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 con elementi di geometria e algebra lineare”, Ed. Zanichelli, 2014. (The text indicated is the same that will be adopted for the course of Mathematics I). - F. Gazzola, "Analisi Matematica 2", Ed. Esculapio, 2022. - M. Bramanti, C. D. Pagani, S. Salsa, "Analisi Matematica 2", Ed. Zanichelli, 2009. For further information, please also refer to the recent edition of the following text: C. D. Pagani, S. Salsa, "Analisi Matematica 2", Ed. Zanichelli, 2016. Workbook: S. Salsa, A. Squellati, ”Esercizi di Analisi Matematica 2”, Ed. Zanichelli, 2011.

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 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 Codice Descrizione 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 exam covering the topics of both the Geometry and Analysis II modules. Each part will be assigned a separate evaluation. The written test typically consists of 4-6 exercises on various course topics. Most of the course topics are covered in each written test. Questions on the theoretical part may also be included. The oral exam consists of a discussion of the exercises included in the written test and a few final questions on the statements and proofs of the main theoretical results. Admission to the oral exam for each of the two Geometry and Analysis II modules is contingent on a minimal mark of 16/30 or higher in each of the respective parts in the written test; the final mark for each of the two modules will be obtained using the written test mark as a starting point, to which points will be added or subtracted based on the performance of the oral exam. The highest mark, corresponding to 30 cum laude, will be awarded if a decidedly high level is achieved in both the written and oral tests. Passing the written exam, even with top marks, does not guarantee passing the exam if the oral exam is insufficient. The final grade for the entire Mathematics II exam will be awarded after an overall assessment based on the results achieved in the individual modules.

Vector spaces and basis, operations between vectors, matrices and operations between matrices, linear systems, linear maps, eigenvalues and eigenvectors, diagonalization of symmetric operators.

- Knowledge and understanding: acquisition of the main notions of matrix and vector calculus. - Ability to apply knowledge and understanding: being able to deduce the main properties of vectors, matrices and functions between vector spaces. - Ability to learn: the student will have to acquire a certain mastery in the use of logical reasoning and in the application of methodological rigor needed to tackle problems based on mathematical modeling.