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| 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 | |
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