Ateneo - UPO

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 several 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.

The contents of the class Matematica I.

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 rounded-up average of the two marks. Students with physical disabilities, Learning Disabilities or Special Education Needs can request specific services and tools via the Staff Sviluppo e Coordinamento Carriere e Servizi alle Studentesse e agli Studenti, consulting the University webpage: https://www.uniupo.it/en/services/services-students-physical-or-learning-disabilities Students with disabilities, learning disabilities or special education needs, once they have contacted the University Staff, can refer to the tutor in charge of the course to define the examination modalities, concerning academic aspects.

The exam consists of a written test and an oral exam. The written test consists of exercises related to the syllabus covered during class and lasts 3 hours. Each exercise carries a maximum score for a total of 30, and the grade is calculated by adding the points received, which reflect the accuracy of the exam. Students who achieve a score of 16/30 on the written test are admitted to the oral exam. The oral exam consists of a discussion based on the written test and is intended to assess the knowledge and skills actually acquired. The exam results highlight the student's understanding of theoretical concepts and their ability to use them to solve new problems. Excellence is achieved if the knowledge and skills acquired are comprehensive and thorough. 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 rounded-up average of the two marks. The student is assured the possibility of taking the entire exam (i.e. Mathematics I and Mathematics II) in a single solution: in this case the written test will last 3 hours and the oral discussion will focus on the entire paper.

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 several real variables (functions in several 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.