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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 | |