The course aims to provide the student with fundamental methods and techniques of Mathematics, with particular reference to the foundations of linear algebra, to differential and integral calculus for the functions of a real variable, to the study of sequences and numerical series. The course provides knowledge of the main notions of mathematical analysis and mastery in the management of calculus. The student will be able to formulate a basic logical and deductive reasoning, with particular reference to mathematics in general. A further objective is the preparation of the student for the application of analytical techniques to other scientific disciplines. The acquired notions will allow the student to understand which mathematical models and techniques are most appropriate for the description of engineering systems. The student will acquire the ability to communicate what has been learned and processed and also express and argue the choice of one methodology over another for solving a mathematical problem.
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. Rouché-Capelli 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 non-aligned 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. Argand-Gauss plane. Trigonometric form of a complex number. Power and n-th 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. Bolzano-Weierstrass 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. 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.
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
Dates of beginning and end of teaching activities
Objectives of the course
Università degli Studi della Tuscia - Rettorato, Via S.M. in Gradi n.4, 01100 Viterbo, ITALY - Tel. 0761.3571