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| First-Order Differential Equations | |
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| Terminology and Separable Equations | |
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| Linear Equations | |
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| Exact Equations | |
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| Homogeneous, Bernoulli and Riccsti Equations | |
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| Additional Applications | |
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| Existence and Uniqueness Questions | |
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| Linear Second-Order Equations | |
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| The Linear Second-Order Equations | |
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| The Constant Coefficient Case | |
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| The Nonhomogeneous Equation | |
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| Spring Motion | |
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| Euler's Differential Equation | |
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| The Laplace Transform Definition and Notation | |
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| Solution of Initial Value Problems | |
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| Shifiting and the Heaviside Function | |
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| Convolution | |
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| Impulses and the Delta Function | |
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| Solution of Systems | |
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| Polynomial Coefficients | |
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| Appendix on Partial Fractions Decompositions | |
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| Series Solutions | |
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| Power Series Solutions | |
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| Frobenius Solutions | |
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| Approximation Of Solutions Direction Fields | |
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| Euler's Method | |
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| Taylor and Modified Euler Methods | |
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| Vectors And Vector Spaces | |
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| Vectors in the Plane and 3 - Space | |
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| The Dot Product | |
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| The Cross Product | |
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| The Vector Space Rn | |
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| Orthogonalization | |
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| Orthogonal Complements and Projections | |
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| The Function Space C[a,b] | |
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| Matrices And Linear Systems | |
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| Matrices | |
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| Elementary Row Operations | |
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| Reduced Row Echelon Form | |
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| Row and Column Spaces | |
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| Homogeneous Systems | |
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| Nonhomogeneous Systems | |
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| Matrix Inverses | |
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| Least Squares Vectors and Data Fitting | |
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| LU - Factorization | |
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| Linear Transformations | |
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| Determinants | |
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| Definition of the Determinant | |
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| Evaluation of Determinants | |
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| Evaluationof Determinants | |
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| A Determinant Formula for A-1 | |
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| Cramer's Rule | |
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| The Matrix Tree Theorem | |
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| Eigenvalues, Diagonalization And Special Matrices | |
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| Diagonalization | |
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| Some Special Types of Matrices | |
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| Systems Of Linear Differential Equations | |
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| Linear Systems | |
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| Solution of X'=AX for Constant A. Solution of X'=AX+G | |
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| Exponential Matrix Solutions | |
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| Applications and Illustrations of Techniques | |
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| Phase Portaits | |
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| Vector Differential Calculu.S. Vector Functions of One Variable | |
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| Velocity and Curvature | |
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| Vector Fields and Streamlines | |
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| The Gradient Field | |
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| Divergence and Curl | |
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| Vector Integral Calculu.S | |
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| Line Integrals | |
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| Green's Theorem | |
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| An Extension of Green's Theorem | |
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| Independence of Path and Potential Theory | |
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| Surface Integrals | |
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| Applications of Surface Integrals | |
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| Lifting Green's Theorem to R3 | |
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| The Divergence Theorem of Gauss | |
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| Stokes's Theorem | |
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| Curvilinear Coordinates | |
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| Fourier Series | |
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| Why Fourier Series? | |
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| The Fourier Series of a Function | |
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| Sine and Cosine Series | |
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| Integration and Differentiation of Fourier Series | |
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| Phase Angle Form | |
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| Complex Fourier Series | |
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| Filtering of Signals | |
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| The Fourier Integral And Transforms | |
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| The Fourier Integral | |
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| Fourier Cosine and Sine Integrals | |
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| The Fourier Transform | |
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| Fourier Cosine and Sine Transforms | |
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| The Discrete Fourier Transform | |
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| Sampled Fourier Series | |
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| DFT Approximation of the Fourier Transform | |
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| Special Functions And Eigenfunction Expansions | |
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| Eigenfunction Expansions | |
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| Legendre Polynomials | |
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| Bessel Functions | |
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| Part V | |
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| The Wave Equation | |
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| Derivation of the Wave Equation | |
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| Wave Motion on an Interval | |
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| Wave Motion in an Infinite Medium | |
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| Wave Motion in a Semi-Infinite Medium | |
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| Laplace Transform Techniques | |
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| Characteristics and d'Alembert's Solution | |
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| Vibrations in a Circular Membrane | |
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| Vibrationsin a Circular Membrane | |
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| Vibrations in a Rectangular Membrane | |
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| The Heat Equation | |
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| Initial and Boundary Conditions | |
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| The Heat Equation on [0, L] | |
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| Solutions in an Infinite Medium | |
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| Laplace Transform Techniques | |
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| Heat Conduction in an Infinite Cylinder | |
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| Heat Conduction in a Rectangular Plate | |
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| The Potential Equation | |
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| Laplace's Equation | |
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| Dirichlet Problem for a Rectangle | |
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| Dirichlet Problem for a Disk | |
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| Poisson's Integral Formula | |
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| Dirichlet Problem for Unbounded Regions | |
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| A Dirichlet Problem for a Cube | |
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| Steady-State Equation for a Sphere | |
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| The Neumann Problem | |
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| Part VI | |
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| Complex Numbers And Functions | |
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| Geometry and Arithmetic of Complex Numbers | |
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| Complex Functions | |
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| The Exponential and Trigonometric Functions | |
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| The Complex Logarithm | |
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| Powers | |
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| Complex Integration | |
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| The Integral of a Complex Function | |
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| Cauchy's Theorem | |
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| Consequences of Cauchy's Theorem | |
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| Series Representations Of Functions | |
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| Power Series | |
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| The Laurent Expansion | |
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| Singularities And The Residue Theorem | |
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| Singularities | |
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| The Residue Theorem | |
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| Evaluation of Real Integrals | |
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| Residues and the Inverse Laplace Transform | |
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| Conformal Mappings And Applications | |
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| Conformal Mappings | |
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| Construction of Conformal Mappings | |
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| Conformal Mappings and Solutions of Dirichlet Problems | |
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| Models of Plane Fluid Flow | |
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| Appendix: A Maple Primer | |
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| Answers to Selected Problems | |