Ateneo - UPO

The course aims to provide the basic notions of linear algebra and differential and integral calculus for functions of several real variables. The main topics that will be covered in this course are the following: vector spaces and basis, operations between vectors, matrices and operations between matrices, linear systems, linear maps, eigenvalues and eigenvectors, diagonalization of symmetric operators, partial differentiation, multiple integration, line and surface integrals , differential and potential forms, Gauss-Green, divergence and curl formulas.

Recommended text for the Geometry module:

Bramanti, Pagani, Salsa, “Mathematical Analysis 1 with elements of geometry and linear algebra”, Ed. Zanichelli

Recommended texts for the Analysis II module:

F. Gazzola, "Mathematical Analysis 2", Ed. Esculapio

M. Bramanti, C. D. Pagani, S. Salsa, "Mathematical Analysis 2", Ed. Zanichelli

For a deeper analysis of the subject we also suggest the more recent edition of the following textbook:

C. D. Pagani, S. Salsa, "Mathematical Analysis 2", Ed. Zanichelli

Provide a solid foundation in linear algebra and the theory of functions of multiple real variables and of differential and integral calculus. Those who intend to pass the exam of this course must be able to tackle a problem with the right methodological rigor, both in presenting a statement and a proof of a theoretical result and in solving the exercises.

Contents of the Mathematics I course on differential and integral calculus for real functions of one real variable. In particular, people who intend to attend this course and pass the related exam, must achieve the expected learning outcomes for the Mathematics I course which we recall below:
- know the basic notions of Mathematical Analysis in one real variable (functions in one variable, limits, derivatives, integrals);
- ability to apply knowledge and understanding: you must have developed a mastery of the notions learned which allows you to connect them together independently and use them jointly in solving simple theoretical problems;
- ability to communicate what has been learned: you must be able to clearly explain both the notions themselves and their use in general terms or in the specific case of a problem.

The course is organized with lectures with theoretical part and exercises. Each topic of the course is introduced through a discussion which aims to
make it understandable to students as much as possible. In a second moment the basic notions of each topic are introduced; they are subsequently followed by examples in order to clarify their meaning. The third step is dedicated to the statements of the main theorems and their proofs. The last part is dedicated to exercises. Active participation in lessons is stimulated through direct questions to students which also aims to understand the level of difficulty encountered by them in following the lessons themselves; students are also invited to propose exercises on topics which, in their view, require further clarification.

In addition to the books suggested for the theory, further material for the preparation of the exam will be provided during the course.

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-learningdisabilities
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 a subsequent oral test including the topics of both modules of Geometry and Analysis II. The written test usually consists of 4-6 exercises on different topics of the course. In each written test, most of the topics contained in the course are covered. The presence of questions on the theoretical part is not excluded. The oral exam consists of a discussion on the exercises contained in the written test and some final questions on the statements and demonstrations of the main theoretical results. Admission to the oral test is subject to passing the written test with a grade greater than or equal to 16/30; the final grade will be obtained having as a starting point the grade of the written test to which points will be added or subtracted based on the progress of the oral test. Passing the written test, even with full marks, does not guarantee passing the exam in the event of an insufficient oral test.

The course aims to provide the basic notions of vector and matrix calculus and of differential and integral calculus for functions of several real variables. The main topics that will be covered in this course are the following: vector spaces and basis, operations between vectors, matrices and operations between matrices, linear systems, linear applications, eigenvalues ​​and eigenvectors, diagonalization of symmetric operators, numerical series, partial derivation, multiple integration, line and surface integrals, differential and potential forms, Gauss-Green, divergence and curl formulas.

- Knowledge and understanding: acquisition of the main notions of vector and matrix calculus and of differential and integral calculus for functions of several real variables.
- Ability to apply knowledge and understanding: being able to deduce the main qualitative and quantitative properties of vectors, matrices, functions between vector spaces and functions of several real variables.
- Ability to learn: the student will have to acquire a certain mastery in the use of logical reasoning and in the application of the methodological rigor necessary to face problems based on mathematical modelling.