Chapter 1: First-Order Differential EquationsSection 1-1:
Terminology and Separable Equations
Section 1-2:
Linear Equations
Section 1-3:
Exact Equations
Section 1-4:
Homogeneous, Bernoulli, and Riccati Equations
Section 1-5:
Additional Applications
Section 1-6:
Existence and Uniqueness Questions
Chapter 2: Linear Second-Order EquationsSection 2-1:
The Linear Second-Order Equation
Section 2-2:
The Constant Coefficient Case
Section 2-3:
The Nonhomogeneous Equation
Section 2-4:
Spring Motion
Section 2-5:
Euler's Differential Equation
Chapter 3: The Laplace TransformSection 3-1:
Definition and Notation
Section 3-2:
Solution of Initial Value Problems
Section 3-3:
Shifting and the Heaviside Function
Section 3-4:
Convolution
Section 3-5:
Impulses and the Delta Function
Section 3-6:
Solution of Systems
Section 3-7:
Polynomial Coefficients
Chapter 4: Series SolutionsSection 4-1:
Power Series Solutions
Section 4-2:
Frobenius Solutions
Chapter 5: Approximation of SolutionsSection 5-1:
Direction Fields
Section 5-2:
Euler's Method
Section 5-3:
Taylor and Modified Euler Methods
Chapter 6: Vectors and Vector SpacesSection 6-1:
Vectors in the Plane and 3-Space
Section 6-2:
The Dot Product
Section 6-3:
The Cross Product
Section 6-4:
The Vector Space Rn
Section 6-5:
Orthogonalization
Section 6-6:
Orthogonal Complements and Projections
Section 6-7:
The Function Space C[a, b]
Chapter 7: Matrices and Linear SystemsSection 7-1:
Matrices
Section 7-2:
Elementary Row Operations
Section 7-3:
Reduced Row Echelon Form
Section 7-4:
Row and Column Spaces
Section 7-5:
Homogeneous Systems
Section 7-6:
Nonhomogeneous Systems
Section 7-7:
Matrix Inverses
Section 7-8:
Least Squares Vectors and Data Fitting
Section 7-9:
LU Factorization
Section 7-10:
Linear Transformations
Exercise 26
Exercise 27
Exercise 28
Section 8-1:
Definition of the Determinant
Section 8-2:
Evaluation of Determinants I
Section 8-3:
Evaluation of Determinants II
Section 8-4:
A Determinant Formula for A−1
Section 8-5:
Cramer's Rule
Section 8-6:
The Matrix Tree Theorem
Chapter 9: Eigenvalues, Diagonalization, and Special MatricesSection 9-1:
Eigenvalues and Eigenvectors
Section 9-2:
Diagonalization
Section 9-3:
Some Special Types of Matrices
Chapter 10: Systems of Linear Differential EquationsSection 10-1:
Linear Systems
Section 10-2:
Solution of X' = AX for Constant A
Section 10-3:
Solution of X' = AX+G
Section 10-4:
Exponential Matrix Solutions
Section 10-5:
Applications and Illustrations of Techniques
Section 10-6:
Phase Portraits
Chapter 11: Vector Differential CalculusSection 11-1:
Vector Functions of One Variable
Section 11-2:
Velocity and Curvature
Section 11-3:
Vector Fields and Streamlines
Section 11-4:
The Gradient Field
Section 11-5:
Divergence and Curl
Chapter 12: Vector Integral CalculusSection 12-1:
Line Integrals
Section 12-2:
Green's Theorem
Section 12-3:
An Extension of Green's Theorem
Section 12-4:
Independence of Path and Potential Theory
Section 12-5:
Surface Integrals
Section 12-6:
Applications of Surface Integrals
Section 12-7:
Lifting Green's Theorem to R3
Section 12-8:
The Divergence Theorem of Gauss
Section 12-9:
Stoke's Theorem
Section 12-10:
Curvilinear Coordinates
Chapter 13: Fourier SeriesSection 13-1:
Why Fourier Series?
Section 13-2:
The Fourier Series of a Function
Section 13-3:
Sine and Cosine Series
Section 13-4:
Integration and Differentiation of Fourier Series
Section 13-5:
Phase Angle Form
Section 13-6:
Complex Fourier Series
Section 13-7:
Filtering of Signals
Chapter 14: The Fourier Integral and TransformsSection 14-1:
The Fourier Integral
Section 14-2:
Fourier Cosine and Sine Integrals
Section 14-3:
The Fourier Transform
Section 14-4:
Fourier Cosine and Sine Transforms
Section 14-5:
The Discrete Fourier Transform
Section 14-6:
Sampled Fourier Series
Section 14-7:
DFT Approximation of the Fourier Transform
Chapter 15: Special Functions and Eigenfunction ExpansionsSection 15-1:
Eigenfunction Expansions
Section 15-2:
Legendre Polynomials
Section 15-3:
Bessel Functions
Chapter 16: The Wave EquationSection 16-1:
Derivation of the Wave Equation
Section 16-2:
Wave Motion on an Interval
Section 16-3:
Wave Motion in an Infinite Medium
Section 16-4:
Wave Motion in a Semi-Infinite Medium
Section 16-5:
Laplace Transform Techniques
Section 16-6:
Characteristics and d'Alembert's Solution
Section 16-7:
Vibrations in a Circular Membrane I
Section 16-8:
Vibrations in a Circular Membrane II
Section 16-9:
Vibrations in a Rectangular Membrane
Chapter 17: The Heat EquationSection 17-1:
Initial and Boundary Conditions
Section 17-2:
The Heat Equation on [0, L]
Section 17-3:
Solutions in an Infinite Medium
Section 17-4:
Laplace Transform Techniques
Section 17-5:
Heat Conduction in an Infinite Cylinder
Section 17-6:
Heat Conduction in a Rectangular Plate
Chapter 18: The Potential EquationSection 18-1:
Laplace's Equation
Section 18-2:
Dirichlet Problem for a Rectangle
Section 18-3:
Dirichlet Problem for a Disk
Section 18-4:
Poisson's Integral Formula
Section 18-5:
Dirichlet Problem for Unbounded Regions
Section 18-6:
A Dirichlet Problem for a Cube
Section 18-7:
Steady-State Equation for a Sphere
Section 18-8:
The Neumann Problem
Chapter 19: Complex Numbers and FunctionsSection 19-1:
Geometry and Arithmetic of Complex Numbers
Section 19-2:
Complex Functions
Section 19-3:
The Exponential and Trigonometric Functions
Section 19-4:
The Complex Logarithm
Section 19-5:
Powers
Chapter 20: Complex IntegrationSection 20-1:
The Integral of a Complex Functions
Section 20-2:
Cauchy's Theorem
Section 20-3:
Consequences of Cauchy's Theorem
Chapter 21: Series Representations of FunctionsSection 21-1:
Power Series
Section 21-2:
The Laurent Expansion
Chapter 22: Singularities and the Residue TheoremSection 22-1:
Singularities
Section 22-2:
The Residue Theorem
Section 22-3:
Evaluation of Real Integrals
Section 22-4:
Residues and the Inverse Laplace Transform
Chapter 23: Conformal Mappings and ApplicationsSection 23-1:
Conformal Mappings
Section 23-2:
Construction of Conformal MappingsSection 23-3:
Conformal Mapping Solutions of Dirichlet ProblemsSection 23-4:
Models of Plane Fluid FlowExercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 11
Exercise 12
Exercise 13
Exercise 14
Exercise 15
Exercise 16
Exercise 22
Exercise 23
Exercise 24
Exercise 25
Exercise 26
Exercise 27
Exercise 28
Exercise 29
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