@@@PHYSICS+MATHS
: BIENNIO
: 0) MATHEMATICS FOR PHYSICS
: 1 Equivalences between units of measurement
2 Calculations with powers
3 Proportions and percentages
4 The equations: obtaining an unknown quantity from a formula
5 Relationships between the sides of a triangle
6 Trigonometric relations
1) MEASUREMENT IN PHYSICS
: 1 What does physics do?
physical observables
the experimental method
Galileo Galilei
Physical law and mathematical language
Physical Theories
Physics and natural sciences
Physics and technology
2 Physical quantities
measurement of a physical quantity
operational definition of physical quantities
Correct expression of a measured physical quantity
Multiple and sub-multiples of physical quantities
Prefixes of physical quantities
Fundamental quantities and derived quantities
The physical dimensions of the quantities
Physical dimension of a derived quantity
3 The international system and the fundamental dimensions of mechanics
Absolute units of measure systems
The International System
Measure and unit of time
Measurement and unit of lengths
Measurement and unity of mass
The radiant
The steradian
4 Big numbers and small numbers
Power properties
Properties of powers of 10
The scientific notation
Examples of time intervals
Examples of lengths
Examples of masses
prefixes and powers of 10
Order of sizes
5 Direct and indirect measures
Length measurements
Area measurements
Volume measurements
The density
Density measurement
6 Making estimates: Fermi's problems
estimate of the order of magnitude
examples of Fermi estimates
2) DATA PROCESSING IN PHYSICS
: 1 Measurement errors
2 Estimate of the error
3 Error propagation and significant figures
4 The construction of a Cartesian graph
5 Representations of experimental data
6 Mathematical and graphic representation of physical laws
3) SCALAR SIZES AND VECTOR SIZES
: 1 Displacement: a physical quantity to describe movement
2 Sum of trips
3 Scalars and vectors
4 Some operations on carriers
5 Decomposition of a carrier
6 Scalar product and vector product
4) FORCES AS VECTORS
: 1 Forces
2 The weight force
3 Reaction to a deformation: the elastic force
4 The forces of constraint and friction
3 The weight force
5) THE BALANCE OF SOLIDS
: 1 The balance of a material point
2 Moment of a force and a system of forces
3 The balance of a rigid body
4 Center of gravity and balance stability
5 Simple machines: levers and pulleys
6) PRESSURE AND BALANCE OF FLUIDS
: 1 Fluids and pressure
2 Pressure in liquids
3 Atmospheric pressure
4 The flotation of bodies
7) THE UNIFORM MOTION
: 1 The description of the motion
2 Speed
3 The space-time graph
4 The uniform rectilinear motion
8) THE UNIFORMLY ACCELERATED MOTION
: 1 Acceleration
2 The speed-time graph
3 Uniformly accelerated motion
4 bodies in free fall
9) MOTIONS IN THE PLAN AND HARMONIC MOTION
: 1 The motions in the plane
2 The motion of projectiles
3 Uniform circular motion
4 Displacement and angular velocity
5 Harmonic motion
10) THE NEWTONIAN DYNAMICS
: 1 From the description of motion to its causes
2 The first principle of dynamics
3 The second principle of dynamics
4 The third principle of dynamics
5 Applications: falling motions
6 Applications: harmonic motion
11) WORK AND ENERGY
: 1 The work of a constant force
2 The work of the weight force
3 The work of a variable force
4 The power
5 Kinetic energy
6 The potential energy
7 Energy conservation
12) TEMPERATURE AND HEAT
: 1 Temperature and balance
2 Thermal expansion
3 Heat as energy in transit
4 Heat propagation
5 states of aggregation and status changes
13) GEOMETRIC OPTICS
: 1) Light sources and light rays
2) Reflection
3) Refraction
4) Total reflection
5) Spherical mirrors
6) The lenses
THREE-YEAR PERIOD
: 1) THE LAWS OF DYNAMICS AND BALANCE
: 1 The magnitudes of dynamics: a reminder
2 The Cartesian representation of vectors
3 Operations with vectors in Cartesian representation
4 Newton's laws
5 Equilibrium of the material point and of the rigid body
2) MOTIONS AS A RESULT OF THE LAWS OF DYNAMICS
: 1 Uniform rectilinear motion
2 The accelerated uniform straight motion
3 The use of derivatives in physics: velocity and acceleration
4 Motion in two and three dimensions
5 Parabolic motion
6 Circular motion
7 The vector quantities of circular motion
8 Harmonic motion and the pendulum
3) INERTIAL AND NON-INERIAL REFERENCE SYSTEMS
: 1 Classical composition of displacements, velocities and accelerations
2 The principle of classical relativity
3 Galilean transformations
4 Apparent forces in reference systems in accelerated translational motion
5 apparent forces in the reference systems in circular motion
4) MECHANICAL ENERGY
: 1 Work as a scalar product
2 The work of a constant force: the case of weight force
3 The work of a variable force: the case of elastic force
4 Kinetic energy
5 Conservative forces and potential energy
6 Energy conservation
7 Average and instantaneous power
5) DYNAMICS OF FLUIDS
: 1 Ideal fluids and real fluids
2 The continuity equation
3 The Bernoulli equation
4 The viscosity of fluids
6) THE QUANTITY OF MOTION AND IMPACT
: 1 Amount of motion and impulse
2 The conservation of momentum
3 The bumps
4 Elastic collisions in one and two dimensions
5 Center of mass and motion of a particle system
7) ANGULAR MOMENT AND RIGID BODIES
: 1 The angular momentum
2 The variation of the angular momentum
3 Moment of inertia and angular momentum of an extended body
4 Preservation of angular momentum
5 The rotational dynamics of a rigid body
6 Kinetic energy, work and power in rotary motion
8) UNIVERSAL GRAVITATION
: 1 The orbits of the planets
2 The law of universal gravitation
3 The gravitational field
4 The gravitational potential energy
5 Speed, period and energy of planets and satellites
9) GAS LAWS
: 1 Temperature and thermometric scales
2 The gas laws
3 Boyle's law and the two laws of Gay-Lussac
4 The constant-volume gas thermometer and absolute zero
5 A simpler form for the laws of Gay-Lussac
The equation of state of perfect gases
10) THE KINETIC THEORY OF GAS
: 1 Molecular model of perfect gases
2 Molecular shocks and pressure
3 Average quadratic speed and temperature
4 The Maxwell distribution
5 The average kinetic energy
6 The properties of real gases
11) The first principle of dynamics
: 1 Heat, thermal equilibrium and state changes
2 Heat propagation
3 Thermodynamic systems and transformations
4 Thermodynamic work
5 The first principle: energy conservation
6 Internal energy and specific heats of a perfect gas
7 The first principle and adiabatic transformations
12) SECOND PRINCIPLE OF THERMODYNAMICS AND ENTROPY
: 1 Thermal machines
2 The second principle: the privileged verse
3 The Carnot cycle and the maximum efficiency of thermal machines
4 Refrigerating machines
5 The entropy of Clausius
6 The second principle is a principle of "non conservation"
7 Entropy and disorder: the Boltzmann equation
13) THE PROPERTIES OF THE WAVING MOTORS
: 1 The undulatory motions
2 The wave function
3 The overlapping principle: interference
4 The beats
5 Reflection and standing waves
6 Wave diffraction and the Huygens principle
14) SOUND
: 1 Sources and propagation of sound waves
2 The characteristics of sound
3 The perception of sound
4 The Doppler effect
15) THE WAVY PROPERTIES OF LIGHT
: 1 The refraction of light
2 Light interference
3 Interference on a thin layer
4 Young's double slit interferometer
5 The diffraction of light
6 The polarization of light
7 Energy transported by light
16) THE ELECTRIC CHARGE AND THE LAW OF COULOMB
: 1 The electric charge and the interactions between electrified bodies
2 Conductors and insulators. Contact electrification
3 Electrostatic induction
4 The polarization of dielectrics
5 The law of Coulomb
17) THE ELECTRIC FIELD
: 1 The concept of the electric field
2 The electric field generated by point charges
3 The flow of the electric field and the Gauss theorem
4 Applications of the Gauss theorem
18) POTENTIAL AND CAPACITY
: 1 The potential electric energy
2 The electric potential and the potential difference
3 The electric field circuitry
4 The potential of a conductor in electrostatic equilibrium
5 Capacitors and capacitance
6 Capacitor systems
7 The accumulation of electricity in a capacitor
19) THE ELECTRICITY AND THE OHM LAWS
: 1 The electric current
2 Electric resistance and Ohm's first law
3 The second law of Ohm
4 Microscopic interpretation of Ohm's laws
20) ELECTRICAL CIRCUITS
: 1 The electromotive force
2 Direct current electrical circuits: Kirchhoof's laws
3 Resistance systems
4 RC circuits
5 Electric power
6 Instruments for electrical quantities
7 The extraction of electrons from a metal
21) THE ELECTRIC CURRENT IN THE FLUIDS AND THE VACUUM
: 1 Batteries and accumulators
2 Electrolytic solutions and electrolysis
3 Faraday's laws
4 Electrical conduction in gases
5 Electric currents through the vacuum
22) THE MAGNETISM
: 1 The magnets and the electric field
2 The magnetic induction
3 The magnetic fields generated by currents
4 The flow and circulation of the magnetic field
5 Magnetic forces on currents
6 The magnetic force on a moving electric charge
7 The action of a magnetic field on a loop covered by current
8 The magnetic properties of matter
23) CHARGES IN ELECTRIC AND MAGNETIC FIELDS
: 1 The motion of a charge in an electric field
2 Applications: Millikan's experiment
3 The motion of a charge in a magnetic field
4 Applications: Thomson's experiment
5 Applications: the mass spectrograph
6 Applications: the Hall effect
24) ELECTROMAGNETIC INDUCTION
: 1 Induced current
2 Faraday-Neumann's law and Lenz's law
3 Mutual induction and self-induction
4 RL Circuits and inductor energy
25) THE ALTERNATING CURRENT
: 1 The alternator
2 Electric circuits in alternating current
3 The RLC circuit
4 The power absorbed by a circuit
5 The transformer
26) ELECTROMAGNETIC WAVES
: 1 The electromagnetic field and the speed of light
2 The displacement current
3 Maxwell's equations
4 The propagation of electromagnetic waves
5 The energy and momentum transported by an electromagnetic wave
6 Production and reception of electromagnetic waves
7 The electromagnetic spectrum
27) KINEMATICS IN RESTRICTED RELATIVITY
: 1 The crisis of the principle of classical relativity
2 The postulates of special relativity
3 Lorentz transformations
4 A new concept of simultaneity
5 Time dilation
6 The contraction of lengths
7 Relativistic composition of velocities
8 Space time
28) RELATIVIST DYNAMICS AND GENERAL RELATIVITY
: 1 Mass and quantity of motion in the relativistic dynamics
2 Mass-energy equivalence
3 General relativity: a new principle of equivalence
4 Gravity and the curvature of space-time
5 Experimental verifications of general relativity
6 Gravitational wave research
29) ORIGINS OF PHYSICS OF THOSE
: 1 Black-body radiation and Planck quanta
2 The light quanta and the photoelectric effect
3 The Compton effect
4 The characteristic spectra of atoms
5 The first atomic models
6 The Bohr model
7 Quantized orbits and spectral lines of atoms
30) WAVES, BODIES AND INDETERMINATION
: 1 The wave-corpuscle duality
2 Wave mechanics
3 The quantum numbers of the hydrogen atom
4 The electronic configuration of complex atoms
5 The Heisenberg uncertainty principle
6 The Tunnel effect
31.A) THE PHYSICS OF THE SOLID STATE
: 1 The molecules and crystalline solids
2 The energy band theory
3 The doping of semiconductors
4 The pn junction
5 Transistors and integrated circuits
6 Superconductivity
31.B) THE NUCLEUS AND RADIOACTIVITY
: 1 The structure of the atomic nucleus
2 Natural radioactivity
3 Radioactive decay
4 The biological effects of ionizing radiation
5 Artificial transmutations and synthetic elements
6 Nuclear fission
7 Nuclear fusion
31.C) THE ELEMENTARY PARTICLES AND THEIR INTERACTIONS
: 1 The ultimate constituents of the subject
2 The neutrinos
3 Conservation laws and quantum numbers in particle physics
4 The charm of quarks
5 The Standard Model and the frontiers of physics
31.D) ASTROPHYSICS AND COSMOLOGY
: 1 The Sun, the stars and the galaxies
2 Stellar evolution: birth, life and death of the stars
3 Radio astronomy and the mysterious objects of distant galaxies
4 The expanding universe
5 The Big Bang hypothesis
6 The future of the universe
MATHEMATICS
: FIRST YEAR
: 1) NATURAL NUMBERS
: 1 What are natural numbers
2 The four operations
3 The powers
4 Expressions with natural numbers
5 The properties of operations
6 The properties of powers
7 The multiples and divisors of a number
8 The greatest common divisor is the common minimum
9 Numbering systems
2) INTERNAL NUMBERS:
1 What are integers
2 Addition and subtraction
3 Multiplication, division and power
4 The laws of monotony
3) RATIONAL NUMBERS AND REAL NUMBERS:
1 From fractions to rational numbers
2 The comparison of rational numbers
3 Operations in Q
4 Powers with negative integer exponent
5 Rational numbers and decimal numbers
6 The real numbers
7 Fractions and proportions
8 Percentages
9 The approximate calculation
10 The scientific notation and the order of magnitude
4) THE SETS AND THE LOGIC:
1 What is a whole
2 The representations of a whole
3 The subsets
4 Operations with sets
5 The set of parts and the partition of a set
6 The logical propositions
7 Logical connectives and expressions
8 Valid forms of reasoning
9 Logic and sets
10 The quantifiers
5) RELATIONS AND FUNCTIONS:
1 Binary relations
2 The relationships defined in a whole and their properties
3 Equivalence relations
4 Ordering reports
5 The functions
6 Numerical functions
7 The Cartesian plane and the graph of a function
8 Special numerical functions
9 The circular functions
6) THE MONOMES:
1 What are monomials
2 Operations with monomials
3 Maximum common divisor and least common multiple of monomials
7) PYRINOMICS:
1 What are polynomials
2 Operations with polynomials
3 Notable products
4 Polynomial functions
5 The division between polynomials
6 Ruffini's rule
7 The theorem of the rest
8 Ruffini's theorem
8) BREAKDOWN IN FACTORS:
1 Factor decomposition of polynomials Summary: Polynomial decomposition
2 The MCD and the mcm between polynomials
9) ALGEBRIC FRACTIONS:
1 Algebraic fractions
2 The calculation with algebraic fractions
10) LINEAR EQUATIONS:
1 Identities
2 The equations
3 The principles of equivalence
4 The whole numerical equations
5 Equations and problems
6 The fractional equations
7 Literal equations
11) LINEAR DISEQUATIONS:
1 Numerical inequalities Inequalities
2 The whole inequalities
3 The systems of inequalities
4 Equations with absolute values
5 Inequalities with absolute values The study of the sign of a product
6 Fractional inequalities
12) ELEMENTS OF COMPUTER SCIENCE:
1 Numbers and digital information
2 Problems and algorithms
3 Program with Python
13) INTRODUCTION TO THE STATISTICS:
1 Statistical data
2 The graphical representation of the data
3 The central position indices
4 Variability indices
G1) THE GEOMETRY OF THE PLAN:
1 Geometrical objects and properties
2 The postulates of belonging and order
3 The fundamental bodies
4 Operations with segments and angles Figures and demonstrations
5 Lengths, widths, measures
G2) THE TRIANGLES:
1 First definitions on triangles
2 The first criterion of congruence
3 The second criterion of congruence
4 The properties of the isosceles triangle
5 The third criterion of congruence; Congruence criteria and isosceles and equilateral triangles
6 Inequalities in triangles
G3) PERPENDICULAR AND PARALLEL:
1 The perpendicular lines
2 The parallel lines
3 The properties of polygon angles
4 The criteria of congruence of the right-angled triangles
G4) PARALLELOGRAMS AND TRAPEZIES:
1 The parallelogram
2 The rectangle
3 The turbot
4 The square
5 The trapeze
6 The correspondences in a bundle of parallel lines
SECOND YEAR
: 13) LINEAR SYSTEMS:
1 The systems of two equations in two unknowns
2 The replacement method
3 Determined, impossible, indeterminate systems
4 The comparison method
5 The reduction method
6 The matrices and determinants
7 The Cramer method
8 The systems of three equations in three unknowns
9 The literal and fract systems
10 Linear systems and problems
14) THE RADICALS:
1 Real numbers
2 Square roots and cubic roots
3 The nth root
4 Simplification and comparison of radicals
15) OPERATIONS WITH RADICALS:
1 Multiplication and division of roots
2 The transport of a factor outside or inside the root sign
3 The power and root of a radical
4 Addition and subtraction of radicals
5 The rationalization of the denominator of a fraction
16) THE CARTESIAN PLAN AND THE RIGHT:
1 Points and segments
2 The distance between two points is the midpoint
3 The equation of a line passing through the origin
4 The general equation of the line
5 The straight lines and the linear systems
6 Parallel lines and perpendicular lines
7 The bundles of lines
8 How to determine the equation of a line
9 The distance of a point from a line
10 The parts of the plan and the line
17) THE SECOND-GRADE EQUATIONS AND THE PARABOLA:
1 The second degree equations: definitions
2 The resolution of a second degree equation
3 The whole numerical equations
4 The quadratic function and the parabola
5 The relations between the roots and the coefficients
6 The rule of Descartes
7 The decomposition of a second degree trinomial
8 Secondary equations and problems
18) THE APPLICATIONS OF SECOND-GRADE EQUATIONS:
1 The fractional and literal equations
2 Equations and problems
3 The second degree parametric equations
4 Equations higher than the second degree
19) SECOND DEGREE SYSTEMS AND UPPER DEGREE:
1 The second degree systems
2 The graphic interpretation of second degree systems
3 Systems higher than the second degree
4 Problems with higher than second degree systems
20) THE SECOND-DEGREE DISEQUATIONS AND UPPER DEGREE:
1 Linear inequalities
2 The sign of entire second degree inequalities
3 The resolution of entire second degree inequalities
4 The whole inequalities of degree higher than the second
5 The broken inequalities
6 The systems of inequalities
7 Problems with inequalities
21) APPLICATIONS OF DISEQUATIONS:
1 Parametric equations
2 The irrational equations
3 Irrational inequalities
4 Equations with absolute values
5 Inequalities with absolute values
6 Function graphs with absolute values
Beta) INTRODUCTION TO PROBABILITY:
1 Events and the sample space
2 The classical definition of probability
3 Operations with events
4 Theorems related to probability calculus
5 Other definitions of probability
G5) THE CIRCUMFERENCE:
1 The geometric places
2 The circumference and the circle
3 Theorems on the strings
4 Circumferences and lines
5 The mutual positions between two circles
6 The angles on the circumference
G6) THE INSCRIBED AND CIRCUMSCRIBED POLIGONES:
1 The inscribed polygons
2 The circumscribed polygons
3 The triangles and the remarkable points
4 The quadrilaterals inscribed and circumscribed
5 The regular polygons
G7) EQUIVALENT SURFACES AND AREAS:
1 The equivalence of surfaces
2 The equivalence of parallelograms
3 Triangles and equivalence
4 The equivalence between a circumscribed polygon and a triangle
5 The construction of equivalent polygons
6 Measurement of polygon areas
G8) THE THEOREMES OF EUCLIDE AND PITAGORA:
1 Euclid's first theorem
2 The Pythagorean theorem
3 Applications of the Pythagorean theorem
4 Euclid's second theorem
G9) PROPORTIONALITY:
1 Geometric quantities
2 The commensurable and incommensurable magnitudes
G10) THE SIMILITUDE:
1 The simile and the triangles
2 The triangularity similarity criteria
3 The simile and the theorems of Euclid
4 Similarity and polygons
5 Similarity and circumference
6 The golden section and its applications
7 The length of the circumference and the area of the circle
G11) GEOMETRIC TRANSFORMATIONS:
1 Geometric transformations and isometries
2 Translation
3 The rotation
4 The central symmetry
5 Axial symmetry
6 A non-isometric transformation: the homothetic
THIRD YEAR
: 1) EQUATIONS AND DISEQUATIONS:
1 Inequalities and principles of equivalence
2 First-degree inequalities
3 Second Degree Inequalities
4 Full second inequalities
5 Inequalities of a grade higher than the second
6 Fractional inequalities
7 Systems of inequalities
8 Equations and inequalities with absolute values
9 Irrational equations and inequalities
2) FUNCTIONS:
1 Functions and their characteristics
2 Injective, surjective and bijective functions
3 Inverse function
4 Properties of the functions
5 Compound functions
6 Geometric and graphic transformations
3) SUCCESSIONS AND PROGRESSIONS:
1 Numerical sequences
2 Induction principle
3 Arithmetic progressions
4 Geometric progressions
4) CARTESIAN AND STRAIGHT PLAN:
1 Coordinates in the plan
2 Length of a segment
3 Midpoint of a segment, center of gravity of a triangle
4 Parallel lines and perpendicular lines
5 Distance of a point from a line
6 Geometric and straight places
7 Bundles of lines
8 Problems with straight lines
5) PARABOLA:
1 Parable and its equation
2 Parabola with axis parallel to the axis x Parabola and functions
3 Parabola and geometric transformations
4 Lines and parabolas
5 Determine the equation of a parabola
6 Search for the equation of a parabola
7 Bundles of parabolas
6) CIRCUMFERENCE:
1 Circumference and its equation
2 Lines and circumferences
3 Determine the equation of a circle
4 Position of two circles
5 Bundles of circumferences
7) ELLIPSE:
1 Ellipse and its equation
2 Ellipses and lines
3 Determine the equation of an ellipse
4 Ellipse and geometric transformations
8) HYPERBOLE:
1 Hyperbole and its equation
2 Hyperbolas and lines
3 Determine the equation of a hyperbola
4 Hyperbola translated
5 Equilateral hyperbole
9) CONIC:
1 Definition of a conic by eccentricity
2 Second degree inequalities in two unknowns
3 Conics and geometric problems
FOURTH YEAR
: 10) EXPONENTIALS:
1 Powers with real exponent
2 Exponential function
3 Exponential equations
4 Exponential inequalities
11) LOGARITMI:
1 Definition of logarithm
2 Logarithm properties
3 Logarithmic function
4 Logarithmic equations
5 Logarithmic equations
6 Logarithmic inequalities
7 Logarithmic inequalities
8 Logarithms and exponential equations and inequalities
9 Domain and sign of functions with exponentials and logarithms
10 Logarithmic equations and inequalities that can be solved only graphically
11 Logarithmic and semilogarithmic coordinates
BETA_1) UNIQUE STATISTICS:
1 Statistical data
2 Indices of position and variability
3 Gaussian distribution
4 Statistical reports
5 Effectiveness, efficiency, quality
6 Indicators of effectiveness, efficiency, quality
7 Reports and indicators
BETA_2) BIVARIATE STATISTICS:
1 Introduction to bivariate statistics
2 Regression
3 Correlation
C1) POLAR COORDINATES IN THE PLAN:
1 Polar coordinates
2 Curve equations
3 Uniform circular motion
C2) APPROXIMATE CALCULATION:
1 The approximations
2 Error propagation
C3) CARRIERS:
1 Vectors in the plan
2 Linearly dependent and independent vectors
3 Scalar product and vector product
4 Cartesian representation of vectors
C4) MATRICES AND DETERMINANTS:
1 Matrices
2 Square matrices
3 Operations with matrices
4 Determinants
5 Properties of determinants
6 Rank
7 Reverse matrix
8 Some applications of matrices and determinants
C5) LINEAR EQUATION SYSTEMS:
1 What are linear systems
2 Inverse matrix method
3 Cramer's rule
4 Reduction method
5 Theorem of Rouché - Capelli
6 Homogeneous linear systems of n equations in n unknowns
C6) CONICAL SECTIONS: THE SYNTHETIC POINT OF VIEW:
1 The Dandelin theorems
2 The parabolic segment
C7) SPEED OF VARIATION OF A GREATNESS:
1 Average speed and instantaneous variation
12) GONIOMETRIC FUNCTIONS:
1 Angle measurement
2 Sine and cosine functions
3 Tangent function
4 Secant and cosecant functions
5 Cotangent function
6 Goniometric functions of particular angles
7 Associated corners
8 Inverse goniometric functions
9 Goniometric functions and geometric transformations
13) GONIOMETRIC FORMULAS:
1 Addition and subtraction formulas
2 Duplication formulas
3 Bisection formulas
4 Parametric formulas
5 Prosthesis and Werner formulas
14) GONIOMETRIC EQUATIONS AND DISEQUATIONS:
1 Elementary goniometric equations
2 Linear equations in sine and cosine
3 Homogeneous second degree equations in the sine and cosine
4 Systems of goniometric equations
5 Goniometric inequalities
15) TRIGONOMETRY:
1 Rectangular triangles
2 Applications of theorems on right-angled triangles
3 Triangles any
4 Applications to trigonometry
5 Cosine theorem
16) COMPLEX NUMBERS:
1 Algebraic form of complex numbers
2 Operations with imaginary numbers
3 Operations with complex numbers in algebraic form
4 Algebraic representation of complex numbers
5 Trigonometric form of a complex number
6 Operations between complex numbers in trigonometric form
7 Nth roots of the unit
17) CARRIERS, MATRICES, DETERMINANTS:
1 Vectors in the plan
2 Vectors in the Cartesian plane
3 Matrices
4 Operations with matrices
5 Determinants
6 Reverse matrix
7 Matrices and analytical geometry
18) GEOMETRIC TRANSFORMATIONS:
1 Translation
2 Rotation
3 Central symmetry
4 Axial symmetry
5 Isometries
6 Omotetia
7 Similarity
8 Affinity
19) EUCLIDEA GEOMETRY IN SPACE:
1 Points, lines, planes in space
2 Perpendicularity and parallelism
3 Distances and angles in space
4 Geometric transformations
5 Polyhedra
6 Rotating solids
7 Solids areas
8 Extension and equivalence of solids
9 Volumes of solids
20) ANALYTICAL GEOMETRY IN THE SPACE:
1 Coordinates in space
2 Vectors in space
3 Plan and its equation
4 Line and its equation
5 Mutual position of a line and a plane
6 Some notable surfaces
Alfa1) COMBINATORIAL CALCULATION:
1 What is combinatorial calculus
2 Provisions
3 Permutations
4 Combinations
5 Newton's binomial
Alfa2) PROBABILITY:
1 Events
2 Classical concept of probability
3 Logical sum of events
4 Conditional probability
5 Logical product of events
6 Bayes theorem
7 Statistical concept of probability
8 Subjective conception of probability
9 Axiomatic setting of probability
C8) TRANSMISSION NUMBERS:
1 Rational numbers and irrational numbers
2 Algebraic numbers and transcendent numbers
C9) NUMBER OF SOLUTIONS OF A POLYNOMIAL EQUATION:
C10) LANGUAGE AND REASONING IN MATHEMATICS:
FIFTH YEAR
: 21) FUNCTIONS AND THEIR PROPERTIES:
1 Real functions of real variable
2 Domain of a function
3 Properties of the functions
4 Inverse function
5 Composite function
22) FUNCTION LIMITS:
1 Sets of real numbers
2 finite limits for x that tends to a finite value
3 infinite limits for x that tends to a finite value
4 finite limits for x that tends to an infinite value
5 infinite limits for x that tends to an infinite value
6 First theorems about limits
23) CALCULATION OF LIMITS AND CONTINUITY OF FUNCTIONS:
1 Limit operations
2 Indeterminate forms
3 Notable limits
4 Calculation of limits
5 Infinitesimals, infinities and their comparison
6 Continuous functions
7 Theorems on continuous functions
8 Points of discontinuity of a function
9 Asymptotes
10 Search for asymptotes
11 Probable graph of a function
24) SUCCESSIONS AND SERIES:
1 Numerical sequences
2 progressions
3 Some properties of the sequences
4 Limit of a succession
5 Calculation of the limit of a succession
6 Induction principle
7 What is a series
8 Converging, divergent, indeterminate series
25) DERIVATIVES:
1 Derivative of a function
2 Fundamental derivatives,
3 Operations with derivatives
4 Derivative of a composite function
5 Derivative of [f (x)] ^ g (x)
26) DIFFERENTIAL CALCULATION THEOREMS:
1 Rolle's theorem
2 Lagrange theorem
3 Consequences of the Lagrange theorem
4 Cauchy theorem
5 De l'Hospìtal theorem
27) MAXIMUM, MINIMUM AND FLEXI:
1 Definitions
2 Maximum, minimum, horizontal inflections and first derivative
3 Flexes and second derivative
4 Next maxima, minima, inflections and derivatives
5 Optimization problems
28) FUNCTION STUDY:
1 Study of a function
2 Graphs of a function and its derivative
3 Applications of the study of a function
4 Approximate resolution of an equation
29) INDEFINITE INTEGRALS:
1 Immediate indefinite integrals
2 Integration by substitution
3 Integration by parts
4 Integration of fractional rational functions
30) INTEGRALS · DEFINED:
1 Fundamental theorem of integral calculus
2 Area calculation
3 Calculation of volumes
4 Volume of a rotating solid
5 Improper integrals
6 Applications of integrals to physics
7 Numerical integration
31) DIFFERENTIAL EQUATIONS:
1 What is a differential equation
2 First order differential equations
3 Second order differential equations
4 Differential equations and physics
32) PROBABILITY DISTRIBUTIONS:
1 Discrete random variables and probability distributions
2 Characterizing values of a discrete random variable
3 Frequently used probability distributions
4 Random games
5 Standardized random variables
6 Continuous random variables
C8) TRANSMISSION NUMBERS:
1 Rational numbers and irrational numbers
2 Algebraic numbers and transcendent numbers
C9) NUMBER OF SOLUTIONS OF A POLYNOMIAL EQUATION:
1 Polynomial functions and equations
2 Approximate calculation of a solution
C10) LANGUAGE AND REASONING IN MATHEMATICS:
1 Demonstrations and patterns of reasoning
2 Validity of reasoning schemes
C11) GEOMETRIES AND FOUNDATIONS:
1 Elements of Euclid
2 Non-Euclidean geometries
3 Fundamentals of mathematics