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 texts:

Bramanti, Pagani, Salsa, "Analisi Matematica 1 with elements of geometry and linear algebra", Ed. Zanichelli (for the Geometry module).

F. Gazzola, "Analisi Matematica 2", Ed. Esculapio (for the Analysis II module).

Provide solid foundations on linear algebra and on the theory of functions of several real variables and of differential and integral calculus. The student must be able to face a problem with the right methodological rigour, both in presenting a statement and a proof of a theoretical result and in solving exercises.

Contents of the Mathematics I course on the theory of sets and functions and in particular, in the case of real functions of real variables, on the differential and integral calculus for these particular functions.

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.

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.