Description

1. First-Order Ordinary Differential Equations (ODEs)
   1. Basic Concepts, Modeling
   2. Geometric meaning of derivative, Direction Fields, Euler’s method
   3. Separable ODEs. Modeling
   4. Exact ODEs, Integrating Factors
   5. Linear ODEs. Bernoulli Equation. Population Dynamics
   6. Orthogonal trajectories. optional
   7. Existence and uniqueness of solutions for initial value problems
   8. Chapter 1 reviews questions and problems
   9. summary of chapter 1
2. Second-Order Linear ODEs
   1. Homogeneous linear ODEs of second order
   2. Homogeneous linear ODEs with constant coefficients
   3. Differential operators. optional
   4. modelling of free oscillations of a mass-spring system
   5. Euler-cauchy equations
   6. Existence and uniqueness of solutions. Wronskian.
   7. Non-homogeneous ODEs
   8. Modelling: forced oscillations. Resonance
   9. Modeling: Electric Circuits
   10. Solution by variation of parameters
   11. Chapter 2 review questions and problems
   12. Summary of chapter 2
3. Higher Order Linear ODEs
   1. Homogeneous linear ODEs
   2. Homogeneous linear ODEs with constant coefficients
   3. Nonhomogeneous linear ODEs
   4. Chapter 3 review questions and problems
   5. Summary of chapter 3
4. Systems of ODEs. Phase Plane. Qualitative Methods
   1. For Reference: Basics of Matrices and vectors
   2. System of ODEs as models in Engineering Applications
   3. Basic Theory of Systems of ODEs. Wronskian
   4. Constant Coefficient Systems. phase plane method
   5. Criteria for Critical points. Stability
   6. Qualitative methods for nonlinear system
   7. Nonhomogeneous linear systems of ODEs
   8. Chapter 4 review questions and problems
   9. Summary of chapter 4
5. Series Solutions of ODEs. Special Functions
   1. Power series method
   2. Legendre’s equations. Legendre polynomials
   3. Extended power series method. Frobenius method
   4. Bessel’s equations. Bessel functions.
   5. Bessel functions. General solutions.
6. Laplace Transforms
   1. Laplace transform. linearity. First shifting theorem s-shifting.
   2. Transforms of derivatives and integrals. ODEs
   3. Unit step function (heaviside function). Second shifting theorem (t-shifting).
   4. Short impulses. Dirac’s delta functions. Partial functions.
   5. Convolution. Integral equations.
   6. Differentiation and integration of transforms. ODEs with variable coefficients.
   7. System of ODEs.
   8. Laplace transforms. general formulas.
   9. Table of laplace transforms.
7. Linear Algebra: Matrices, Vectors, Determinants. Linear Systems
   1. Matrices, Vectors: Addition and scalar multiplication
   2. Matrix multiplication
   3. linear systems of equations. gauss elimination
   4. linear Independence. rank of a matrix. vector space.
   5. solutions of linear systems. existence, uniqueness.
   6. For reference: second and third-order determinants
   7. determinants. Cramer’s rule
   8. inverse of a matrix. gauss-jordan elimination
   9. vector spaces. inner product spaces. linear transformations.
   10. Chapter 7 review questions and problems
   11. Summary of chapter 7
8. Linear Algebra: Matrix Eigenvalue Problems
   1. The matrix eigenvalue problem. Determining eigenvalue and eigenvectors
   2. Some applications of eigenvalue problems
   3. Symmetric, skew-symmetric, and orthogonal matrices
   4. Eigenbases. Diagonization. Quadratic forms
   5. Complex matrices and forms. optional
   6. Chapter 8 review questions and problems
   7. summary of chapter 8
9. Vector Differential Calculus. Grad, Div, Curl
   1. Vectors in 2-space and 3-space
   2. inner product (dot product)
   3. Vector product (cross product)
   4. vector and scalar functions and their fields. vector calculus. derivatives.
   5. Curves. arc length. curvature. torsion
   6. calculus review. functions of several variables. optional
   7. gradient of a scalar field. directional derivative.
   8. divergence of a vector field.
   9. Curl of a vector field
   10. Chapter 9 review questions and problems
   11. summary of chapter 9
10. Vector Integral Calculus. Integral Theorems
    1. line integrals
    2. path independence of line integrals
    3. calculus review. double integrals. optional
    4. green’s theorem in the plane
    5. surfaces for surface integral
    6. surface integrals
    7. triple integrals. divergence theorem of gauss
    8. Further applications of the divergence theorem
    9. Stoke’s theorem
    10. chapter 10 review questions and the problems
    11. Summary of Chapter 10
11. Fourier Analysis
    1. Fourier Series
    2. Arbitrary Period. Even and Odd Functions. Half-Range Expansions
    3. Forced Oscillations
    4. Approximation by Trigonometric Polynomials
    5. Sturm–Liouville Problems. Orthogonal Functions
    6. Orthogonal Series. Generalized Fourier Series
    7. Fourier Integral
    8. Fourier Cosine and Sine Transforms
    9. Fourier Transform. Discrete and Fast Fourier Transforms
    10. Tables of Transforms
    11. Chapter 11 Review Questions and Problems
    12. Summary of Chapter 11
12. Partial Differential Equations (PDEs)
    1. Basic Concepts of PDEs
    2. Modeling: Vibrating String, Wave Equation
    3. Solution by Separating Variables. Use of Fourier Series
    4. D’Alembert’s Solution of the Wave Equation. Characteristics
    5. Modeling: Heat Flow from a Body in Space. Heat Equation
    6. Heat Equation: Solution by Fourier Series.
       Steady Two-Dimensional Heat Problems. Dirichlet Problem
    7. Heat Equation: Modeling Very Long Bars.
       Solution by Fourier Integrals and Transforms
    8. Modeling: Membrane, Two-Dimensional Wave Equation
    9. Rectangular Membrane. Double Fourier Series
    10. Laplacian in Polar Coordinates. Circular Membrane. Fourier–Bessel Series
    11. Laplace’s Equation in Cylindrical and Spherical Coordinates. Potential
    12. Solution of PDEs by Laplace Transforms
    13. Chapter 12 Review Questions and Problems
    14. Summary of Chapter 12
13. Complex Numbers and Functions. Complex Differentiation
    1. Complex Numbers and Their Geometric Representation
    2. Polar Form of Complex Numbers. Powers and Roots
    3. Derivative. Analytic Function
    4. Cauchy–Riemann Equations. Laplace’s Equation
    5. Exponential Function
    6. Trigonometric and Hyperbolic Functions. Euler’s Formula
    7. Logarithm. General Power. Principal Value
    8. Chapter 13 Review Questions and Problems
    9. Summary of Chapter 13
14. Complex Integration
    1. Line Integral in the Complex Plane
    2. Cauchy’s Integral Theorem
    3. Cauchy’s Integral Formula
    4. Derivatives of Analytic Functions
    5. Chapter 14 Review Questions and Problems
    6. Summary of Chapter 14
15. Power Series, Taylor Series
    1. Sequences, Series, Convergence Tests
    2. Power Series
    3. Functions Given by Power Series
    4. Taylor and Maclaurin Series
    5. Uniform Convergence. Optional
    6. Chapter 15 Review Questions and Problems
    7. Summary of Chapter 15
16. Laurent Series. Residue Integration
    1. Laurent Series
    2. Singularities and Zeros. Infinity
    3. Residue Integration Method
    4. Residue Integration of Real Integrals
    5. Chapter 16 Review Questions and Problems
    6. Summary of Chapter 16
17. Conformal Mapping
    1. Geometry of Analytic Functions: Conformal Mapping
    2. Linear Fractional Transformations (Möbius Transformations)
    3. Special Linear Fractional Transformations
    4. Conformal Mapping by Other Functions
    5. Riemann Surfaces. Optional
    6. Chapter 17 Review Questions and Problems
    7. Summary of Chapter 17
18. Complex Analysis and Potential Theory
    1. Electrostatic Fields
    2. Use of Conformal Mapping. Modeling
    3. Heat Problems
    4. Fluid Flow
    5. Poisson’s Integral Formula for Potentials
    6. General Properties of Harmonic Functions.
       Uniqueness Theorem for the Dirichlet Problem
    7. Chapter 18 Review Questions and Problems
    8. Summary of Chapter 18
19. Numerics in General
    1. Introduction
    2. Solution of Equations by Iteration
    3. Interpolation
    4. Spline Interpolation
    5. Numeric Integration and Differentiation
    6. Chapter 19 Review Questions and Problems
    7. Summary of Chapter 19
20. Numeric Linear Algebra
    1. Linear Systems: Gauss Elimination
    2. Linear Systems: LU-Factorization, Matrix Inversion
    3. Linear Systems: Solution by Iteration
    4. Linear Systems: Ill-Conditioning, Norms
    5. Least Squares Method
    6. Matrix Eigenvalue Problems: Introduction
    7. Inclusion of Matrix Eigenvalues
    8. Power Method for Eigenvalues
    9. Tridiagonalization and QR-Factorization
    10. Chapter 20 Review Questions and Problems
    11. Summary of Chapter 20
21. Numerics for ODEs and PDEs
    1. Methods for First-Order ODEs
    2. Multistep Methods
    3. Methods for Systems and Higher Order ODEs
    4. Methods for Elliptic PDEs
    5. Neumann and Mixed Problems. Irregular Boundary
    6. Methods for Parabolic PDEs
    7. Method for Hyperbolic PDEs
    8. Chapter 21 Review Questions and Problems
    9. Summary of Chapter 21
22. Unconstrained Optimization. Linear Programming
    1. Basic Concepts. Unconstrained Optimization: Method of Steepest Descent
    2. Linear Programming
    3. Simplex Method
    4. Simplex Method: Difficulties
    5. Chapter 22 Review Questions and Problems
    6. Summary of Chapter 22
23. Graphs. Combinatorial Optimization
    1. Graphs and Digraphs
    2. Shortest Path Problems. Complexity
    3. Bellman’s Principle. Dijkstra’s Algorithm
    4. Shortest Spanning Trees: Greedy Algorithm
    5. Shortest Spanning Trees: Prim’s Algorithm
    6. Flows in Networks
    7. Maximum Flow: Ford–Fulkerson Algorithm
    8. Bipartite Graphs. Assignment Problems
    9. Chapter 23 Review Questions and Problems
    10. Summary of Chapter 23
24. Data Analysis. Probability Theory
    1. Data Representation. Average. Spread
    2. Experiments, Outcomes, Events
    3. Probability
    4. Permutations and Combinations
    5. Random Variables. Probability Distribution
    6. Mean and Variance of a Distribution
    7. Binomial, Poisson, and Hypergeometric Distributions
    8. Normal Distribution
    9. Distributions of Several Random Variables
    10. Chapter 24 Review Questions and Problems
    11. Summary of Chapter 24
25. Mathematical Statistics
    1. Introduction. Random Sampling
    2. Point Estimation of Parameters
    3. Confidence Intervals
    4. Testing Hypotheses. Decisions
    5. Quality Control
    6. Acceptance Sampling
    7. Goodness of Fit. 2
       -Test
    8. Nonparametric Tests
    9. Regression. Fitting Straight Lines. Correlation
    10. Chapter 25 Review Questions and Problems
    11. Summary of Chapter 25

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