2020/2021

2020/2021

DM270

MAT/05 (MATHEMATICAL ANALYSIS)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

1°

2nd semester (01/03/2021 - 11/06/2021)

9

84 lesson hours

Italian

WRITTEN AND ORAL TEST

VITALI ENRICO (titolare) - 9 ECTS

Basic results from the courses Mathematical Analysis 1 and Linear Algebra, i.e.: elementary functions, fundamentals of differential and integral calculus in one variable, basic results about vector spaces and matrix calculus.

The principal aim is to lead students to master the basic elements in the fields of Mathematical Analysis which are the natural continuation of the contents of the course Mathematical Analysis 1; hence the core of the course deals with differential and integral calculus in several variables. Within these areas the student will be expected to know the main theoretical features and to be able to suitably select and apply the principal techniques from Analysis for the resolution of the proposed problems.

The course will focus on the theoretical fundamentals and the key analytical techniques related to the study of functions between Euclidean spaces. In particular, the following topics will be dealt with: metric and topological structure of euclidean spaces; partial derivatives, differential, gradient and Jacobian matrix, Taylor formula and the Mean Value Theorem; extrema with and without constraints; implicit functions and local invertibility results; Peano-Jordan measure and Riemann multiple integral; Fubini’s Theorem; change of variables in multiple integrals; improper integrals; regular surfaces; integration on lines and surfaces; differential forms.

Notice: some in-depth parts will not be required to students of the course "Complementi di analisi matematica I" (Corso di Laurea in Fisica) which shares 6 CFU with the course.

Teaching will be mainly carried out through traditional classes. Whenever possible, personal supervised work will be organized within these classes to help students with exercises.
A tutoring support will be available to strengthen practice skills.

The main topics are contained in:
a) “Analisi Matematica 1” and “Analisi Matematica 2” by M. Bramanti, C.D. Pagani and S. Salsa (Zanichelli)
b) “Analisi Matematica 1” and “Analisi Matematica 2” by C.D. Pagani and S. Salsa (Zanichelli).
References (b) are more extensive and in-depth.

Further readings:
“Lezioni di Analisi Matematica" vol. 1 and vol. 2”, by Giovanni Prodi (Boringhieri).
In particular, volume 1 deals with the general notion of limit in metric and topological spaces.

Some references for exercises:
'Esercizi di Analisi Matematica 2' by S. Salsa and A. Squellati (Zanichelli, 2011)
"Esercitazioni di Analisi Matematica Due" (prima e seconda parte) by P. Marcellini and C. Sbordone (Zanichelli, 2017)

The exam consists in a written test and in an oral part. The first one mainly aims to check the level of knowledge of the principal analytical methods dealt with in the course, together with the ability to face a mathematical problem in the field. A threshold mark is needed to pass to the oral part, which intends to verify the global understanding of the theoretical framework.