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Ordinary Differential Equations (ODEs) | |
First-Order ODEs | |
Basic Concepts. Modeling | |
Geometric Meaning of y' = f(x, y). Direction Fields | |
Separable ODEs. Modeling | |
Exact ODEs. Integrating Factors | |
Linear ODEs. Bernoulli Equation. Population Dynamics | |
Orthogonal Trajectories. Optional | |
Existence and Uniqueness of Solutions | |
Chapter 1 Review Questions and Problems | |
Summary of Chapter 1 | |
Second-Order Linear ODEs | |
Homogeneous Linear ODEs of Second Order | |
Homogeneous Linear ODEs with Constant Coefficients | |
Differential Operators. Optional | |
Modeling: Free Oscillations. (Mass-Spring System) | |
Euler-Cauchy Equations | |
Existence and Uniqueness of Solutions. Wronskian | |
Nonhomogeneous ODEs | |
Modeling: Forced Oscillations. Resonance | |
Modeling: Electric Circuits | |
Solution by Variation of Parameters | |
Chapter 2 Review Questions and Problems | |
Summary of Chapter 2 | |
Higher Order Linear ODEs | |
Homogeneous Linear ODEs | |
Homogeneous Linear ODEs with Constant Coefficients | |
Nonhomogeneous Linear ODEs | |
Chapter 3 Review Questions and Problems | |
Summary of Chapter 3 | |
Systems of ODEs. Phase Plane. Qualitative Methods | |
Basics of Matrices and Vectors | |
Systems of ODEs as Models | |
Basic Theory of Systems of ODEs | |
Constant-Coefficient Systems. Phase Plane Method | |
Criteria for Critical Points. Stability | |
Qualitative Methods for Nonlinear Systems | |
Nonhomogeneous Linear Systems of ODEs | |
Chapter 4 Review Questions and Problems | |
Summary of Chapter 4 | |
Series Solutions of ODEs. Special Functions | |
Power Series Method | |
Legendre's Equation. Legendre Polynomials Pn(x) | |
Frobenius Method | |
Bessel's Equation. Bessel Functions Jv(x) | |
Bessel Functions of the Second Kind Yv(x) | |
Chapter 5 Review Questions and Problems | |
Summary of Chapter 5 | |
Laplace Transforms | |
Laplace Transform. Inverse Transform. Linearity. ^-Shifting | |
Transforms of Derivatives and Integrals. ODEs | |
Unit Step Function. f-Shifting | |
Short Impulses. Dirac's Delta Function. Partial Fractions | |
Convolution. Integral Equations | |
Differentiation and Integration of Transforms | |
Systems of ODEs | |
Laplace Transform: General Formulas | |
Table of Laplace Transforms | |
Chapter 6 Review Questions and Problems | |
Summary of Chapter 6 | |
Linear Algebra. Vector Calculus | |
Linear Algebra: Matrices, Vectors, Determinants. Linear Systems | |
Matrices, Vectors: Addition and Scalar Multiplication | |
Matrix Multiplication | |
Linear Systems of Equations. Gauss Elimination | |
Linear Independence. Rank of a Matrix. Vector Space | |
Solutions of Linear Systems: Existence, Uniqueness | |
For Reference: Second- and Third-Order Determinants | |
Determinants. Cramer's Rule | |
Inverse of a Matrix. Gauss-Jordan Elimination | |
Vector Spaces, Inner Product Spaces. Linear Transformations Optional | |
Chapter 7 Review Questions and Problems | |
Summary of Chapter 7 | |
Linear Algebra: Matrix Eigenvalue Problems | |
Eigenvalues, Eigenvectors | |
Some Applications of Eigenvalue Problems | |
Symmetric, Skew-Symmetric, and Orthogonal Matrices | |
Eigenbases. Diagonalization. Quadratic Forms | |
Complex Matrices and Forms. Optional | |
Chapter 8 Review Questions and Problems | |
Summary of Chapter 8 | |
Vector Differential Calculus. Grad, Div, Curl | |
Vectors in 2-Space and 3-Space | |
Inner Product | |
Vector Product | |
Vector and Scalar Functions and Fields. Derivatives | |
Curves. Arc Length. Curvature. Torsion | |
Calculus Review: Functions of Several Variables. Optional | |
Gradient of a Scalar Field. Directional Derivative | |
Divergence of a Vector Field | |
Curl of a Vector Field | |
Chapter 9 Review Questions and Problems | |
Summary of Chapter 9 | |
Vector Integral Calculus. Integral Theorems | |
Line Integrals | |
Path Independence of Line Integrals | |
Calculus Review: Double Integrals. Optional | |
Green's Theorem in the Plane | |
Surfaces for Surface Integrals | |
Surface Integrals | |
Triple Integrals. Divergence Theorem of Gauss | |
Further Applications of the Divergence Theorem | |
Stokes's Theorem | |
Chapter 10 Review Questions and Problems | |
Summary of Chapter 10 | |
Fourier Analysis. Partial Differential Equations (PDEs) | |
Fourier Series, Integrals, and Transforms | |
Fourier Series | |
Functions of Any Period p = 2L. Even and Odd Functions. Half-Range Expansions | |
Forced Oscillations | |
Approximation by Trigonometric Polynomials | |
Sturm-Liouville Problems. Orthogonal Functions | |
Orthogonal Eigenfunction Expansions | |
Fourier Integral | |
Fourier Cosine and Sine Transforms | |
Fourier Transform. Discrete and Fast Fourier Transforms | |
Tables of Transforms | |
Chapter 11 Review Questions and Problems | |
Summary of Chapter 11 | |
Partial Differential Equations (PDEs) | |
Basic Concepts | |
Modeling: Vibrating String, Wave Equation | |
Solution by Separating Variables. Use of Fourier Series | |
D'Alembert's Solution of the Wave Equation. Characteristics | |
Introduction to the Heat Equation | |
Heat Equation: Solution by Fourier Series | |
Heat Equation: Solution by Fourier Integrals and Transforms | |
Modeling: Membrane, Two-Dimensional Wave Equation | |
Rectangular Membrane. Double Fourier Series | |
Laplacian in Polar Coordinates. Circular Membrane. Fourier-Bessel Series | |
Laplace's Equation in Cylindrical and Spherical Coordinates. Potential | |
Solution of PDEs by Laplace Transforms | |
Chapter 12 Review Questions and Problems | |
Summary of Chapter 12 | |
Complex Analysis | |
Complex Numbers and Functions | |
Complex Numbers. Complex Plane | |
Polar Form of Complex Numbers. Powers and Roots | |
Derivative. Analytic Function | |
Cauchy-Riemann Equations. Laplace's Equation | |
Exponential Function | |
Trigonometric and Hyperbolic Functions | |
Logarithm. General Power | |
Chapter 13 Review Questions and Problems | |
Summary of Chapter 13 | |
Complex Integration | |
Line Integral in the Complex Plane | |
Cauchy's Integral Theorem | |
Cauchy's Integral Formula | |
Derivatives of Analytic Functions | |
Chapter 14 Review Questions and Problems | |
Summary of Chapter 14 | |
Power Series, Taylor Series | |
Sequences, Series, Convergence Tests | |
Power Series | |
Functions Given by Power Series | |
Taylor and Maclaurin Series | |
Uniform Convergence. Optional | |
Chapter 15 Review Questions and Problems | |
Summary of Chapter 15 | |
Laurent Series. Residue Integration | |
Laurent Series | |
Singularities and Zeros. Infinity | |
Residue Integration Method | |
Residue Integration of Real Integrals | |
Review Questions and Problems | |
Summary of Chapter 16 | |
Conformal Mapping | |
Geometry of Analytic Functions: Conformal Mapping | |
Linear Fractional Transformations | |
Special Linear Fractional Transformations | |
Conformal Mapping by Other Functions | |
Riemann Surfaces. Optional | |
Chapter 17 Review Questions and Problems | |
Summary of Chapter 17 | |
Complex Analysis and Potential Theory | |
Electrostatic Fields | |
Use of Conformal Mapping. Modeling | |
Heat Problems | |
Fluid Flow | |
Poisson's Integral Formula for Potentials | |
General Properties of Harmonic Functions | |
Chapter 18 Review Questions and Problems | |
Summary of Chapter 18 | |
Numeric Analysis | |
Software | |
Numerics in General | |
Introduction | |
Solution of Equations by Iteration | |
Interpolation | |
Spline Interpolation | |
Numeric Integration and Differentiation | |
Chapter 19 Review Questions and Problems | |
Summary of Chapter 19 | |
Numeric Linear Algebra | |
Linear Systems: Gauss Elimination | |
Linear Systems: LU-Factorization, Matrix Inversion | |
Linear Systems: Solution by Iteration | |
Linear Systems: Ill-Conditioning, Norms | |
Least Squares Method | |
Matrix Eigenvalue Problems: Introduction | |
Inclusion of Matrix Eigenvalues | |
Power Method for Eigenvalues | |
Tridiagonalization and QR-Factorization | |
Chapter 20 Review Questions and Problems | |
Summary of Chapter 20 | |
Numerics for ODEs and PDEs | |
Methods for First-Order ODEs | |
Multistep Methods | |
Methods for Systems and Higher Order ODEs | |
Methods for Elliptic PDEs | |
Neumann and Mixed Problems. Irregular Boundary | |
Methods for Parabolic PDEs | |
Method for Hyperbolic PDEs | |
Chapter 21 Review Questions and Problems | |
Summary of Chapter 21 | |
Optimization, Graphs | |
Unconstrained Optimization. Linear Programming | |
Basic Concepts. Unconstrained Optimization | |
Linear Programming | |
Simplex Method | |
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