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What is included with this book?
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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