CONTENTS
PART A Ordinary Differential Equations (ODEs)
1.2 Geometric Meaning of y′ = f(x, y). Direction Fields, Euler's Method
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
Chapter 1 Review Questions and Problems
CHAPTER 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
2.6 Existence and Uniqueness of Solutions. Wronskian
2.8 Modeling: Forced Oscillations. Resonance
2.9 Modeling: Electric Circuits
2.10 Solution by Variation of Parameters
Chapter 2 Review Questions and Problems
CHAPTER 3 Higher Order Linear ODEs
3.2 Homogeneous Linear ODEs with Constant Coefficients
3.3 Nonhomogeneous Linear ODEs
Chapter 3 Review Questions and Problems
CHAPTER 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
Chapter 4 Review Questions and Problems
CHAPTER 5 Series Solutions of ODEs. Special Functions
5.2 Legendre's Equation. Legendre Polynomials Pn(x)
5.3 Extended Power Series Method: Frobenius Method
5.4 Bessel's Equation. Bessel Functions Jν(x)
5.5 Bessel Functions of the Yν(x). General Solution
Chapter 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
Chapter 6 Review Questions and Problems
PART B Linear Algebra. Vector Calculus
CHAPTER 7 Linear Algebra: Matrices, Vectors, Determinants. Linear Systems
7.1 Matrices, Vectors: Addition and Scalar 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
Chapter 7 Review Questions and Problems
CHAPTER 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
Chapter 8 Review Questions and Problems
CHAPTER 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. Arc 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
Chapter 9 Review Questions and Problems
CHAPTER 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
Chapter 10 Review Questions and Problems
PART 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
Chapter 11 Review Questions and Problems
CHAPTER 12 Partial Differential Equations (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
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
Chapter 12 Review Questions and Problems
CHAPTER 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
13.6 Trigonometric and Hyperbolic Functions. Euler's Formula
13.7 Logarithm. General Power. Principal Value
Chapter 13 Review Questions and Problems
CHAPTER 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
Chapter 14 Review Questions and Problems
CHAPTER 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
Chapter 15 Review Questions and Problems
CHAPTER 16 Laurent Series. Residue Integration
16.2 Singularities and Zeros. Infinity
16.3 Residue Integration Method
16.4 Residue Integration of Real Integrals
Chapter 16 Review Questions and Problems
17.1 Geometry of Analytic Functions: Conformal Mapping
17.2 Linear Fractional Transformations (Möbius Transformations)
17.3 Special Linear Fractional Transformations
17.4 Conformal Mapping by Other Functions
17.5 Riemann Surfaces. Optional
Chapter 17 Review Questions and Problems
CHAPTER 18 Complex Analysis and Potential Theory
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
Chapter 18 Review Questions and Problems
CHAPTER 19 Numerics in General
19.2 Solution of Equations by Iteration
19.5 Numeric Integration and Differentiation
Chapter 19 Review Questions and Problems
CHAPTER 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
20.6 Matrix Eigenvalue Problems: Introduction
20.7 Inclusion of Matrix Eigenvalues
20.8 Power Method for Eigenvalues
20.9 Tridiagonalization and QR-Factorization
Chapter 20 Review Questions and Problems
CHAPTER 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
Chapter 21 Review Questions and Problems
CHAPTER 22 Unconstrained Optimization. Linear Programming
22.1 Basic Concepts. Unconstrained Optimization: Method of Steepest Descent
22.4 Simplex Method: Difficulties
Chapter 22 Review Questions and Problems
CHAPTER 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
Chapter 23 Review Questions and Problems
PART G Probability, Statistics
CHAPTER 24 Data Analysis. Probability Theory
24.1 Data Representation. Average. Spread
24.2 Experiments, Outcomes, Events
24.4 Permutations and Combinations
24.5 Random Variables. Probability Distributions
24.6 Mean and Variance of a Distribution
24.7 Binomial, Poisson, and Hypergeometric Distributions
24.9 Distributions of Several Random Variables
Chapter 24 Review Questions and Problems
CHAPTER 25 Mathematical Statistics
25.1 Introduction. Random Sampling
25.2 Point Estimation of Parameters
25.4 Testing Hypotheses. Decisions
25.9 Regression. Fitting Straight Lines. Correlation
Chapter 25 Review Questions and Problems
APPENDIX 2 Answers to Odd-Numbered Problems
A3.1 Formulas for Special Functions