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Preface | |

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About the Cover | |

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Project for Section 3.7 Road Mirages | |

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Project for Section 3.10 The Ballistic Pendulum | |

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Project for Section 8.1 Two-Ports in Electrical Circuits | |

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Project for Section 8.2 Traffic Flow | |

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Project for Section 8.15 Temperature Dependence of Resistivity | |

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Project for Section 9.16 Minimal Surfaces | |

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Project for Section 14.3 The Hydrogen Atom | |

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Project for Section 15.4 The Uncertainty Inequality in Signal Processing | |

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Project for Section 15.4 Fraunhofer Diffraction by a Circular Aperture | |

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Project for Section 16.2 Instabilities of Numerical Methods | |

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Ordinary Differential Equations | |

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Introduction to Differential Equations | |

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Definitions and Terminology | |

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Initial-Value Problems | |

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Differential Equations as Mathematical Models | |

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Review Exercises | |

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First-Order Differential Equations | |

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Solution Curves Without a Solution | |

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Direction Fields | |

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Autonomous First-Order DEs | |

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Separable Variables | |

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Linear Equations | |

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Exact Equations | |

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Solutions by Substitutions | |

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A Numerical Method | |

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Linear Models | |

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Nonlinear Models | |

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Modeling with Systems of First-Order DEs | |

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Review Exercises | |

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Higher-Order Differential Equations | |

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Preliminary Theory: Linear Equations | |

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Initial-Value and Boundary-Value Problems | |

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Homogeneous Equations | |

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Nonhomogeneous Equations | |

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Reduction of Order | |

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Homogeneous Linear Equations with Constant Coefficients | |

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Undetermined Coefficients | |

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Variation of Parameters | |

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Cauchy-Euler Equation | |

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Nonlinear Equations | |

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Linear Models: Initial-Value Problems | |

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Spring/Mass Systems: Free Undamped Motion | |

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Spring/Mass Systems: Free Damped Motion | |

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Spring/Mass Systems: Driven Motion | |

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Series Circuit Analogue | |

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Linear Models: Boundary-Value Problems | |

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Nonlinear Models | |

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Solving Systems of Linear Equations | |

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Review Exercises | |

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The Laplace Transform | |

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Definition of the Laplace Transform | |

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The Inverse Transform and Transforms of Derivatives | |

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Inverse Transforms | |

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Transforms of Derivatives | |

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Translation Theorems | |

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Translation on the s-axis | |

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Translation on the t-axis | |

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Additional Operational Properties | |

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Derivatives of Transforms | |

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Transforms of Integrals | |

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Transform of a Periodic Function | |

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The Dirac Delta Function | |

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Systems of Linear Differential Equations | |

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Review Exercises | |

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Series Solutions of Linear Differential Equations | |

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Solutions about Ordinary Points | |

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Review of Power Series | |

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Power Series Solutions | |

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Solutions about Singular Points | |

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Special Functions | |

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Bessel Functions | |

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Legendre Functions | |

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Review Exercises | |

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Numerical Solutions of Ordinary Differential Equations | |

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Euler Methods and Error Analysis | |

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Runge-Kutta Methods | |

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Multistep Methods | |

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Higher-Order Equations and Systems | |

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Second-Order Boundary-Value Problems | |

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Review Exercises | |

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Vectors, Matrices, and Vector Calculus | |

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Vectors | |

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Vectors in 2-Space | |

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Vectors in 3-Space | |

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Dot Product | |

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Cross Product | |

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Lines and Planes in 3-Space | |

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Vector Spaces | |

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Gram-Schmidt Orthogonalization Process | |

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Review Exercises | |

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Matrices | |

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Matrix Algebra | |

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Systems of Linear Algebraic Equations | |

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Rank of a Matrix | |

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Determinants | |

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Properties of Determinants | |

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Inverse of a Matrix | |

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Finding the Inverse | |

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Using the Inverse to Solve Systems | |

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Cramer's Rule | |

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The Eigenvalue Problem | |

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Powers of Matrices | |

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Orthogonal Matrices | |

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Approximation of Eigenvalues | |

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Diagonalization | |

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Cryptography | |

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An Error-Correcting Code | |

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Method of Least Squares | |

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Discrete Compartmental Models | |

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Review Exercises | |

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Vector Calculus | |

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Vector Functions | |

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Motion on a Curve | |

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Curvature and Components of Acceleration | |

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Partial Derivatives | |

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Directional Derivatives | |

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Tangent Planes and Normal Lines | |

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Divergence and Curl | |

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Line Integrals | |

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Independence of Path | |

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Double Integrals | |

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Double Integrals in Polar Coordinates | |

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Green's Theorem | |

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Surface Integrals | |

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Stokes' Theorem | |

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Triple Integrals | |

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Divergence Theorem | |

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Change of Variables in Multiple Integrals | |

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Review Exercises | |

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Systems of Differential Equations | |

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Systems of Linear Differential Equations | |

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Preliminary Theory | |

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Homogeneous Linear Systems | |

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Distinct Real Eigenvalues | |

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Repeated Eigenvalues | |

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Complex Eigenvalues | |

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Solution by Diagonalization | |

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Nonhomogeneous Linear Systems | |

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Undetermined Coefficients | |

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Variation of Parameters | |

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Diagonalization | |

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Matrix Exponential | |

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Review Exercises | |

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Systems of Nonlinear Differential Equations | |

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Autonomous Systems | |

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Stability of Linear Systems | |

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Linearization and Local Stability | |

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Autonomous Systems as Mathematical Models | |

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Periodic Solutions, Limit Cycles, and Global Stability | |

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Review Exercises | |

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Fourier Series and Partial Differential Equations | |

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Orthogonal Functions and Fourier Series | |

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Orthogonal Functions | |

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Fourier Series | |

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Fourier Cosine and Sine Series | |

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Complex Fourier Series | |

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Sturm-Liouville Problem | |

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Bessel and Legendre Series | |

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Fourier-Bessel Series | |

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Fourier-Legendre Series | |

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Review Exercises | |

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Boundary-Value Problems in Rectangular Coordinates | |

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Separable Partial Differential Equations | |

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Classical Equations and Boundary-Value Problems | |

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Heat Equation | |

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Wave Equation | |

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Laplace's Equation | |

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Nonhomogeneous BVPs | |

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Orthogonal Series Expansions | |

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Fourier Series in Two Variables | |

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Review Exercises | |

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Boundary-Value Problems in Other Coordinate Systems | |

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Problems in Polar Coordinates | |

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Problems in Cylindrical Coordinates | |

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Problems in Spherical Coordinates | |

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Review Exercises | |

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Integral Transform Method | |

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Error Function | |

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Applications of the Laplace Transform | |

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Fourier Integral | |

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Fourier Transforms | |

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Fast Fourier Transform | |

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Review Exercises | |

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Numerical Solutions of Partial Differential Equations | |

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Laplace's Equation | |

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The Heat Equation | |

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The Wave Equation | |

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Review Exercises | |

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Complex Analysis | |

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Functions of a Complex Variable | |

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Complex Numbers | |

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Powers and Roots | |

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Sets in the Complex Plane | |

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Functions of a Complex Variable | |

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Cauchy-Riemann Equations | |

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Exponential and Logarithmic Functions | |

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Trigonometric and Hyperbolic Functions | |

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Inverse Trigonometric and Hyperbolic Functions | |

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Review Exercises | |

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Integration in the Complex Plane | |

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Contour Integrals | |

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Cauchy-Goursat Theorem | |

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Independence of Path | |

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Cauchy's Integral Formulas | |

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Review Exercises | |

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Series and Residues | |

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Sequences and Series | |

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Taylor Series | |

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Laurent Series | |

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Zeros and Poles | |

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Residues and Residue Theorem | |

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Evaluation of Real Integrals | |

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Review Exercises | |

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Conformal Mappings | |

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Complex Functions as Mappings | |

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Conformal Mappings | |

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Linear Fractional Transformations | |

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Schwarz-Christoffel Transformations | |

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Poisson Integral Formulas | |

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Applications | |

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Review Exercises | |

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Appendices | |

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Some Derivative and Integral Formulas | |

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Gamma Function | |

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Table of Laplace Transforms | |

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Conformal Mappings | |

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Answers for Selected Odd-Numbered Problems | |

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Index | |