		Review
	CHAPTER 19  Series and Residues
		Outline & Intro
		19.1 Sequences and Series
		19.2 Taylor Series
		19.3 Laurent Series
		19.4 Zeros and Poles
		19.5 Residues and Residue Theorem
		19.6 Evaluation of Real Integrals
		Review
	CHAPTER 20  Conformal Mappings
		Outline & Intro
		20.1 Complex Functions as Mappings
		20.2 Conformal Mappings
		20.3 Linear Fractional Transformations
		20.4 Schwarz-Christoffel Transformations
		20.5 Poisson Integral Formulas
		20.6 Applications
		Review
APPENDICES
	I. Derivative and Integral Formulas
	II. Gamma Function
	III. Table of Laplace Transforms
	IV. Conformal Mappings
ANSWERS to Selected Odd-Numbered Problems
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	appII
INDEX
	A
	B
	C
	D
	E
	F
	G-H
	I
	J-K-L
	M
	N
	O
	P
	Q
	R
	S
	T
	U
	V
	W
	X
	Y-Z
Credits
Table of Laplace Transforms
Errata