Mathematical Analysis 3

A.Y. 2020/2021
Overall hours
Learning objectives
The course aims to provide the basics concepts concerning the succession and the series of functions, ordinary differential equations, integration of differential forms along paths in open domain of the R^n space.
Expected learning outcomes
Capability to relate different aspects of the subject, and self-confidence in the use of the main techniques of Calculus.
Course syllabus and organization

Single session

Lesson period
First semester
The didactic activity will be performed on the basis of what is stated by the university. As to what is stated at the moment (july 2020), classes will be given on-line as scheduled in the time-table, as well as the tutoring. More tutoring could be available in presence. Exams will be permorfed following what will be stated by the university. Any on-line activity will be performed by using Zoom. The content of the course as well as the rules and the level of the exams will be independent on the employed methods.
Course syllabus
Sequences and series of functions: pointwise and uniform convergence. Power series: the domain and the radius of convergence. Abel's theorem. Taylor's series.
Functionals spaces connected with the uniform convergence. A fixed point theorem.
Implicit functions: Dini's theorem in the scalar and vector cases. The inverse function theorem. Constrained optimization.
First order differential equations: the Cauchy problem, existence and uniqueness of solutions. Differential equations of higher order. Linear equations.
Curves in R^n: length, integration along curves.
Differential forms, exactness and related results. Curves and differential forms. Conservative fields. Closed and exact forms. Potentials.
Prerequisites for admission
Mathematical Analysis 1 and 2, Geometry 1 and 2 (these courses are not mandatory, but they are strongly recommended).
Teaching methods
Lessons from a teacher and public solution of esercises from a second teacher. A tutor is made available to students in order to help them understand the subjects and solve and discuss exercises that have been proposed by the teachers.
Teaching Resources
-) G. Molteni, "Note del corso", free from the web page of the course;
-) C. Zanco, "Appunti dalle lezioni", free from the web page of the course;
-) G. Molteni, M. Vignati, "Analisi Matematica 3", Città Studi ed.;
-) N. Fusco, P. Marcellini, C. Sbordone "Analisi Matematica due", Zanichelli ed.;
-) C. Maderna, P.M. Soardi "Lezioni di Analisi Matematica II", Città Studi ed.;
-) C.D. Pagani, S. Salsa "Analisi matematica, vol. 2", Zanichelli ed.;
-) C. Maderna, G. Molteni, M. Vignati: "Esercizi di Analisi Matematica 2 e 3", Città Studi ed..
Assessment methods and Criteria
Both a written and an oral exam. The student is elegible for the oral exam only when its written exam is positively evaluated. The written exam mainly deals with exercises at a level comparable to the level of teaching, while the oral part of the exam is essentially a discussion about the theoretical results which are presented in lessons.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 9
Practicals: 48 hours
Lessons: 45 hours