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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 | |