Ateneo - UPO

Basic concepts on sets. Real numbers and complex numbers. Real functions of a real variable. Linear algebra. Advanced analysis. Ordinary differential equations.

Recommended textbooks: + M. Bramanti, C.D. Pagani, S. Salsa. Analisi matematica 1 con elementi di geometria e algebra lineare, Zanichelli, 2014. + M. Bramanti, C.D. Pagani, S. Salsa. Analisi matematica 2, Zanichelli, 2009. + S. Salsa, A. Squellati, Esercizi di Analisi Matematica vol. 1, Zanichelli, 2011. + S. Salsa, A. Squellati, Esercizi di Analisi Matematica vol. 2, Zanichelli, 2011.

The class aims at providing students with the knowledge of the basic notions regarding real functions of one and more real variables, ordinary differential equations and linear algebra. The purpose of the class is that students acquire and know how to use an appropriate mathematical language in relation to topics covered in the course. The educational goal of this class is to develop the ability to apply this knowledge in solving problems encountered in applications in Physics and Chemistry.

Mathematics skills common to all secondary school curricula.

The lessons will take place through lectures on the blackboard. In addition to the theoretical lessons, classroom exercises will be carried out by the teacher with the involvement of the students to delve into the topics covered during the theoretical lessons. The concepts covered by the course will be discussed collectively in the classroom and applied directly during classroom exercises to stimulate the students' critical sense and autonomy of judgment.

The study plan includes a single exam for the courses of Mathematics I and Mathematics II. However, the student is given the opportunity to take the tests separately. In any case, a single grade is recorded for the two courses of Mathematics I and Mathematics II, which will be assigned starting from the average rounded up by the two marks obtained following a collegial evaluation of the student.

The exam consists of a written test and an oral test. The written test consists of exercises related to the program carried out during the lessons and lasts 3 hours. The oral exam is at the teacher's discretion and consists of a discussion on the written test. The student is assured the possibility of taking the two exam modules separately (i.e., Mathematics I and Mathematics II): in this case each written test will last an hour and a half. However, only one vote is recorded for the two courses in Mathematics I and Mathematics II, which will be decided starting from the average rounded up by the two marks obtained following a collegial evaluation of the student. At the end of the written test the teacher corrects it and communicates the results. Afterwards the oral exam generally follows. The results of the examination highlight the degree of understanding of theoretical concepts and the ability to use them to solve new problems.

Basics on set theory. Numeric sets, construction of the real numbers, complex numbers. Functions, composition between functions, injectivity, surjectivity, bijectivity, invertibility, cardinality of a set. Real sequences, limit of a sequence and classical theorems. Limit of a real function of one real variable; right limit and left limit; classical theorems. Continuity for real functions of one real variable; basic properties and classical results. Differential Calculus for real functions of one real variable. Local maxima and minima; monotonicity of a function and sign of the first derivative; de l'Hôpital Theorem; convex and concave functions, flex points and sign of the second derivative. Riemann integral: classical theorems and integral calculus. Numeric series and power series. Vector spaces: dimension, basis; matrices and linear maps, kernel, range; determinants; resolution of linear systems; characteristic polynomial, eigenvalues, eigenvectors, diagonalization of matrices. Real functions of several real variables: domain, limits and continuity for multivariate functions; partial derivatives and tangent plane; differentiable functions and some basic properties; free extremals for a differentiable functions. Integral calculus for multivariate functions: definition of integral, computation of double integrals via successive integrations on rectangles and simple domains; change of variables for multivariate integrals, applications; line integrals. Basics of surface integration. Vector fields: vector operators, Theorems of Green, Gauss, and Stokes. Basics on ordinary differential equations.

Knowledge and understanding: at the end of the course the student must know the basic notions of Linear Algebra (vector spaces, homomorphisms, matrices and related operations) and of Mathematical Analysis in one and more real variables (functions in one and more variables, limits, differentials, integrals). Ability to apply knowledge and understanding: the student must have developed a mastery of the learned notions allowing to connect them independently and to use them jointly in solving simple theoretical problems. Ability to communicate what has been learned: the student must be able to clearly explain both the concepts themselves and their use in general terms or in the specific case of a problem.