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Ordinary Differential Equations | |

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First-Order Differential Equations | |

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Preliminary Concepts | |

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Separable Equations | |

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Linear Differential Equations | |

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Exact Differential Equations | |

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Integrating Factors | |

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Homogeneous, Bernoulli, and Riccati Equations | |

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Applications to Mechanics, Electrical Circuits, and Orthogonal Trajectories | |

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Existence and Uniqueness for Solutions of Initial Value Problems | |

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Second-Order Differential Equations | |

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Preliminary Concepts | |

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Theory of Solutions y" + p(x)y' + q(x)y = f(x) | |

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Reduction of Order | |

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The Constant Coefficient Homogeneous Linear Equation | |

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Euler's Equation | |

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The Nonhomogeneous Equation y" + p(x)y' + q(x)y = f(x) | |

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Application of Second-Order Differential Equations to a Mechanical System | |

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The Laplace Transform | |

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Definition and Basic Properties | |

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Solution of Initial Value Problems Using the Laplace Transform | |

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Shifting Theorems and the Heaviside Function | |

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Convolution | |

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Unit Impulses and the Dirac Delta Function | |

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Laplace Transform Solution of Systems | |

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Differential Equations with Polynomial Coefficients | |

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Series Solutions | |

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Power Series Solutions of Initial Value Problems | |

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Power Series Solutions Using Recurrence Relations | |

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Singular Points and the Method of Frobenius | |

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Second Solutions and Logarithm Factors | |

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Appendix on Power Series | |

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Vectors and Linear Algebra | |

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Vectors and Vector Spaces | |

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The Algebra and Geometry of Vectors | |

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The Dot Product | |

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The Cross Product | |

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The Vector Space R[superscript n] | |

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Linear Independence, Spanning Sets, and Dimension in R[superscript n] | |

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Abstract Vector Spaces | |

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Matrices and Systems of Linear Equations | |

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Matrices | |

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Elementary Row Operations and Elementary Matrices | |

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The Row Echelon Form of a Matrix | |

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The Row and Column Spaces of a Matrix and Rank of a Matrix | |

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Solution of Homogeneous Systems of Linear Equations | |

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The Solution Space of AX = O | |

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Nonhomogeneous Systems of Linear Equations | |

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Summary for Linear Systems | |

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Matrix Inverses | |

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Determinants | |

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Permutations | |

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Definition of the Determinant | |

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Properties of Determinants | |

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Evaluation of Determinants by Elementary Row and Column Operations | |

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Cofactor Expansions | |

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Determinants of Triangular Matrices | |

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A Determinant Formula for a Matrix Inverse | |

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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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Eigenvalues and Eigenvectors | |

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Diagonalization of Matrices | |

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Orthogonal and Symmetric Matrices | |

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Quadratic Forms | |

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Unitary, Hermitian, and Skew-Hermitian Matrices | |

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Systems of Differential Equations and Qualitative Methods | |

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Systems of Linear Differential Equations | |

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Theory of Systems of Linear First-Order Differential Equations | |

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Solution of X' = AX When A Is Constant | |

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Solution of X' = AX + G | |

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Qualitative Methods and Systems of Nonlinear Differential Equations | |

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Nonlinear Systems and Existence of Solutions | |

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The Phase Plane, Phase Portraits, and Direction Fields | |

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Phase Portraits of Linear Systems | |

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Critical Points and Stability | |

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Almost Linear Systems | |

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Predator/Prey Population Models | |

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Competing Species Models | |

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Lyapunov's Stability Criteria | |

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Limit Cycles and Periodic Solutions | |

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Vector Analysis | |

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Vector Differential Calculus | |

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Vector Functions of One Variable | |

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Velocity, Acceleration, Curvature, and Torsion | |

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Vector Fields and Streamlines | |

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The Gradient Field and Directional Derivatives | |

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Divergence and Curl | |

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Vector Integral Calculus | |

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Line Integrals | |

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Green's Theorem | |

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Independence of Path and Potential Theory in the Plane | |

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Surfaces in 3-Space and Surface Integrals | |

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Applications of Surface Integrals | |

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Preparation for the Integral Theorems of Gauss and Stokes | |

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The Divergence Theorem of Gauss | |

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The Integral Theorem of Stokes | |

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Fourier Analysis, Orthogonal Expansions, and Wavelets | |

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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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Convergence of Fourier Series | |

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Fourier Cosine and Sine Series | |

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Integration and Differentiation of Fourier Series | |

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The Phase Angle Form of a Fourier Series | |

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Complex Fourier Series and the Frequency Spectrum | |

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The Fourier Integral and Fourier Transforms | |

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The Fourier Integral | |

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Fourier Cosine and Sine Integrals | |

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The Complex Fourier Integral and the Fourier Transform | |

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Additional Properties and Applications of the Fourier Transform | |

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The Fourier Cosine and Sine Transforms | |

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The Finite Fourier Cosine and Sine Transforms | |

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The Discrete Fourier Transform | |

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Sampled Fourier Series | |

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The Fast Fourier Transform | |

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Special Functions, Orthogonal Expansions, and Wavelets | |

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Legendre Polynomials | |

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Bessel Functions | |

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Sturm-Liouville Theory and Eigenfunction Expansions | |

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Orthogonal Polynomials | |

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Wavelets | |

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Partial Differential Equations | |

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The Wave Equation | |

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The Wave Equation and Initial and Boundary Conditions | |

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Fourier Series Solutions of the Wave Equation | |

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Wave Motion Along Infinite and Semi-infinite Strings | |

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Characteristics and d'Alembert's Solution | |

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Normal Modes of Vibration of a Circular Elastic Membrane | |

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Vibrations of a Circular Elastic Membrane, Revisited | |

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Vibrations of a Rectangular Membrane | |

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The Heat Equation | |

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The Heat Equation and Initial and Boundary Conditions | |

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Fourier Series Solutions of the Heat Equation | |

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Heat Conduction in Infinite Media | |

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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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Harmonic Functions and the Dirichlet Problem | |

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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 for the Disk | |

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Dirichlet Problems in Unbounded Regions | |

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A Dirichlet Problem for a Cube | |

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The Steady-State Heat Equation for a Solid Sphere | |

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The Neumann Problem | |

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Canonical Forms, Existence and Uniqueness of Solutions, and Well-Posed Problems | |

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Canonical Forms | |

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Existence and Uniqueness of Solutions | |

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Well-Posed Problems | |

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Complex Analysis | |

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Geometry and Arithmetic of Complex Numbers | |

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Complex Numbers | |

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Loci and Sets of Points in the Complex Plane | |

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Complex Functions | |

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Limits, Continuity, and Derivatives | |

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Power Series | |

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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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Curves in the Plane | |

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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 Representations | |

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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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Some Applications of the Residue Theorem | |

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Conformal Mappings | |

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Functions as Mappings | |

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Conformal Mappings | |

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Construction of Conformal Mappings Between Domains | |

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Harmonic Functions and the Dirichlet Problem | |

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Complex Function Models of Plane Fluid Flow | |

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Historical Notes | |

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Development of Areas of Mathematics | |

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Ordinary Differential Equations | |

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Matrices and Determinants | |

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Vector Analysis | |

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Fourier Analysis | |

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Partial Differential Equations | |

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Complex Function Theory | |

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Biographical Sketches | |

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Galileo Galilei (1564-1642) | |

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Isaac Newton (1642-1727) | |

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Gottfried Wilhelm Leibniz (1646-1716) | |

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The Bernoulli Family | |

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Leonhard Euler (1707-1783) | |

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Carl Friedrich Gauss (1777-1855) | |

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Joseph-Louis Lagrange (1736-1813) | |

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Pierre-Simon de Laplace (1749-1827) | |

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Augustin-Louis Cauchy (1789-1857) | |

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Joseph Fourier (1768-1830) | |

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Henri Poincare (1854-1912) | |

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Answers and Solutions to Selected Odd-Numbered Problems | |

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Index | |