Teacher
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Marucci Alvaro
(syllabus)
COMBINATORIAL CALCULUS Provisions, permutations and combinations with and without repetition, binominal coefficients, definitions and properties. MATRICES AND DETERMINANTS Definitions and properties of matrices, triangular and diagonal matrices, transposed and opposite matrices, inverse matrices. Product associated with a matrix, determinant, Laplace theorems, rule Sarrus, general properties of determinants, and lesser rank or characteristic of a matrix. Eigenvalues and eigenvectors of a square matrix. SYSTEMS OF LINEAR EQUATIONS General information on systems of linear equations, theorem Rouchè-Capelli, Cramer theorem, systems of m equations in n unknowns, homogeneous systems. FUNCTION OF REAL VARIABLE Relations, applications or functions, definition domain, codomain or target set of a function, image. Graph of a function. Injective function or injection, surjective or a surjection, or bijection bijective function or one-to-one correspondence, invertibility of functions. Rational functions, composite functions, inverse functions, extremes of a function. Limits of a function, theorems about limits, uniqueness theorem of the limit theorem of the permanence of the sign, comparison theorem, the limit of the sum, product, quotient etc .. functions, special limits. CONTINUOUS FUNCTIONS Definitions and properties, continuity of a function, Napier's constant, maximum and minimum of a function. DERIVATIVES Definition of the derivative, geometrical meaning, derivation rules, derivation of trigonometric functions, derivation of composite functions and inverse, the derivative of the sum of functions, derivative of the product of two or more functions, the derivative of the quotient of two functions, etc. derivability and continuity of a function, differential, higher order derivatives, maxima and minima of a function, asymptotes, concavity and convexity, flexed, graphs of functions. Rolle's theorem, Cauchy, Lagrange, De L'Hopital, undetermined forms. Logarithm function, exponential function, hyperbolic functions. FUNCTIONS OF TWO VARIABLES Definitions, domain of definition of a function of two variables, limits and continuity, partial derivatives of first and higher. Hessian matrix, study of the functions of two variables. INTEGRATION Indefinite integral, primitive, property, integration by decomposition, integration by parts and by substitution, integration of rational functions, definite integral, the relationship between integration and derivability, calculating areas, Torricelli's theorem. DIFFERENTIAL EQUATIONS Differential equations of the first order: homogeneous, with separate, separable and linear variables. Cauchy problem. Second order differential equations with constant coefficients.
(reference books)
A.Marucci: Lezioni di matematica generale, Ed. Sette Città (2017)
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