| |

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 | |

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