MATHEMATICS

BASICS:
Sets, elements, membership, inclusion, intersection, union, difference, empty set. Relations, applications or functions, domain, codomain. Numerical sets, intervals, upper and lower endpoints. Natural numbers, relative integers, rational numbers, real numbers, imaginary numbers, complex numbers, logarithms, one-to-one correspondence with the points of a line.
EQUATIONS AND INEQUALITIES:
Basic concepts and elementary algebra overview, analytic geometry overview, equations and inequalities of first and second degree, exercises on the solution of equations and inequalities of second degree and systems of inequalities by analytical and graphical methods.
COORDINATES, TRIGONOMETRY:
Oriented lines, polar coordinates, cartesian coordinates, distance between two points. Unit circle, trigonometric functions, addition and subtraction formulas, duplication and bisection formulas, transformation between cartesian and polar coordinates.
ELEMENTS OF ENUMERATIVE COMBINATORICS:
Simple dispositions, permutations and combinations, binomial coefficients, properties and definitions.
MATRICES AND DETERMINANTS:
Definitions and properties of matrices, diagonal matrices, transpose matrix, inverse matrix. Determinant of a square matrix, Laplace theorems, Sarrus rule, general properties of determinants, minors and rank of a matrix.
Eigenvalues and eigenvectors of a square matrix.
SYSTEMS OF LINEAR EQUATIONS:
Overview on systems of linear equations, Rouché-Capelli theorem, Cramer theorem, systems of m equations in n unknowns, homogeneous systems.
ANALYTIC GEOMETRY OVERVIEW:
Cartesian plane, equation of a straight line, parallelism, orthogonality and intersection of straight lines. Circle, ellipse, hyperbola and parabola. General equation of conic sections.
REAL VARIABLE FUNCTIONS:
Trigonometric functions, composition of functions, inverse functions, inf and sup of a function, limits of a function, theorems about limits, uniqueness of limits theorem, permanence of sign theorem, squeeze theorem, limit of a sum, product, quotient, etc. of functions, notable limits.
CONTNUOUS FUNCTIONS:
Definitions and basic properties, continuity, Neper number, maximum and minimum, inverse function.
DERIVATIVES:
Definition and geometric interpretation of derivatives, operations on derivatives and differentiation rules, differentiation of trigonometric functions, differentiation of combined functions and inverse functions, differentiability and continuity, differential, higher derivatives, relative maxima and minima, asymptotes, concavity and convexity, inflection points, graphs of functions. Rolle, Cauchy, Lagrange, De L'Hopital theorems, indeterminate forms.
TRANSCENDENTAL FUNCTIONS:
Inverse trigonometric functions, logarithm function, exponential function, hyperbolic functions.
FUNCTIONS OF TWO VARIABLES:
Definitions, domain, limits and continuity, partial derivatives and higher partial derivatives. Hessian matrix. Brief notes on function of n variables.
INTEGRALS:
Indefinite integrals, definition and properties, relation between integrability e differentiability, notable integrals, definite integrals, calculation of areas. Double integrals.

ELEMENTS OF PHYSICS

Reality description: models, theories, laws and measurements; the international system of units; scalar quantities and vector quantities; sum and difference of vectors; decomposition of vectors along given directions; scalar product, cross product, mixed product.
KINEMATICS:
Basics. Motion of point particles. Coordinate systems. Linear velocity and acceleration. Uniform and non-uniform circular motion; angular velocity and acceleration.
DYNAMICS:
Newton laws (I, II e III) and their applications. Point particles dynamics. Friction forces. Work and kinetic energy. Conservative forces. Potential energy; energy conservation. Power. Non-conservative forces. Momentum and its conservation; collisions and their classification. Motion of rigid bodies and equilibrium conditions. Moment of inertia and angular momentum for rotations around a fixed axis; momentum of forces; second law of rigid body dynamics. Kinetic energy; rolling and the role of frictions. Work and power.
FLUIDS:
Hydrostatics and applications. Fluid dynamics: mass flow rate, conservation law of mass flow rate. Bernoulli theorem and applications. Real fluids: surface tension and capillarity.

"Lezioni di matematica generale" - Alvaro Marucci - Edizioni Sette Città, Viterbo