| |

| |

| |

| |

| |

| |

First-Order Differential Equations | |

| |

| |

Terminology and Separable Equations | |

| |

| |

Linear Equations | |

| |

| |

Exact Equations | |

| |

| |

Homogeneous, Bernoulli and Riccsti Equations | |

| |

| |

Additional Applications | |

| |

| |

Existence and Uniqueness Questions | |

| |

| |

| |

Linear Second-Order Equations | |

| |

| |

The Linear Second-Order Equations | |

| |

| |

The Constant Coefficient Case | |

| |

| |

The Nonhomogeneous Equation | |

| |

| |

Spring Motion | |

| |

| |

Euler's Differential Equation | |

| |

| |

| |

The Laplace Transform Definition and Notation | |

| |

| |

Solution of Initial Value Problems | |

| |

| |

Shifiting and the Heaviside Function | |

| |

| |

Convolution | |

| |

| |

Impulses and the Delta Function | |

| |

| |

Solution of Systems | |

| |

| |

Polynomial Coefficients | |

| |

| |

Appendix on Partial Fractions Decompositions | |

| |

| |

| |

Series Solutions | |

| |

| |

Power Series Solutions | |

| |

| |

Frobenius Solutions | |

| |

| |

| |

Approximation Of Solutions Direction Fields | |

| |

| |

Euler's Method | |

| |

| |

Taylor and Modified Euler Methods | |

| |

| |

| |

| |

| |

| |

Vectors And Vector Spaces | |

| |

| |

Vectors in the Plane and 3 - Space | |

| |

| |

The Dot Product | |

| |

| |

The Cross Product | |

| |

| |

The Vector Space Rn | |

| |

| |

Orthogonalization | |

| |

| |

Orthogonal Complements and Projections | |

| |

| |

The Function Space C[a,b] | |

| |

| |

| |

Matrices And Linear Systems | |

| |

| |

Matrices | |

| |

| |

Elementary Row Operations | |

| |

| |

Reduced Row Echelon Form | |

| |

| |

Row and Column Spaces | |

| |

| |

Homogeneous Systems | |

| |

| |

Nonhomogeneous Systems | |

| |

| |

Matrix Inverses | |

| |

| |

Least Squares Vectors and Data Fitting | |

| |

| |

LU - Factorization | |

| |

| |

Linear Transformations | |

| |

| |

| |

Determinants | |

| |

| |

Definition of the Determinant | |

| |

| |

Evaluation of Determinants | |

| |

| |

| |

Evaluationof Determinants | |

| |

| |

| |

A Determinant Formula for A-1 | |

| |

| |

Cramer's Rule | |

| |

| |

The Matrix Tree Theorem | |

| |

| |

| |

Eigenvalues, Diagonalization And Special Matrices | |

| |

| |

Diagonalization | |

| |

| |

Some Special Types of Matrices | |

| |

| |

| |

Systems Of Linear Differential Equations | |

| |

| |

Linear Systems | |

| |

| |

Solution of X'=AX for Constant A. Solution of X'=AX+G | |

| |

| |

Exponential Matrix Solutions | |

| |

| |

Applications and Illustrations of Techniques | |

| |

| |

Phase Portaits | |

| |

| |

| |

| |

| |

| |

Vector Differential Calculu.S. Vector Functions of One Variable | |

| |

| |

Velocity and Curvature | |

| |

| |

Vector Fields and Streamlines | |

| |

| |

The Gradient Field | |

| |

| |

Divergence and Curl | |

| |

| |

| |

Vector Integral Calculu.S | |

| |

| |

Line Integrals | |

| |

| |

Green's Theorem | |

| |

| |

An Extension of Green's Theorem | |

| |

| |

Independence of Path and Potential Theory | |

| |

| |

Surface Integrals | |

| |

| |

Applications of Surface Integrals | |

| |

| |

Lifting Green's Theorem to R3 | |

| |

| |

The Divergence Theorem of Gauss | |

| |

| |

Stokes's Theorem | |

| |

| |

Curvilinear Coordinates | |

| |

| |

| |

| |

| |

| |

Fourier Series | |

| |

| |

Why Fourier Series? | |

| |

| |

The Fourier Series of a Function | |

| |

| |

Sine and Cosine Series | |

| |

| |

Integration and Differentiation of Fourier Series | |

| |

| |

Phase Angle Form | |

| |

| |

Complex Fourier Series | |

| |

| |

Filtering of Signals | |

| |

| |

| |

The Fourier Integral And Transforms | |

| |

| |

The Fourier Integral | |

| |

| |

Fourier Cosine and Sine Integrals | |

| |

| |

The Fourier Transform | |

| |

| |

Fourier Cosine and Sine Transforms | |

| |

| |

The Discrete Fourier Transform | |

| |

| |

Sampled Fourier Series | |

| |

| |

DFT Approximation of the Fourier Transform | |

| |

| |

| |

Special Functions And Eigenfunction Expansions | |

| |

| |

Eigenfunction Expansions | |

| |

| |

Legendre Polynomials | |

| |

| |

Bessel Functions | |

| |

| |

Part V | |

| |

| |

| |

The Wave Equation | |

| |

| |

Derivation of the Wave Equation | |

| |

| |

Wave Motion on an Interval | |

| |

| |

Wave Motion in an Infinite Medium | |

| |

| |

Wave Motion in a Semi-Infinite Medium | |

| |

| |

Laplace Transform Techniques | |

| |

| |

Characteristics and d'Alembert's Solution | |

| |

| |

Vibrations in a Circular Membrane | |

| |

| |

| |

Vibrationsin a Circular Membrane | |

| |

| |

| |

Vibrations in a Rectangular Membrane | |

| |

| |

| |

The Heat Equation | |

| |

| |

Initial and Boundary Conditions | |

| |

| |

The Heat Equation on [0, L] | |

| |

| |

Solutions in an Infinite Medium | |

| |

| |

Laplace Transform Techniques | |

| |

| |

Heat Conduction in an Infinite Cylinder | |

| |

| |

Heat Conduction in a Rectangular Plate | |

| |

| |

| |

The Potential Equation | |

| |

| |

Laplace's Equation | |

| |

| |

Dirichlet Problem for a Rectangle | |

| |

| |

Dirichlet Problem for a Disk | |

| |

| |

Poisson's Integral Formula | |

| |

| |

Dirichlet Problem for Unbounded Regions | |

| |

| |

A Dirichlet Problem for a Cube | |

| |

| |

Steady-State Equation for a Sphere | |

| |

| |

The Neumann Problem | |

| |

| |

Part VI | |

| |

| |

| |

Complex Numbers And Functions | |

| |

| |

Geometry and Arithmetic of Complex Numbers | |

| |

| |

Complex Functions | |

| |

| |

The Exponential and Trigonometric Functions | |

| |

| |

The Complex Logarithm | |

| |

| |

Powers | |

| |

| |

| |

Complex Integration | |

| |

| |

The Integral of a Complex Function | |

| |

| |

Cauchy's Theorem | |

| |

| |

Consequences of Cauchy's Theorem | |

| |

| |

| |

Series Representations Of Functions | |

| |

| |

Power Series | |

| |

| |

The Laurent Expansion | |

| |

| |

| |

Singularities And The Residue Theorem | |

| |

| |

Singularities | |

| |

| |

The Residue Theorem | |

| |

| |

Evaluation of Real Integrals | |

| |

| |

Residues and the Inverse Laplace Transform | |

| |

| |

| |

Conformal Mappings And Applications | |

| |

| |

Conformal Mappings | |

| |

| |

Construction of Conformal Mappings | |

| |

| |

Conformal Mappings and Solutions of Dirichlet Problems | |

| |

| |

Models of Plane Fluid Flow | |

| |

| |

Appendix: A Maple Primer | |

| |

| |

Answers to Selected Problems | |