Chapter 1: First-Order Differential EquationsSection 1-1:
Basic Concepts and Ideas
Section 1-2:
Geometrical Meaning of y' = f(x,y) Direction Fields
Section 1-3:
Separable Differential Equations
Section 1-4:
Modeling: Separable Equations
Section 1-5:
Exact Differential Equations Integrating Factors
Section 1-6:
Linear Differential Equations Bernoulli Equation
Section 1-7:
Modeling: Electric Circuits
Section 1-8:
Orthogonal Trajectories of Curves. Optional
Section 1-9:
Existence and Uniqueness of Solution Picard Iteration
Page 59:
Chapter Review
Chapter 2: Linear Differential Equations of Second and Higher OrderSection 2-1:
Homogeneous Linear Equations of Second Order
Section 2-2:
Second-Order Homogeneous Equations with Constant Coefficients
Section 2-3:
Case of Complex Roots Complex Exponential Function
Section 2-4:
Differential Operators. Optional
Section 2-5:
Modeling: Free Oscillation (Mass-Spring System)
Section 2-6:
Euler-Cauchy Equation
Section 2-7:
Existence and Uniqueness Theory. Wronskian
Section 2-8:
Nonhomogeneous Equations
Section 2-9:
Solution by Undetermined Coefficients
Section 2-10:
Solution by Variation of Parameters
Section 2-11:
Modeling: Forced Oscillation. Resonance
Section 2-12:
Modeling of Electric Circuits
Section 2-13:
Higher Order Linear Differential Equations
Section 2-14:
Higher Order Homogeneous Equations with Constant Coefficients
Section 2-15:
Higher Order Nonhomogeneous Equations
Page 142:
Chapter Review
Chapter 3: Systems of Differential Equations, Phase Plane, Qualitive MethodsSection 3-1:
Introductory Examples
Section 3-3:
Homogeneous Systems with Constant Coefficients. Phase Plane, Critical Points
Section 3-4:
Criteria for Critical Points. Stability
Section 3-5:
Qualitive Methods for Nonlinear Systems
Section 3-6:
Nonhomogeneous Linear Systems
Page 190:
Chapter Review
Exercise 6
Exercise 13
Exercise 14
Chapter 4: Series Solutions of Differential Equations. Special FunctionsSection 4-1:
Power Series Method
Section 4-2:
Theory of the Power Series Method
Section 4-3:
Legendre's Equation Legendre Polynomials Pn(X)
Section 4-4:
Frobenius Method
Section 4-5:
Bessel's Equation. Bessel Functions Jv(X)
Section 4-6:
Bessel Functions of the Second Kind Yv(X)
Section 4-7:
Sturm-Liouville Problems. Orthogonal Functions
Section 4-8:
Orthogonal Eigenfunction Expansions
Page 247:
Chapter Review
Chapter 5: Laplace TransformsSection 5-1:
Laplace Transform. Inverse Transform Linearity. Shifting
Section 5-2:
Transforms of Derivatives and Integrals Differential Equations
Section 5-3:
Unit Step Function Second Shifting Theorem Direac's Delta Function
Section 5-4:
Differentiation and Integration of Transforms
Section 5-5:
Convolution. Integral Equations
Section 5-6:
Partial Fractions Differential Equations
Section 5-7:
Systems of Differential Equations
Page 299:
Chapter Review
Chapter 6: Linear Algebra: Matrices, Vectors, Determinants Linear Systems of EquationsSection 6-1:
Basic Concepts Matrix Addition, Scalar Multiplication
Section 6-2:
Matrix Multiplication
Section 6-3:
Linear Systems of Equations Gauss Elimination
Section 6-4:
Rank of a Matrix. Linear Independence. Vector Space
Section 6-6:
Determinants Cramer's Rule
Section 6-7:
Inverse of a Matrix Gauss-Jordan Elimination
Section 6-8:
Vectors Spaces, Inner Product Spaces, Linear Transformations Optional
Page 365:
Chapter Review
Chapter 7: Linear Algebra: Matrix Eigenvalue ProblemsSection 7-1:
Eigenvalues, Eigenvectors
Section 7-2:
Some Applications of Eigenvalue Problems
Section 7-3:
Symmetric, Skew-Symmetric, and Orthogonal Matrices
Section 7-4:
Complex Matrices: Hermitian, Skew-Hermitian, Unitary
Section 7-5:
Similarity of Matrices Basis of Eigenvectors Diagonalization
Page 398:
Chapter Review
Chapter 8: Vectors Differential Calculus. Grad, Div, CurlSection 8-1:
Vector Algebra in 2-Space and 3-Space
Section 8-2:
Inner Product (Dot Product)
Section 8-3:
Vector Product (Cross Product)
Section 8-4:
Vector and Scalar Functions and Fields. Derivatives
Section 8-5:
Curves. Tangents. Arc Length
Section 8-6:
Curves in Mechanics Velocity and Acceleration
Section 8-7:
Curvature and Torsion of a Curve Optional
Section 8-8:
Review from Calculus in Several Variables Optional
Section 8-9:
Gradient of a Scalar Field Directional Derivative
Section 8-10:
Divergence of a Vector Field
Section 8-11:
Curl of a Vector Field
Page 459:
Chapter Review
Chapter 9: Vector Integral Calculus Integral TheoremsSection 9-1:
Line Integrals
Section 9-2:
Line Integrals Independent of Path
Section 9-3:
From Calculus: Double Integrals. Optional
Section 9-4:
Green's Theorem in the Plane
Section 9-5:
Surfaces for Surface Integrals
Section 9-6:
Surface Integrals
Section 9-7:
Triple Integrals Divergence Theorem of Gauss
Section 9-8:
Divergence Theorem: Further Applications
Section 9-9:
Stokes's Theorem
Page 521:
Chapter Review
Chapter 10: Fourier Series, Integrals, and TransformsSection 10-1:
Periodic Functions. Trigonometric Series
Section 10-2:
Fourier Series
Section 10-3:
Functions of Any Period p = 2L
Section 10-4:
Even and Odd Functions Half-Range Expansions
Section 10-5:
Complex Fourier Series. Optional
Section 10-6:
Forced Oscillations
Section 10-7:
Approximation by Trigonometric Polynomials
Section 10-8:
Fourier Integrals
Section 10-9:
Fourier Cosine and Sine Transforms
Section 10-10:
Fourier Transform
Page 579:
Chapter Review
Chapter 11: Partial Differential EquationsSection 11-1:
Basic Concepts
Section 11-3:
Separation of variables Use of Fourier Series
Section 11-4:
D'Alembert's Solution of the Wave Equation
Section 11-5:
Heat Equation: Solution by Fourier Series
Section 11-6:
Heat Equation: Solution by Fourier Integrals and Transforms
Section 11-8:
Rectangular Membrane. Use of Double Fourier Series
Section 11-9:
Laplacian in Polar Coordinates
Section 11-10:
Circular Membrane Use of Fourier-Bessel Series
Section 11-11:
Laplace's Equation in Cylindrical and Spherical Coordinates. Potential
Section 11-12:
Solution by Laplace Transforms
Page 647:
Chapter Review
Chapter 12: Complex Numbers and Functions. Conformal MappingSection 12-1:
Complex Numbers. Complex Plane
Section 12-2:
Polar Form of Complex Numbers Powers and Roots
Section 12-3:
Derivative. Analytic Function
Section 12-4:
Cauchy-Riemann Equations Laplace's Equation
Section 12-5:
Geometry of Analytic Functions: Conformal Mapping
Section 12-6:
Exponential Function
Section 12-7:
Trigonometric Functions
Section 12-8:
Logarithm General Power
Section 12-9:
Linear Fractional Transformation Optional
Section 12-10:
Riemann Surface. OptionalPage 701:
Chapter Review
Chapter 13: Complex IntegrationSection 13-1:
Line Integral in the Complex Plane
Section 13-2:
Cauchy's Integral Theorem
Section 13-3:
Cauchy's Integral Formula
Section 13-4:
Derivatives of Analytic Functions
Page 730:
Chapter Review
Exercise 10
Exercise 11
Exercise 16
Exercise 17
Exercise 24
Exercise 26
Exercise 27
Exercise 29
Exercise 30
Chapter 14: Power Series, Taylor SeriesSection 14-1:
Sequences, Series, Convergence Tests
Section 14-2:
Power Series
Section 14-3:
Functions Given by Power Series
Section 14-4:
Taylor Series and Maclaurin Series
Section 14-5:
Uniform Convergence. Optional
Page 767:
Chapter Review
Chapter 15: Laurent Series Residue IntegrationSection 15-1:
Laurent Series
Section 15-2:
Singularities and Zeros. Infinity
Section 15-3:
Residue Integration Method
Section 15-4:
Evaluation of Real Integrals
Page 794:
Chapter Review
Chapter 16: Complex Analysis Applied to Potential TheorySection 16-1:
Electrostatic Fields
Section 16-2:
Use of Conformal Mapping
Section 16-3:
Heat Problems
Section 16-4:
Fluid Flow
Section 16-5:
Poisson's integral Formula
Section 16-6:
General Properties of Harmonic Functions
Page 826:
Chapter Review
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 12
Exercise 14
Exercise 15
Exercise 16
Chapter 17: Numerical Methods in General Section 17-1:
Introduction
Section 17-2:
Solutions by Equations by Iteration
Section 17-3:
Interpolation
Section 17-4:
Splines
Section 17-5:
Numerical Integration and Differentiation
Page 882:
Chapter Review
Exercise 11
Exercise 13
Exercise 14
Exercise 15
Exercise 16
Exercise 25
Chapter 18: Numerical Methods in Linear AlgebraSection 18-1:
Linear Systems: Gauss Elimination
Section 18-2:
Linear Systems: LU-Factorization, Matrix Inversion
Section 18-3:
Linear Systems: Solution by Iteration
Section 18-4:
Linear Systems: Ill-Conditioning, Norms
Section 18-5:
Method of Least Squares
Section 18-7:
Inclusion of Matrix Eigenvalues
Section 18-8:
Eigenvalues by Iteration (Power Method)
Section 18-9:
Tridiagonalization and QR-Factorization
Page 938:
Chapter Review
Chapter 19: Numerical Methods for Differential EquationsSection 19-1:
Methods for First-Order Differential Equations
Section 19-2:
Multistep Methods
Section 19-3:
Methods for Systems and Higher Order Equations
Section 19-4:
Methods for Elliptic Partial Differential Equations
Section 19-5:
Neumann and Mixed Problems. Irregular Boundary
Section 19-6:
Methods for Parabolic Equations
Section 19-7:
Methods for Hyperbolic EquationsPage 984:
Chapter Review
Chapter 20: Unconstrained Optimization, Linear ProgrammingSection 20-1:
Basic Concepts Unconstrained Optimization
Section 20-2:
Linear Programming
Section 20-3:
Simplex Method
Section 20-4:
Simplex Method: Degeneracy, Difficulties in Starting
Page 1007:
Chapter Review
Exercise 1
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Chapter 21: Graphs and Combinatorial OptimizationSection 21-1:
Graphs and Digraphs
Section 21-2:
Shortest Path Problems Complexity
Section 21-3:
Bellman's Optimality Principle Dijkstra's Algorithm
Section 21-4:
Shortest Spanning Tress Kruskal's Greedy Algorithm
Section 21-5:
Prim's Algorithm for Shortest Spanning Trees
Section 21-6:
Networks. Flow Augmenting Parts
Section 21-7:
Ford-Fulkerson Algorithm for Maximum Flow
Section 21-8:
Assignment Problems Bipartite Matching
Page 1046:
Chapter Review
Exercise 1
Exercise 2
Exercise 4
Exercise 5
Exercise 8
Exercise 10
Exercise 11
Exercise 12
Exercise 14
Exercise 15
Exercise 18
Exercise 19
Chapter 22: Data Analysis. Probability TheorySection 22-1:
Data: Representation, Average, Spread
Section 22-2:
Experiments, Outcomes, Events
Section 22-3:
Probability
Section 22-4:
Permutations and Combination
Section 22-5:
Random Variables, Probability Distributions
Section 22-6:
Mean and Variance of a Distribution
Section 22-7:
Binomial, Poisson, and Hypergeometric Distributions
Section 22-8:
Normal Distribution
Section 22-9:
Distributions of Several Random Variables
Page 1100:
Chapter Review
Chapter 23: Mathematical StatisticsSection 23-2:
Estimation of Parameters
Section 23-3:
Confidence Intervals
Section 23-4:
Testing of Hypotheses, Decisions
Section 23-5:
Quality Control
Section 23-6:
Acceptance Sampling
Section 23-7:
Goodness of Fit. X2 - Test
Section 23-8:
Nonparametric Tests
Section 23-9:
Regression Analysis Fitting Straight Lines
Section 23-10:
Correlation Analysis
Exercise 5
Exercise 7
Exercise 8
Exercise 13
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