| |

| |

| |

Ordinary Differential Equations (ODEs) | |

| |

| |

| |

First-Order ODEs | |

| |

| |

| |

Basic Concepts. Modeling | |

| |

| |

| |

Geometric Meaning of y' = f(x, y). Direction Fields | |

| |

| |

| |

Separable ODEs. Modeling | |

| |

| |

| |

Exact ODEs. Integrating Factors | |

| |

| |

| |

Linear ODEs. Bernoulli Equation. Population Dynamics | |

| |

| |

| |

Orthogonal Trajectories. Optional | |

| |

| |

| |

Existence and Uniqueness of Solutions | |

| |

| |

Chapter 1 Review Questions and Problems | |

| |

| |

Summary of Chapter 1 | |

| |

| |

| |

Second-Order Linear ODEs | |

| |

| |

| |

Homogeneous Linear ODEs of Second Order | |

| |

| |

| |

Homogeneous Linear ODEs with Constant Coefficients | |

| |

| |

| |

Differential Operators. Optional | |

| |

| |

| |

Modeling: Free Oscillations. (Mass-Spring System) | |

| |

| |

| |

Euler-Cauchy Equations | |

| |

| |

| |

Existence and Uniqueness of Solutions. Wronskian | |

| |

| |

| |

Nonhomogeneous ODEs | |

| |

| |

| |

Modeling: Forced Oscillations. Resonance | |

| |

| |

| |

Modeling: Electric Circuits | |

| |

| |

| |

Solution by Variation of Parameters | |

| |

| |

Chapter 2 Review Questions and Problems | |

| |

| |

Summary of Chapter 2 | |

| |

| |

| |

Higher Order Linear ODEs | |

| |

| |

| |

Homogeneous Linear ODEs | |

| |

| |

| |

Homogeneous Linear ODEs with Constant Coefficients | |

| |

| |

| |

Nonhomogeneous Linear ODEs | |

| |

| |

Chapter 3 Review Questions and Problems | |

| |

| |

Summary of Chapter 3 | |

| |

| |

| |

Systems of ODEs. Phase Plane. Qualitative Methods | |

| |

| |

| |

Basics of Matrices and Vectors | |

| |

| |

| |

Systems of ODEs as Models | |

| |

| |

| |

Basic Theory of Systems of ODEs | |

| |

| |

| |

Constant-Coefficient Systems. Phase Plane Method | |

| |

| |

| |

Criteria for Critical Points. Stability | |

| |

| |

| |

Qualitative Methods for Nonlinear Systems | |

| |

| |

| |

Nonhomogeneous Linear Systems of ODEs | |

| |

| |

Chapter 4 Review Questions and Problems | |

| |

| |

Summary of Chapter 4 | |

| |

| |

| |

Series Solutions of ODEs. Special Functions | |

| |

| |

| |

Power Series Method | |

| |

| |

| |

Legendre's Equation. Legendre Polynomials Pn(x) | |

| |

| |

| |

Frobenius Method | |

| |

| |

| |

Bessel's Equation. Bessel Functions Jv(x) | |

| |

| |

| |

Bessel Functions of the Second Kind Yv(x) | |

| |

| |

Chapter 5 Review Questions and Problems | |

| |

| |

Summary of Chapter 5 | |

| |

| |

| |

Laplace Transforms | |

| |

| |

| |

Laplace Transform. Inverse Transform. Linearity. ^-Shifting | |

| |

| |

| |

Transforms of Derivatives and Integrals. ODEs | |

| |

| |

| |

Unit Step Function. f-Shifting | |

| |

| |

| |

Short Impulses. Dirac's Delta Function. Partial Fractions | |

| |

| |

| |

Convolution. Integral Equations | |

| |

| |

| |

Differentiation and Integration of Transforms | |

| |

| |

| |

Systems of ODEs | |

| |

| |

| |

Laplace Transform: General Formulas | |

| |

| |

| |

Table of Laplace Transforms | |

| |

| |

Chapter 6 Review Questions and Problems | |

| |

| |

Summary of Chapter 6 | |

| |

| |

| |

Linear Algebra. Vector Calculus | |

| |

| |

| |

Linear Algebra: Matrices, Vectors, Determinants. Linear Systems | |

| |

| |

| |

Matrices, Vectors: Addition and Scalar Multiplication | |

| |

| |

| |

Matrix Multiplication | |

| |

| |

| |

Linear Systems of Equations. Gauss Elimination | |

| |

| |

| |

Linear Independence. Rank of a Matrix. Vector Space | |

| |

| |

| |

Solutions of Linear Systems: Existence, Uniqueness | |

| |

| |

| |

For Reference: Second- and Third-Order Determinants | |

| |

| |

| |

Determinants. Cramer's Rule | |

| |

| |

| |

Inverse of a Matrix. Gauss-Jordan Elimination | |

| |

| |

| |

Vector Spaces, Inner Product Spaces. Linear Transformations Optional | |

| |

| |

Chapter 7 Review Questions and Problems | |

| |

| |

Summary of Chapter 7 | |

| |

| |

| |

Linear Algebra: Matrix Eigenvalue Problems | |

| |

| |

| |

Eigenvalues, Eigenvectors | |

| |

| |

| |

Some Applications of Eigenvalue Problems | |

| |

| |

| |

Symmetric, Skew-Symmetric, and Orthogonal Matrices | |

| |

| |

| |

Eigenbases. Diagonalization. Quadratic Forms | |

| |

| |

| |

Complex Matrices and Forms. Optional | |

| |

| |

Chapter 8 Review Questions and Problems | |

| |

| |

Summary of Chapter 8 | |

| |

| |

| |

Vector Differential Calculus. Grad, Div, Curl | |

| |

| |

| |

Vectors in 2-Space and 3-Space | |

| |

| |

| |

Inner Product | |

| |

| |

| |

| |

Vector Product | |

| |

| |

| |

| |

Vector and Scalar Functions and Fields. Derivatives | |

| |

| |

| |

Curves. Arc Length. Curvature. Torsion | |

| |

| |

| |

Calculus Review: Functions of Several Variables. Optional | |

| |

| |

| |

Gradient of a Scalar Field. Directional Derivative | |

| |

| |

| |

Divergence of a Vector Field | |

| |

| |

| |

Curl of a Vector Field | |

| |

| |

Chapter 9 Review Questions and Problems | |

| |

| |

Summary of Chapter 9 | |

| |

| |

| |

Vector Integral Calculus. Integral Theorems | |

| |

| |

| |

Line Integrals | |

| |

| |

| |

Path Independence of Line Integrals | |

| |

| |

| |

Calculus Review: Double Integrals. Optional | |

| |

| |

| |

Green's Theorem in the Plane | |

| |

| |

| |

Surfaces for Surface Integrals | |

| |

| |

| |

Surface Integrals | |

| |

| |

| |

Triple Integrals. Divergence Theorem of Gauss | |

| |

| |

| |

Further Applications of the Divergence Theorem | |

| |

| |

| |

Stokes's Theorem | |

| |

| |

Chapter 10 Review Questions and Problems | |

| |

| |

Summary of Chapter 10 | |

| |

| |

| |

Fourier Analysis. Partial Differential Equations (PDEs) | |

| |

| |

| |

Fourier Series, Integrals, and Transforms | |

| |

| |

| |

Fourier Series | |

| |

| |

| |

Functions of Any Period p = 2L. Even and Odd Functions. Half-Range Expansions | |

| |

| |

| |

Forced Oscillations | |

| |

| |

| |

Approximation by Trigonometric Polynomials | |

| |

| |

| |

Sturm-Liouville Problems. Orthogonal Functions | |

| |

| |

| |

Orthogonal Eigenfunction Expansions | |

| |

| |

| |

Fourier Integral | |

| |

| |

| |

Fourier Cosine and Sine Transforms | |

| |

| |

| |

Fourier Transform. Discrete and Fast Fourier Transforms | |

| |

| |

| |

Tables of Transforms | |

| |

| |

Chapter 11 Review Questions and Problems | |

| |

| |

Summary of Chapter 11 | |

| |

| |

| |

Partial Differential Equations (PDEs) | |

| |

| |

| |

Basic Concepts | |

| |

| |

| |

Modeling: Vibrating String, Wave Equation | |

| |

| |

| |

Solution by Separating Variables. Use of Fourier Series | |

| |

| |

| |

D'Alembert's Solution of the Wave Equation. Characteristics | |

| |

| |

| |

Introduction to the Heat Equation | |

| |

| |

| |

Heat Equation: Solution by Fourier Series | |

| |

| |

| |

Heat Equation: Solution by Fourier Integrals and Transforms | |

| |

| |

| |

Modeling: Membrane, Two-Dimensional Wave Equation | |

| |

| |

| |

Rectangular Membrane. Double Fourier Series | |

| |

| |

| |

Laplacian in Polar Coordinates. Circular Membrane. Fourier-Bessel Series | |

| |

| |

| |

Laplace's Equation in Cylindrical and Spherical Coordinates. Potential | |

| |

| |

| |

Solution of PDEs by Laplace Transforms | |

| |

| |

Chapter 12 Review Questions and Problems | |

| |

| |

Summary of Chapter 12 | |

| |

| |

| |

Complex Analysis | |

| |

| |

| |

Complex Numbers and Functions | |

| |

| |

| |

Complex Numbers. Complex Plane | |

| |

| |

| |

Polar Form of Complex Numbers. Powers and Roots | |

| |

| |

| |

Derivative. Analytic Function | |

| |

| |

| |

Cauchy-Riemann Equations. Laplace's Equation | |

| |

| |

| |

Exponential Function | |

| |

| |

| |

Trigonometric and Hyperbolic Functions | |

| |

| |

| |

Logarithm. General Power | |

| |

| |

Chapter 13 Review Questions and Problems | |

| |

| |

Summary of Chapter 13 | |

| |

| |

| |

Complex Integration | |

| |

| |

| |

Line Integral in the Complex Plane | |

| |

| |

| |

Cauchy's Integral Theorem | |

| |

| |

| |

Cauchy's Integral Formula | |

| |

| |

| |

Derivatives of Analytic Functions | |

| |

| |

Chapter 14 Review Questions and Problems | |

| |

| |

Summary of Chapter 14 | |

| |

| |

| |

Power Series, Taylor Series | |

| |

| |

| |

Sequences, Series, Convergence Tests | |

| |

| |

| |

Power Series | |

| |

| |

| |

Functions Given by Power Series | |

| |

| |

| |

Taylor and Maclaurin Series | |

| |

| |

| |

Uniform Convergence. Optional | |

| |

| |

Chapter 15 Review Questions and Problems | |

| |

| |

Summary of Chapter 15 | |

| |

| |

| |

Laurent Series. Residue Integration | |

| |

| |

| |

Laurent Series | |

| |

| |

| |

Singularities and Zeros. Infinity | |

| |

| |

| |

Residue Integration Method | |

| |

| |

| |

Residue Integration of Real Integrals | |

| |

| |

| |

Review Questions and Problems | |

| |

| |

Summary of Chapter 16 | |

| |

| |

| |

Conformal Mapping | |

| |

| |

| |

Geometry of Analytic Functions: Conformal Mapping | |

| |

| |

| |

Linear Fractional Transformations | |

| |

| |

| |

Special Linear Fractional Transformations | |

| |

| |

| |

Conformal Mapping by Other Functions | |

| |

| |

| |

Riemann Surfaces. Optional | |

| |

| |

Chapter 17 Review Questions and Problems | |

| |

| |

Summary of Chapter 17 | |

| |

| |

| |

Complex Analysis and Potential Theory | |

| |

| |

| |

Electrostatic Fields | |

| |

| |

| |

Use of Conformal Mapping. Modeling | |

| |

| |

| |

Heat Problems | |

| |

| |

| |

Fluid Flow | |

| |

| |

| |

Poisson's Integral Formula for Potentials | |

| |

| |

| |

General Properties of Harmonic Functions | |

| |

| |

Chapter 18 Review Questions and Problems | |

| |

| |

Summary of Chapter 18 | |

| |

| |

| |

Numeric Analysis | |

| |

| |

Software | |

| |

| |

| |

Numerics in General | |

| |

| |

| |

Introduction | |

| |

| |

| |

Solution of Equations by Iteration | |

| |

| |

| |

Interpolation | |

| |

| |

| |

Spline Interpolation | |

| |

| |

| |

Numeric Integration and Differentiation | |

| |

| |

Chapter 19 Review Questions and Problems | |

| |

| |

Summary of Chapter 19 | |

| |

| |

| |

Numeric Linear Algebra | |

| |

| |

| |

Linear Systems: Gauss Elimination | |

| |

| |

| |

Linear Systems: LU-Factorization, Matrix Inversion | |

| |

| |

| |

Linear Systems: Solution by Iteration | |

| |

| |

| |

Linear Systems: Ill-Conditioning, Norms | |

| |

| |

| |

Least Squares Method | |

| |

| |

| |

Matrix Eigenvalue Problems: Introduction | |

| |

| |

| |

Inclusion of Matrix Eigenvalues | |

| |

| |

| |

Power Method for Eigenvalues | |

| |

| |

| |

Tridiagonalization and QR-Factorization | |

| |

| |

Chapter 20 Review Questions and Problems | |

| |

| |

Summary of Chapter 20 | |

| |

| |

| |

Numerics for ODEs and PDEs | |

| |

| |

| |

Methods for First-Order ODEs | |

| |

| |

| |

Multistep Methods | |

| |

| |

| |

Methods for Systems and Higher Order ODEs | |

| |

| |

| |

Methods for Elliptic PDEs | |

| |

| |

| |

Neumann and Mixed Problems. Irregular Boundary | |

| |

| |

| |

Methods for Parabolic PDEs | |

| |

| |

| |

Method for Hyperbolic PDEs | |

| |

| |

Chapter 21 Review Questions and Problems | |

| |

| |

Summary of Chapter 21 | |

| |

| |

| |

Optimization, Graphs | |

| |

| |

| |

Unconstrained Optimization. Linear Programming | |

| |

| |

| |

Basic Concepts. Unconstrained Optimization | |

| |

| |

| |

Linear Programming | |

| |

| |

| |

Simplex Method | |