1

First-Order ODEs

1

First-Order ODEs

1.1

Basic Concepts. Modeling

1.2

Geometric Meaning of y' = f(x, y). Direction Fields

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

---

2

Second-Order Linear ODEs

2

Second-Order Linear ODEs

2.1

Homogeneous Linear ODEs of Second Order

2.10

Solution by Variation of Parameters

2.2

Homogeneous Linear ODEs with Constant Coefficients

2.3

Differential Operators. Optional

2.4

Modeling: Free Oscillations. (Mass-Spring System)

2.5

Euler-Cauchy Equations

2.6

Existence and Uniqueness of Solutions. Wronskian

2.7

Nonhomogeneous ODEs

2.8

Modeling: Forced Oscillations. Resonance

2.9

Modeling: Electric Circuits

---

3

Higher Order Linear ODEs

3.1

Homogeneous Linear ODEs

3.2

Homogeneous Linear ODEs with Constant Coefficients

3.3

Nonhomogeneous Linear ODEs

---

4

Systems of ODEs. Phase Plane. Qualitative Methods

4

Systems of ODEs. Phase Plane. Qualitative Methods

4.1

Systems of ODEs as Models

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

---

5

Series Solutions of ODEs. Special Functions

5

Series Solutions of ODEs. Special Functions

5.1

Power Series Method

5.2

Theory of the Power Series Method

5.3

Legendre's Equation. Legendre Polynomials Pn{x)

5.4

Frobenius Method

5.5

Bessel's Equation. Bessel Functions Jv(x)

5.6

Bessel Functions of the Second Kind Yv(x)

5.7

Sturm-Liouville Problems. Orthogonal Functions

5.8

Orthogonal Eigenfunction Expansions

---

6

Laplace Transforms

6.1

Laplace Transform. Inverse Transform. Linearity. s-Shifting

6.2

Transforms of Derivatives and Integrals. ODEs

6.3

Unit Step Function. t-Shifting

6.4

Short Impulses. Dirac's Delta Function. Partial Fractions

6.5

Convolution. Integral Equations

6.6

Differentiation and Integration of Transforms.

6.7

Systems of ODEs

6.8

Laplace Transform: General Formulas

---

7

Linear Algebra: Matrices, Vectors, Determinants. Linear Systems

7

Linear Algebra: Matrices, Vectors, Determinants. Linear Systems

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

7.7

Determinants. Cramer's Rule

7.8

Inverse of a Matrix. Gauss-Jordan Elimination

7.9

Vector Spaces, Inner Product Spaces. Linear Transformations. Optional

---

8

Linear Algebra: Matrix Eigenvalue Problems

8

Linear Algebra: Matrix Eigenvalue Problems

8.1

Eigenvalues, 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

---

9

Vector Differential Calculus. Grad, Div, Curl

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 Fields. 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

9.9

Curl of a Vector Field

---

10

Vector Integral Calculus. Integral Theorems

10

Vector Integral Calculus. Integral Theorems

10.1

Line Integrals

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

10.6

Surface Integrals

10.7

Triple Integrals. Divergence Theorem of Gauss

10.8

Further Applications of the Divergence Theorem

10.9

Stokes's Theorem

---

11

Fourier Series, Integrals, and Transforms

11.1

Fourier Series

11.10

Tables of Transforms

11.2

Functions of Any Period p = 2L

11.3

Even and Odd Functions. Half-Range Expansions

11.4

Complex Fourier Series. Optional

11.5

Forced Oscillations

11.6

Approximation by Trigonometric Polynomials

11.7

Fourier Integral

11.8

Fourier Cosine and Sine Transforms

11.9

Fourier Transform. Discrete and Fast Fourier Transforms

---

12

Partial Differential Equations (PDEs)

12

Partial Differential Equations (PDEs)

12.1

Basic Concepts

12.10

Laplace's Equation in Cylindrical and Spherical Coordinates. Potential

12.11

Solution of PDEs by Laplace Transforms

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

Heat Equation: Solution by Fourier Series

12.6

Heat Equation: Solution by Fourier Integrals and Transforms

12.7

Modeling: Membrane, Two-Dimensional Wave Equation

12.8

Rectangular Membrane. Double Fourier Series

12.9

Laplacian in Polar Coordinates. Circular Membrane. Fourier-Bessel Series

---

13

Complex Numbers and Functions

13

Complex Numbers and Functions

13.1

Complex Numbers. Complex Plane

13.2

Polar Form of Complex Numbers. Powers and Roots

13.3

Derivative. Analytic Function

13.4

Cauchy-Riemann Equations. Laplace's Equation

13.5

Exponential Function

13.6

Trigonometric and Hyperbolic Functions

13.7

Logarithm. General Power

---

14

Complex Integration

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

---

15

Power Series, Taylor Series

15

Power Series, Taylor Series

15.1

Sequences, Series, Convergence Tests

15.2

Power Series

15.3

Functions Given by Power Series

15.4

Taylor and Maclaurin Series

15.5

Uniform Convergence. Optional

---

16

Laurent Series. Residue Integration

16

Laurent Series. Residue Integration

16.1

Laurent Series

16.2

Singularities and Zeros. Infinity

16.3

Residue Integration Method

16.4

Residue Integration of Real Integrals

---

17

Conformal Mapping

17

Conformal Mapping

17.1

Geometry of Analytic Functions: Conformal Mapping

17.2

Linear Fractional Transformations

17.3

Special Linear Fractional Transformations

17.4

Conformal Mapping by Other Functions

17.5

Riemann Surfaces. Optional

---

18

Complex Analysis and Potential Theory

18

Complex Analysis and Potential Theory

18.1

Electrostatic Fields

18.2

Use of Conformal Mapping. Modeling

18.3

Heat Problems

18.4

Fluid Flow

18.5

Poisson's Integral Formula for Potentials

18.6

General Properties of Harmonic Functions

---

19

Numerics in General

19

Numerics in General

19.1

Introduction

19.2

Solution of Equations by Iteration

19.3

Interpolation

19.4

Spline Interpolation

19.5

Numeric Integration and Differentiation

---

20

Numeric Linear Algebra

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: iII-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

---

21

Numerics for ODEs and PDEs

21

Numerics for ODEs and PDEs

21.1

Methods for First-Order ODEs

21.2

Multistep Methods

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

---

22

Unconstrained Optimization. Linear Programming

22

Unconstrained Optimization. Linear Programming

22.1

Basic Concepts. Unconstrained Optimization

22.2

Linear Programming

22.3

Simplex Method

22.4

Simplex Method: Difficulties

---

23

Graphs. Combinatorial Optimization

23

Graphs. Combinatorial Optimization

23.1

Graphs and Digraphs

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

23.6

Flows in Networks

23.7

Maximum Flow: Ford-Fulkerson Algorithm

23.8

Bipartite Graphs. Assignment Problems

---

24

Data Analysis. Probability Theory

24

Data Analysis. Probability Theory

24.1

Data Representation. Average. Spread

24.2

Experiments, Outcomes, Events

24.3

Probability

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

24.8

Normal Distribution

24.9

Distributions of Several Random Variables

---

25

Mathematical Statistics

25

Mathematical Statistics

25.2

Point Estimation of Parameters

25.3

Confidence Intervals

25.4

Testing Hypotheses. Decisions

25.5

Quality Control

25.6

Acceptance Sampling

25.7

Goodness of Fit. x2-Test

25.8

Nonparametric Tests

25.9

Regression. Fitting Straight Lines. Correlation