Teacher
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Capuani Rossana
(syllabus)
NUMERICAL SETS Definition of set and operations between sets. Properties. Cartesian product, equivalence and order relations. Equinumerous sets. Set of natural, integer, rational, real numbers and their properties. Infimum and supremum of a set. Combinatorial calculation: permutations and simple combinations. Binomial expansion.
SET OF COMPLEX NUMBERS Definition of the set of complex numbers. Algebraic representation of a complex number. Opposite, conjugate, modulus of a complex number and their properties. Operations between complex numbers (sum, difference, product and quotient). Graphical representation of a complex number. Trigonometric form of a complex number. Power and n-th root of a complex number. Exponential representation and Euler's formulas. Algebraic equations in C.
VECTORS Definitions and examples of vectors. Operations on vectors. Versor. Scalar product. Cross product of vectors. Examples.
VECTOR SPACE Space R^n. Definition and basic properties. Subspaces. Linear dependence and independence. Bases and dimensions. Examples. Normed vector spaces and inner product spaces.
INTRODUCTION TO MATRIX CALCULATIONS Definition of matrix. Sum between matrices, matrix product and their properties. Transposed matrix. Definition of determinant and its properties. Calculation through the Laplace formula. Inverse matrix. Rank of a matrix. Examples.
LINEAR SYSTEM Linear maps. Linear systems. Examples. Homogeneous systems. Cramer's rule. Rouché-Capelli theorem. Matrix diagonalization. Introduction to eigenvalues and eigenvectors. Algebraic multiplicity and geometric multiplicity.
INTRODUCTION OF SPACE GEOMETRY Equation of straight line in space. Parallelism and orthogonality between straight lines. Plane in space. Parallelism and orthogonality between straight line and plane. Orthogonality between planes. Examples.
ELEMENTARY FUNCTIONS Definition of function. Injective, surjective, bijective and invertible functions. Monotonic functions. Composition between functions. Exponentiation with natural and real exponent and its properties. Root extraction and its properties. Exponential function and its properties. Logarithm function and its properties. Trigonometric functions and properties. Inverse trigonometric functions. Function graphs.
SEQUENCES Definition of sequence. Convergence and divergence. Uniqueness of the limit. Operations with limits. Comparison theorems. Remarkable limits. Monotonic sequences. Extract of a sequence. Bolzano-Weierstrass theorem. Cauchy convergence criterion.
SERIES Definition of series. Sequence of partial sums, series with positive terms, harmonic series, geometric series, telescoping series. Necessary condition of convergence. Comparison criterion, ratio criterion, root criterion, infinitesimal criterion. Alternating series. Absolute convergence. Leibniz criterion.
LIMITS AND CONTINUITY FOR REAL FUNCTIONS OF A REAL VARIABLE Limits of functions of a real variable and properties. Operations with limits. Remarkable limits. Continuity and theorems on continuous functions. Monotonic functions. Maximum and minimum of function. Asymptotes. Discontinuity.
DERIVATIVES FOR REAL FUNCTIONS OF A REAL VARIABLE Difference quotient. Geometric interpretation of the derivative. Derivative of elementary functions. Differentiation rules. Derivative of composed function. Derivative of the inverse function. Rolle's theorem, Lagrange's theorem. Non-derivability. Critical points, monotony, concavity and convexity. The de l'Hôpital theorem. Study of the graph of a function. Infinitesimal and infinite concept. Applications to the calculation of limits. The concept of differential. Taylor-MacLaurin formula with Peano remainder and with Lagrange remainder.
INTEGRAL Definition of integral. Classes of integrable functions. Integral properties. Integral mean theorem. Fundamental theorem of integral calculus. Primitives and calculation of Riemann integrals. Immediate integrals by decomposition, by replacement. Integration of rational functions. Integration by parts. Integration of trigonometric functions. Integration of irrational functions.
(reference books)
Analisi Matematica 1 con elementi di geometria e algebra lineare. Bramanti, Pagani, Salsa. Zanichelli (ed. 2014) Elementi di analisi matematica 1. Versione semplificata per i nuovi corsi di laurea. Marcellini, Sbordone. Liguori (ed. 2002) Esercitazione di matematica Vol 1. Marcellini, Sbordone. Liguori Esercitazione di matematica Vol 2. Marcellini, Sbordone. Liguori
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