**REUSABLE LEARNING OBJECTs**

**Fundamental mathematics**

On creative mathematical problems at school

Linear cyclic systems

Using knowledge of planimetry for solving some systems of algebraic equations

Solving algebraic equations rationally

Solving problems using substitutions

Application of the derivative for proving identities, inequalities and solving equations

Function limits

**Algebra**

Matrices and systems

Elementary row operations

Row echelon rows

System of linear equations

Gauss method for Systems of linear equations

Gauss and Jordan elimination methods

Dijkstra algorithm for finding the shortest paths in a graph

Matrices, Determinants, Systems of Linear Equations

**Calculus**

What is a sequence?

Bounded Sequences

Increasing and Decreasing Sequences

Converging and Diverging Sequences

Some Basic Properties of Sequences

Calculating the Limit of a Sequence

What is a Series

Understanding Sigma Notation

The Sn Notation

Converging and Diverging Series

Application of Series - Estimating the square root of a number

Functions of a real Variable

Limits and Continuity

Properties of Continuous Functions, Asymptotes of Graphs, Derivatives

Rules of Differentiation, Derivatives of Elementary Functions, Differential

L'Hospital's Rule, Monotonicity, Local Extrema

Global Extrema, Convexity and Concavity, Investigation of the Behaviour of a Function

Indefinite Integrals, Integration by Parts, The Substitution Method

Integration of Some Rational, Irrational and Trigonometric Functions

**Difference and differential equations**

Difference equations - basic definitions and properties

Differential Equations of the 1st order

Linear Second Order Homogeneous Differential Equations with Constant Coefficients

Linear Second Order Non-homogeneous Differential Equations with Constant Coefficients

**Geometry**

Cartesian coordinates

Polar coordinates

Cylindrical coordinates

Spherical coordinates

Homogeneous coordinates

Geometric transformations

Euclidean transformations in the plane

Euclidean transformations in the space

Composite Euclidean transformations in the space

Affine transformations

Axial affinity

Projective transformations

Central collineation

Matrices of transformations

Creative space

Monge method

Orthogonal axonometry

Matrices of projection methods

Superposition of geometric figures

Metric problems in projection methods

Views of planar figures in projection methods

Curves in the extended Euclidean space

Curves in plane

Curves in space

Approximation and interpolation curves

Surfaces in the extended Euclidean space

Translation surfaces - theory

Homothetical surfaces - theory

Prismatic and pyramidal surfaces - views

Cylindrical and conical surfaces - views

Intersection points of lines and surfaces - views

Developable surfaces - theory

Surfaces of revolution - theory

Special types of surfaces of revolution - theory

Quadratic surfaces of revolution - theory

Helical surfaces - theory

Envelope surfaces - theory

Intersections of surfaces - theory

Intersections of elementary surfaces - views

Notes on Geometry of Elementary Solid Cells

In-homogeneous Bezier Solid Cells

One-Parametric Deformations of Solids

Free-Form Deformations of Solid Cells

**Multivariable calculus**

Limit and Continuity

Vector algebra

Functions of Two and More Variables, Domains of Definition, Graphs, Limits

Continuity, Partial Derivatives of Functions of Two Variables, Total Dierentials, Tangent Planes

Local, Constrained and Global Extrema for Functions of Two Variables

Double Integrals, Basic Properties, Fubini´s Theorems

Double Integrals in Polar Coordinates

Triple Integrals, Basic Properties, Fubini´s Theorems

Triple Integrals in Cylindrical and Spherical Coordinates

**Numerical analysis**

Basic notions: numerical methods, modeling, types of errors, correctness, stability

Introduction to numerical solving of algebraic equations

Function approximation (by Lagrange's interpolation polynomial)

Least square method

Spline interpolation

Solving systems of linear equations - exact methods. Gauss's method (Gauss's elimination)

Gauss-Jordan method

Calculating determinants

Matrix inversion

Preparation of
system of linear algebraic equations with positive defined matrix in order to apply the iterative approximation method

Bisection method for solving equations with a single unknown

Power method for solving the partial eigenvalue problem

Jacobi's method (method of rotation)

Numerical differentiation

Numerical integration (Newton-Cotes quadrature formulas)

Example
of coefficient instability of the Gauss elimination method for solving linear system of algebraic equations

Gauss-Zeidel method

Method of simple
iteration for solving systems of linear algebraic equations (Jacobi method)

Double
sweep method for tri-diagonal system of linear equations (Thompson's method)

Method of consecutive iterations for solving nonlinear algebraic equations

Introduction to the numerical integration

Euler method for numerical
solving of ordinary differential equations and systems

Modified Euler method

Euler-Cauchy method

Euler's Method for ordinary differential equations using Mathematica

Modified Euler's method for ordinary difference

Euler's-Cauchy Method using Mathematica

Euler's Method for systems of ordinary differential equations using Mathematica

Mesh method for solving boundary value problem of the mixed type using Mathematica

Two-order Runge-Kutta Method using Mathematica

Two-order Runge-Kutta Method for systems of ordinary differential equations using Mathematica

Fourth-order Runge-Kutta Method using Mathematica

Fourth-order Runge-Kutta Method using Mathematica

Optimising the Parameters of Polarised Electromagnetic Construction Using Mathematica

Examples on Variational Methods for Solving Boundary Problems for Ordinary Differential Equations (ODE) of the Second Order (Galerkin and Ritz methods)

Some more about the systems of linear equations

The square root method

Solving systems of linear equations - approximate methods

Introduction to finite difference method for solving partial differential equations

Finite difference method for solving PDE of parabolic type

Examples for finite difference method for solving PDE of parabolic type

**Optimisation**

Basic concepts in mathematical optimization

Elements of dynamical optimization - basic notions

Graphs, definitions, representations

Network planning (problem of finding the shortest and longest path in a network)

One-dimensional resource distribution problem. Bellman's functional equations

Bellman's optimality principle

Bellman-Ford algorithm

Dijkstra algorithm for finding the shortest paths in a graph

**Probability and statistics**

Rlo

**History of mathematics**

How the Greeks might have
discovered and approximate irrational numbers

Development of the School of Mathematics in Cluj-Kolozsvár-Klausenburg (1872-1919)