## Description

- First-Order Ordinary Differential Equations (ODEs)
- Basic Concepts, Modeling
- Geometric meaning of derivative, Direction Fields, Euler’s method
- Separable ODEs. Modeling
- Exact ODEs, Integrating Factors
- Linear ODEs. Bernoulli Equation. Population Dynamics
- Orthogonal trajectories. optional
- Existence and uniqueness of solutions for initial value problems
- Chapter 1 reviews 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
- modelling of free oscillations of a mass-spring system
- Euler-cauchy equations
- Existence and uniqueness of solutions. Wronskian.
- Non-homogeneous ODEs
- Modelling: 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
- For Reference: Basics of Matrices and vectors
- System of ODEs as models in Engineering Applications
- Basic Theory of Systems of ODEs. Wronskian
- Constant Coefficient Systems. phase plane method
- Criteria for Critical points. Stability
- Qualitative methods for nonlinear system
- 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 equations. Legendre polynomials
- Extended power series method. Frobenius method
- Bessel’s equations. Bessel functions.
- Bessel functions. General solutions.

- Laplace Transforms
- Laplace transform. linearity. First shifting theorem s-shifting.
- Transforms of derivatives and integrals. ODEs
- Unit step function (heaviside function). Second shifting theorem (t-shifting).
- Short impulses. Dirac’s delta functions. Partial functions.
- Convolution. Integral equations.
- Differentiation and integration of transforms. ODEs with variable coefficients.
- System of ODEs.
- Laplace transforms. general formulas.
- Table of laplace transforms.

- 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.
- Chapter 7 review questions and problems
- Summary of chapter 7

- Linear Algebra: Matrix Eigenvalue Problems
- The matrix eigenvalue problem. Determining eigenvalue and eigenvectors
- Some applications of eigenvalue problems
- Symmetric, skew-symmetric, and orthogonal matrices
- Eigenbases. Diagonization. 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 (dot product)
- Vector product (cross product)
- vector and scalar functions and their fields. vector calculus. 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 integral
- surface integrals
- triple integrals. divergence theorem of gauss
- Further applications of the divergence theorem
- Stoke’s theorem
- chapter 10 review questions and the problems
- Summary of Chapter 10

- Fourier Analysis
- Fourier Series
- Arbitrary Period. Even and Odd Functions. Half-Range Expansions
- Forced Oscillations
- Approximation by Trigonometric Polynomials
- Sturm–Liouville Problems. Orthogonal Functions
- Orthogonal Series. Generalized Fourier Series
- 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 of PDEs
- Modeling: Vibrating String, Wave Equation
- Solution by Separating Variables. Use of Fourier Series
- D’Alembert’s Solution of the Wave Equation. Characteristics
- Modeling: Heat Flow from a Body in Space. Heat Equation
- Heat Equation: Solution by Fourier Series.

Steady Two-Dimensional Heat Problems. Dirichlet Problem - Heat Equation: Modeling Very Long Bars.

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 Numbers and Functions. Complex Differentiation
- Complex Numbers and Their Geometric Representation
- Polar Form of Complex Numbers. Powers and Roots
- Derivative. Analytic Function
- Cauchy–Riemann Equations. Laplace’s Equation
- Exponential Function
- Trigonometric and Hyperbolic Functions. Euler’s Formula
- Logarithm. General Power. Principal Value
- 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
- Chapter 16 Review Questions and Problems
- Summary of Chapter 16

- Conformal Mapping
- Geometry of Analytic Functions: Conformal Mapping
- Linear Fractional Transformations (Möbius 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.

Uniqueness Theorem for the Dirichlet Problem - Chapter 18 Review Questions and Problems
- Summary of Chapter 18

- 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

- Unconstrained Optimization. Linear Programming
- Basic Concepts. Unconstrained Optimization: Method of Steepest Descent
- Linear Programming
- Simplex Method
- Simplex Method: Difficulties
- Chapter 22 Review Questions and Problems
- Summary of Chapter 22

- Graphs. Combinatorial Optimization
- Graphs and Digraphs
- Shortest Path Problems. Complexity
- Bellman’s Principle. Dijkstra’s Algorithm
- Shortest Spanning Trees: Greedy Algorithm
- Shortest Spanning Trees: Prim’s Algorithm
- Flows in Networks
- Maximum Flow: Ford–Fulkerson Algorithm
- Bipartite Graphs. Assignment Problems
- Chapter 23 Review Questions and Problems
- Summary of Chapter 23

- Data Analysis. Probability Theory
- Data Representation. Average. Spread
- Experiments, Outcomes, Events
- Probability
- Permutations and Combinations
- Random Variables. Probability Distribution
- Mean and Variance of a Distribution
- Binomial, Poisson, and Hypergeometric Distributions
- Normal Distribution
- Distributions of Several Random Variables
- Chapter 24 Review Questions and Problems
- Summary of Chapter 24

- Mathematical Statistics
- Introduction. Random Sampling
- Point Estimation of Parameters
- Confidence Intervals
- Testing Hypotheses. Decisions
- Quality Control
- Acceptance Sampling
- Goodness of Fit. 2

-Test - Nonparametric Tests
- Regression. Fitting Straight Lines. Correlation
- Chapter 25 Review Questions and Problems
- Summary of Chapter 25

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