Teacher

LUPICA ANTONELLA
(syllabus)
NUMERICAL SETS Definition of set and operations between sets. Properties and De Morgan laws. Cartesian product, equivalence and order relations. Definition of function. Injective, surjective, bijective and invertible functions. Monotonic functions. Composition between functions. Equinumerous sets. Cantor's theorem. Set of natural, integer, rational, real numbers and their properties. Infimum and supremum of a set. Combinatorial calculation: permutations and simple combinations. Binomial expansion. Mathematical induction. Logic elements.
ELEMENTARY 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. Hyperbolic functions and properties. Inverse hyperbolic functions. Function graphs.
LINEAR ALGEBRA Vectors and vector spaces on R. Vector subspaces. Canonical vectors, linear dependence and independence. Bases and dimensions. Definition of matrix. Transposed, symmetric matrix. Sum between matrices, matrix product and their properties. Definition of determinant and its properties. Calculation through the Laplace formula. Calculation of the determinant of matrices of order 3 with the Sarrus rule. Inverse matrix. Rank of a matrix. Linear maps. Image and kernel of a linear map. Matrix representation of linear maps. Linear systems. RouchéCapelli theorem. Free parameters theorem. Cramer's rule. Homogeneous systems. Introduction to eigenvalues and eigenvectors.
PLAN AND SPACE GEOMETRY Cross, dot and triple product between vectors. Cartesian Euclidean space. Three points alignment. Plane in space: plane passing through 3 nonaligned points, plane passing through a point and orthogonal to a vector, plane passing through a point and parallel to two vectors. Planes in a particular position. Intersection of planes. Straight line in space: straight line as intersection of planes, straight line passing through a point and parallel to a vector, straight line passing through two distinct points. Parallelism and orthogonality between straight lines. Parallelism and orthogonality between straight line and plane. Orthogonality between planes. Plane algebraic curves. Canonical equation of ellipse, hyperbola and parabola. Conic sections.
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. ArgandGauss plane. Trigonometric form of a complex number. Power and nth root of a complex number. De Moivre formula. Exponential representation and Euler's formulas. Algebraic equations in C and fundamental theorem of algebra.
SEQUENCES Definition of sequence. Convergence and divergence. Uniqueness of the limit. Operations with limits. Comparison theorems. Remarkable limits. Monotonic sequences. Sequence of Euler. Extract of a sequence. BolzanoWeierstrass theorem. Cauchy convergence criterion.
SERIES Definition of series. Sequence of partial sums, series with positive terms, harmonic series, geometric series, exponential series. Necessary condition of convergence. Comparison criterion, ratio criterion, root criterion, infinitesimal criterion. Alternating series. Absolute convergence. Leibniz criterion. Cauchy theorem.
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, Cauchy's theorem, Lagrange's theorem. Nonderivability. 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. TaylorMacLaurin 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)
1. Analisi Matematica 1 con elementi di geometria e algebra lineare. Bramanti, Pagani, Salsa. Zanichelli (ed. 2014) 2. Elementi di analisi matematica 1. Versione semplificata per i nuovi corsi di laurea. Marcellini, Sbordone. Liguori (ed. 2002) 3. Analisi Matematica 1. Bramanti, Pagani, Salsa. Zanichelli (ed. 2008) 4. Esercitazione di matematica Vol 1. Marcellini, Sbordone. Liguori 5. Dispense del docente
