Ateneo - UPO

Elements of general topology in n-dimensional Euclidean space, limits and continuity for functions of several real variables, partial differentiation, multiple integration, line and surface integrals, differential forms and vector fields, Gauss-Green, divergence and curl formulas.

Recommended textbooks:

F. Gazzola, "Mathematical Analysis 2", Ed. Esculapio, 2022.

M. Bramanti, C. D. Pagani, S. Salsa, "Mathematical Analysis 2", Ed. Zanichelli, 2009.

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, 2016.

Workbook:

S. Salsa, A. Squellati, ”Esercizi di Analisi Matematica 2”, Ed. Zanichelli, 2011.

Provide a solid foundation on 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-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 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.

The course aims to provide the basic notions of differential and integral calculus for functions of several real variables. The main topics that will be covered in this course are the following: partial differentiation, stationary points, relative maximum and minimum points, differential forms and vector fields and related line integrals, multiple integration, line and surface integrals, Gauss-Green, divergence and curl formulas.

- Knowledge and understanding: acquisition of the main notions 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 for 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.