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| Ordinary Differential Equations (ODEs) | |
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| First-Order ODEs | |
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| Basic Concepts. Modeling | |
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| Geometric Meaning of y' = f(x, y). Direction Fields | |
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| Separable ODEs. Modeling | |
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| Exact ODEs. Integrating Factors | |
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| Linear ODEs. Bernoulli Equation. Population Dynamics | |
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| Orthogonal Trajectories. Optional | |
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| Existence and Uniqueness of Solutions | |
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| Chapter 1 Review Questions and Problems | |
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| Summary of Chapter 1 | |
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| Second-Order Linear ODEs | |
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| Homogeneous Linear ODEs of Second Order | |
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| Homogeneous Linear ODEs with Constant Coefficients | |
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| Differential Operators. Optional | |
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| Modeling: Free Oscillations. (Mass-Spring System) | |
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| Euler-Cauchy Equations | |
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| Existence and Uniqueness of Solutions. Wronskian | |
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| Nonhomogeneous ODEs | |
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| Modeling: Forced Oscillations. Resonance | |
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| Modeling: Electric Circuits | |
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| Solution by Variation of Parameters | |
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| Chapter 2 Review Questions and Problems | |
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| Summary of Chapter 2 | |
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| Higher Order Linear ODEs | |
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| Homogeneous Linear ODEs | |
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| Homogeneous Linear ODEs with Constant Coefficients | |
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| Nonhomogeneous Linear ODEs | |
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| Chapter 3 Review Questions and Problems | |
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| Summary of Chapter 3 | |
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| Systems of ODEs. Phase Plane. Qualitative Methods | |
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| Basics of Matrices and Vectors | |
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| Systems of ODEs as Models | |
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| Basic Theory of Systems of ODEs | |
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| Constant-Coefficient Systems. Phase Plane Method | |
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| Criteria for Critical Points. Stability | |
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| Qualitative Methods for Nonlinear Systems | |
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| Nonhomogeneous Linear Systems of ODEs | |
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| Chapter 4 Review Questions and Problems | |
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| Summary of Chapter 4 | |
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| Series Solutions of ODEs. Special Functions | |
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| Power Series Method | |
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| Theory of the Power Series Method | |
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| Legendre's Equation. Legendre Polynomials P[subscript n](x) | |
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| Frobenius Method | |
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| Bessel's Equation. Bessel Functions J[subscript v](x) | |
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| Bessel Functions of the Second Kind Y[subscript v](x) | |
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| Sturm-Liouville Problems. Orthogonal Functions | |
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| Orthogonal Eigenfunction Expansions | |
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| Chapter 5 Review Questions and Problems | |
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| Summary of Chapter 5 | |
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| Laplace Transforms | |
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| Laplace Transform. Inverse Transform. Linearity. s-Shifting | |
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| Transforms of Derivatives and Integrals. ODEs | |
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| Unit Step Function. t-Shifting | |
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| Short Impulses. Dirac's Delta Function. Partial Fractions | |
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| Convolution. Integral Equations | |
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| Differentiation and Integration of Transforms | |
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| Systems of ODEs | |
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| Laplace Transform: General Formulas | |
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| Table of Laplace Transforms | |
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| Chapter 6 Review Questions and Problems | |
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| Summary of Chapter 6 | |
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| Linear Algebra. Vector Calculus | |
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| Linear Algebra: Matrices, Vectors, Determinants, Linear Systems | |
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| Matrices, Vectors: Addition and Scalar Multiplication | |
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| Matrix Multiplication | |
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| Linear Systems of Equations. Gauss Elimination | |
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| Linear Independence. Rank of a Matrix. Vector Space | |
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| Solutions of Linear Systems: Existence, Uniqueness | |
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| For Reference: Second- and Third-Order Determinants | |
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| Determinants. Cramer's Rule | |
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| Inverse of a Matrix. Gauss-Jordan Elimination | |
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| Vector Spaces, Inner Product Spaces. Linear Transformations. Optional | |
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| Chapter 7 Review Questions and Problems | |
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| Summary of Chapter 7 | |
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| Linear Algebra: Matrix Eigenvalue Problems | |
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| Eigenvalues, Eigenvectors | |
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| Some Applications of Eigenvalue Problems | |
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| Symmetric, Skew-Symmetric, and Orthogonal Matrices | |
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| Eigenbases. Diagonalization. Quadratic Forms | |
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| Complex Matrices and Forms. Optional | |
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| Chapter 8 Review Questions and Problems | |
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| Summary of Chapter 8 | |
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| Vector Differential Calculus. Grad, Div, Curl | |
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| Vectors in 2-Space and 3-Space | |
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| Inner Product (Dot Product) | |
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| Vector Product (Cross Product) | |
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| Vector and Scalar Functions and Fields. Derivatives | |
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| Curves. Arc Length. Curvature. Torsion | |
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| Calculus Review: Functions of Several Variables. Optional | |
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| Gradient of a Scalar Field. Directional Derivative | |
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| Divergence of a Vector Field | |
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| Curl of a Vector Field | |
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| Chapter 9 Review Questions and Problems | |
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| Summary of Chapter 9 | |
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| Vector Integral Calculus. Integral Theorems | |
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| Line Integrals | |
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| Path Independence of Line Integrals | |
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| Calculus Review: Double Integrals. Optional | |
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| Green's Theorem in the Plane | |
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| Surfaces for Surface Integrals | |
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| Surface Integrals | |
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| Triple Integrals. Divergence Theorem of Gauss | |
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| Further Applications of the Divergence Theorem | |
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| Stokes's Theorem | |
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| Chapter 10 Review Questions and Problems | |
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| Summary of Chapter 10 | |
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| Fourier Analysis. Partial Differential Equations (PDEs) | |
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| Fourier Series, Integrals, and Transforms | |
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| Fourier Series | |
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| Functions of Any Period p = 2L | |
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| Even and Odd Functions. Half-Range Expansions | |
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| Complex Fourier Series. Optional | |
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| Forced Oscillations | |
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| Approximation by Trigonometric Polynomials | |
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| Fourier Integral | |
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| Fourier Cosine and Sine Transforms | |
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| Fourier Transform. Discrete and Fast Fourier Transforms | |
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| Tables of Transforms | |
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| Chapter 11 Review Questions and Problems | |
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| Summary of Chapter 11 | |
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| Partial Differential Equations (PDEs) | |
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| Basic Concepts | |
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| Modeling: Vibrating String, Wave Equation | |
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| Solution by Separating Variables. Use of Fourier Series | |
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| D'Alembert's Solution of the Wave Equation. Characteristics | |
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| Heat Equation: Solution by Fourier Series | |
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| Heat Equation: Solution by Fourier Integrals and Transforms | |
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| Modeling: Membrane, Two-Dimensional Wave Equation | |
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| Rectangular Membrane. Double Fourier Series | |
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| Laplacian in Polar Coordinates. Circular Membrane. Fourier-Bessel Series | |
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| Laplace's Equation in Cylindrical and Spherical Coordinates. Potential | |
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| Solution of PDEs by Laplace Transforms | |
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| Chapter 12 Review Questions and Problems | |
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| Summary of Chapter 12 | |
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| Complex Analysis | |
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| Complex Numbers and Functions | |
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| Complex Numbers. Complex Plane | |
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| Polar Form of Complex Numbers. Powers and Roots | |
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| Derivative. Analytic Function | |
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| Cauchy-Riemann Equations. Laplace's Equation | |
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| Exponential Function | |
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| Trigonometric and Hyperbolic Functions | |
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| Logarithm. General Power | |
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| Chapter 13 Review Questions and Problems | |
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| Summary of Chapter 13 | |
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| Complex Integration | |
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| Line Integral in the Complex Plane | |
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| Cauchy's Integral Theorem | |
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| Cauchy's Integral Formula | |
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| Derivatives of Analytic Functions | |
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| Chapter 14 Review Questions and Problems | |
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| Summary of Chapter 14 | |
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| Power Series, Taylor Series | |
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| Sequences, Series, Convergence Tests | |
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| Power Series | |
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| Functions Given by Power Series | |
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| Taylor and Maclaurin Series | |
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| Uniform Convergence. Optional | |
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| Chapter 15 Review Questions and Problems | |
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| Summary of Chapter 15 | |
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| Laurent Series. Residue Integration | |
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| Laurent Series | |
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| Singularities and Zeros. Infinity | |
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| Residue Integration Method | |
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| Residue Integration of Real Integrals | |
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| Chapter 16 Review Questions and Problems | |
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| Summary of Chapter 16 | |
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| Conformal Mapping | |
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| Geometry of Analytic Functions: Conformal Mapping | |
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| Linear Fractional Transformations | |
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| Special Linear Fractional Transformations | |
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| Conformal Mapping by Other Functions | |
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| Riemann Surfaces. Optional | |
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| Chapter 17 Review Questions and Problems | |
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| Summary of Chapter 17 | |
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| Complex Analysis and Potential Theory | |
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| Electrostatic Fields | |
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| Use of Conformal Mapping. Modeling | |
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| Heat Problems | |
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| Fluid Flow | |
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| Poisson's Integral Formula for Potentials | |
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| General Properties of Harmonic Functions | |
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| Chapter 18 Review Questions and Problems | |
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| Summary of Chapter 18 | |
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| Numeric Analysis | |
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| Software | |
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| Numerics in General | |
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| Introduction | |
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| Solution of Equations by Iteration | |
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| Interpolation | |
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| Spline Interpolation | |
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| Numeric Integration and Differentiation | |
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| Chapter 19 Review Questions and Problems | |
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| Summary of Chapter 19 | |
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| Numeric Linear Algebra | |
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| Linear Systems: Gauss Elimination | |
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| Linear Systems: LU-Factorization, Matrix Inversion | |
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| Linear Systems: Solution by Iteration | |
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| Linear Systems: Ill-Conditioning, Norms | |
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| Least Squares Method | |
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| Matrix Eigenvalue Problems: Introduction | |
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| Inclusion of Matrix Eigenvalues | |
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| Power Method for Eigenvalues | |
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| Tridiagonalization and QR-Factorization | |
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| Chapter 20 Review Questions and Problems | |
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| Summary of Chapter 20 | |
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| Numerics for ODEs and PDEs | |
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| Methods for First-Order ODEs | |
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| Multistep Methods | |
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| Methods for Systems and Higher Order ODEs | |
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| Methods for Elliptic PDEs | |
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| Neumann and Mixed Problems. Irregular Boundary | |
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| Methods for Parabolic PDEs | |
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| Method for Hyperbolic PDEs | |
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| Chapter 21 Review Questions and Problems | |
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| Summary of Chapter 21 | |
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| Optimization, Graphs | |
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| Unconstrained Optimization. Linear Programming | |
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| Basic Concepts. Unconstrained Optimization | |
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| Linear Programming | |
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| Simplex Method | |
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| Simplex Method: Difficulties | |
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| Chapter 22 Review Questions and Problems | |
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| Summary of Chapter 22 | |
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| Graphs. Combinatorial Optimization | |
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| Graphs and Digraphs | |
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| Shortest Path Problems. Complexity | |
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| Bellman's Principle. Dijkstra's Algorithm | |
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| Shortest Spanning Trees: Greedy Algorithm | |
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| Shortest Spanning Trees: Prim's Algorithm | |
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| Flows in Networks | |
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| Maximum Flow: Ford-Fulkerson Algorithm | |
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| Bipartite Graphs. Assignment Problems | |
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| Chapter 23 Review Questions and Problems | |
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| Summary of Chapter 23 | |
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| Probability, Statistics | |
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| Data Analysis. Probability Theory | |
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| Data Representation. Average. Spread | |
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| Experiments, Outcomes, Events | |
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| Probability | |
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| Permutations and Combinations | |
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| Random Variables. Probability Distributions | |
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| Mean and Variance of a Distribution | |
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| Binomial, Poisson, and Hypergeometric Distributions | |
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| Normal Distribution | |
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| Distributions of Several Random Variables | |
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| Chapter 24 Review Questions and Problems | |
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| Summary of Chapter 24 | |
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| Mathematical Statistics | |
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| Introduction. Random Sampling | |
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| Point Estimation of Parameters | |
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| Confidence Intervals | |
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| Testing Hypotheses. Decisions | |
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| Quality Control | |
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| Acceptance Sampling | |
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| Goodness of Fit. x[superscript 2]-Test | |
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| Nonparametric Tests | |
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| Regression. Fitting Straight Lines. Correlation | |
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| Chapter 25 Review Questions and Problems | |
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| Summary of Chapter 25 | |
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| References | |
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| Answers to Odd-Numbered Problems | |
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| Auxiliary Material | |
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| Formulas for Special Functions | |
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| Partial Derivatives | |
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| Sequences and Series | |
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| Grad, Div, Curl, [down triangle, open] [superscript 2] in Curvilinear Coordinates | |
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| Additional Proofs | |
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| Tables | |
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| Photo Credits | |
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| Index | |