Machine generated contents note: pt. A Ordinary Differential Equations (ODEs)
1.1.Basic Concepts. Modeling
1.2.Geometric Meaning of y' = f(x, y). Direction Fields, Euler's Method
1.3.Separable ODEs. Modeling
1.4.Exact ODEs. Integrating Factors
1.5.Linear ODEs. Bernoulli Equation. Population Dynamics
1.6.Orthogonal Trajectories. Optional
1.7.Existence and Uniqueness of Solutions for Initial Value Problems
ch. 1 Review Questions and Problems
ch. 2 Second-Order Linear ODEs
2.1.Homogeneous Linear ODEs of Second Order
2.2.Homogeneous Linear ODEs with Constant Coefficients
2.3.Differential Operators. Optional
2.4.Modeling of Free Oscillations of a Mass-Spring System
2.5.Euler-Cauchy Equations
2.6.Existence and Uniqueness of Solutions. Wronskian
2.8.Modeling: Forced Oscillations. Resonance
2.9.Modeling: Electric Circuits
Note continued: 2.10.Solution by Variation of Parameters
ch. 2 Review Questions and Problems
ch. 3 Higher Order Linear ODEs
3.1.Homogeneous Linear ODEs
3.2.Homogeneous Linear ODEs with Constant Coefficients
3.3.Nonhomogeneous Linear ODEs
ch. 3 Review Questions and Problems
ch. 4 Systems of ODEs. Phase Plane. Qualitative Methods
4.0.For Reference: Basics of Matrices and Vectors
4.1.Systems of ODEs as Models in Engineering Applications
4.2.Basic Theory of Systems of ODEs. Wronskian
4.3.Constant-Coefficient Systems. Phase Plane Method
4.4.Criteria for Critical Points. Stability
4.5.Qualitative Methods for Nonlinear Systems
4.6.Nonhomogeneous Linear Systems of ODEs
ch. 4 Review Questions and Problems
ch. 5 Series Solutions of ODEs. Special Functions
5.2.Legendre's Equation. Legendre Polynomials Pn(x)
Note continued: 5.3.Extended Power Series Method: Frobenius Method
5.4.Bessel's Equation. Bessel Functions Jv(x)
5.5.Bessel Functions of the Yv(x). General Solution
ch. 5 Review Questions and Problems
6.1.Laplace Transform. Linearity. First Shifting Theorem (s-Shifting)
6.2.Transforms of Derivatives and Integrals. ODEs
6.3.Unit Step Function (Heaviside Function). Second Shifting Theorem (t-Shifting)
6.4.Short Impulses. Dirac's Delta Function. Partial Fractions
6.5.Convolution. Integral Equations
6.6.Differentiation and Integration of Transforms. ODEs with Variable Coefficients
6.8.Laplace Transform: General Formulas
6.9.Table of Laplace Transforms
ch. 6 Review Questions and Problems
pt. B Linear Algebra. Vector Calculus
ch. 7 Linear Algebra: Matrices, Vectors, Determinants. Linear Systems
Note continued: 7.1.Matrices, Vectors: Addition and Scalar Multiplication
7.2.Matrix Multiplication
7.3.Linear Systems of Equations. Gauss Elimination
7.4.Linear Independence. Rank of a Matrix. Vector Space
7.5.Solutions of Linear Systems: Existence, Uniqueness
7.6.For Reference: Second- and Third-Order Determinants
7.7.Determinants. Cramer's Rule
7.8.Inverse of a Matrix. Gauss-Jordan Elimination
7.9.Vector Spaces, Inner Product Spaces. Linear Transformations. Optional
ch. 7 Review Questions and Problems
ch. 8 Linear Algebra: Matrix Eigenvalue Problems
8.1.The Matrix Eigenvalue Problem. Determining Eigenvalues and Eigenvectors
8.2.Some Applications of Eigenvalue Problems
8.3.Symmetric, Skew-Symmetric, and Orthogonal Matrices
8.4.Eigenbases. Diagonalization. Quadratic Forms
8.5.Complex Matrices and Forms. Optional
ch. 8 Review Questions and Problems
Note continued: ch. 9 Vector Differential Calculus. Grad, Div, Curl
9.1.Vectors in 2-Space and 3-Space
9.2.Inner Product (Dot Product)
9.3.Vector Product (Cross Product)
9.4.Vector and Scalar Functions and Their Fields. Vector Calculus: Derivatives
9.5.Curves. Are Length. Curvature. Torsion
9.6.Calculus Review: Functions of Several Variables. Optional
9.7.Gradient of a Scalar Field. Directional Derivative
9.8.Divergence of a Vector Field
9.9.Curl of a Vector Field
ch. 9 Review Questions and Problems
ch. 10 Vector Integral Calculus. Integral Theorems
10.2.Path Independence of Line Integrals
10.3.Calculus Review: Double Integrals. Optional
10.4.Green's Theorem in the Plane
10.5.Surfaces for Surface Integrals
10.7.Triple Integrals. Divergence Theorem of Gauss
10.8.Further Applications of the Divergence Theorem
Note continued: ch. 10 Review Questions and Problems
pt. C Fourier Analysis. Partial Differential Equations (PDEs)
11.2.Arbitrary Period. Even and Odd Functions. Half-Range Expansions
11.4.Approximation by Trigonometric Polynomials
11.5.Sturm-Liouville Problems. Orthogonal Functions
11.6.Orthogonal Series. Generalized Fourier Series
11.8.Fourier Cosine and Sine Transforms
11.9.Fourier Transform. Discrete and Fast Fourier Transforms
11.10.Tables of Transforms
ch. 11 Review Questions and Problems
ch. 12 Partial Differential Equations (PDEs)
12.1.Basic Concepts of PDEs
12.2.Modeling: Vibrating String, Wave Equation
12.3.Solution by Separating Variables. Use of Fourier Series
12.4.D'Alembert's Solution of the Wave Equation. Characteristics
Note continued: 12.5.Modeling: Heat Flow from a Body in Space, Heat Equation
12.6.Heat Equation: Solution by Fourier Series. Steady Two-Dimensional Heat Problems. Dirichlet Problem
12.7.Heat Equation: Modeling Very Long Bars. Solution by Fourier Integrals and Transforms
12.8.Modeling: Membrane, Two-Dimensional Wave Equation
12.9.Rectangular Membrane. Double Fourier Series
12.10.Laplacian in Polar Coordinates. Circular Membrane. Fourier-Bessel Series
12.11.Laplace's Equation in Cylindrical and Spherical Coordinates. Potential
12.12.Solution of PDEs by Laplace Transforms
ch. 12 Review Questions and Problems
ch. 13 Complex Numbers and Functions. Complex Differentiation
13.1.Complex Numbers and Their Geometric Representation
13.2.Polar Form of Complex Numbers. Powers and Roots
13.3.Derivative. Analytic Function
13.4.Cauchy-Riemann Equations. Laplace's Equation
Note continued: 13.5.Exponential Function
13.6.Trigonometric and Hyperbolic Functions. Euler's Formula
13.7.Logarithm. General Power. Principal Value
ch. 13 Review Questions and Problems
ch. 14 Complex Integration
14.1.Line Integral in the Complex Plane
14.2.Cauchy's Integral Theorem
14.3.Cauchy's Integral Formula
14.4.Derivatives of Analytic Functions
ch. 14 Review Questions and Problems
ch. 15 Power Series, Taylor Series
15.1.Sequences, Series, Convergence Tests
15.3.Functions Given by Power Series
15.4.Taylor and Maclaurin Series
15.5.Uniform Convergence. Optional
ch. 15 Review Questions and Problems
ch. 16 Laurent Series. Residue Integration
16.2.Singularities and Zeros. Infinity
16.3.Residue Integration Method
16.4.Residue Integration of Real Integrals
Note continued: ch. 16 Review Questions and Problems
17.1.Geometry of Analytic Functions: Conformal Mapping
17.2.Linear Fractional Transformations (Mobius Transformations)
17.3.Special Linear Fractional Transformations
17.4.Conformal Mapping by Other Functions
17.5.Riemann Surfaces. Optional
ch. 17 Review Questions and Problems
ch. 18 Complex Analysis and Potential Theory
18.1.Electrostatic Fields
18.2.Use of Conformal Mapping. Modeling
18.5.Poisson's Integral Formula for Potentials
18.6.General Properties of Harmonic Functions. Uniqueness Theorem for the Dirichlet Problem
ch. 18 Review Questions and Problems
ch. 19 Numerics in General
19.2.Solution of Equations by Iteration
19.4.Spline Interpolation
Note continued: 19.5.Numeric Integration and Differentiation
ch. 19 Review Questions and Problems
ch. 20 Numeric Linear Algebra
20.1.Linear Systems: Gauss Elimination
20.2.Linear Systems: LU-Factorization, Matrix Inversion
20.3.Linear Systems: Solution by Iteration
20.4.Linear Systems: Ill-Conditioning, Norms
20.5.Least Squares Method
20.6.Matrix Eigenvalue Problems: Introduction
20.7.Inclusion of Matrix Eigenvalues
20.8.Power Method for Eigenvalues
20.9.Tridiagonalization and QR-Factorization
ch. 20 Review Questions and Problems
ch. 21 Numerics for ODEs and PDEs
21.1.Methods for First-Order ODEs
21.3.Methods for Systems and Higher Order ODEs
21.4.Methods for Elliptic PDEs
21.5.Neumann and Mixed Problems. Irregular Boundary
21.6.Methods for Parabolic PDEs
21.7.Method for Hyperbolic PDEs
ch. 21 Review Questions and Problems
Note continued: Summary of Chapter 21
pt. F Optimization, Graphs
ch. 22 Unconstrained Optimization. Linear Programming
22.1.Basic Concepts. Unconstrained Optimization: Method of Steepest Descent
22.4.Simplex Method: Difficulties
ch. 22 Review Questions and Problems
ch. 23 Graphs. Combinatorial Optimization
23.2.Shortest Path Problems. Complexity
23.3.Bellman's Principle. Dijkstra's Algorithm
23.4.Shortest Spanning Trees: Greedy Algorithm
23.5.Shortest Spanning Trees: Prim's Algorithm
23.7.Maximum Flow: Ford-Fulkerson Algorithm
23.8.Bipartite Graphs. Assignment Problems
ch. 23 Review Questions and Problems
pt. G Probability, Statistics
ch. 24 Data Analysis. Probability Theory
24.1.Data Representation. Average. Spread
24.2.Experiments, Outcomes, Events
