Mathematical Analysis 3 | Università degli Studi di Milano Statale

A.Y. 2024/2025

Max ECTS

Overall hours

SSD

MAT/05

Language

Italian

Included in the following degree programmes

Physics (Classe L-30)-Enrolled from 2012/2013 Academic Year

Learning objectives

The course aims to complete the basic Knowledge of Mathematical Analysis for the students of the three-year degree in Physics. In particular the course will provide the student with the knowledge and the mastery of the following concepts, widely used in Physics:

· implicit functions and constrained maxima and minima;
· differential forms, exact and closed forms;
· Lebesgue theory for functions of several variables;
· surfaces and surface integral;
· Gauss-Green formulas, divergence and Stokes theorem.

Expected learning outcomes

At the end of the course the students will have acquired the following knowledge and skills.

1. Knowledge and understanding of the following subjects:
· implicit functions;
· constrained maxima and minima;
· differential forms, exact and closed forms;
· Lebesgue theory for functions of several variables;
· surfaces and surface integral;
· Gauss-Green formulas, divergence and Stokes theorem.

2. Ability to present and discuss in a critical way the concepts studied;

3. Ability to solve problems by using the results and the tools, they have learnt.
In particular they will be able
· to study implicit functions
· to minimize functions of several variables with constraints
· to calculate integrals in several variables;
· to understand whether a differential form is exact and to determine a primitive;
· to apply the divergence and Stokes theorems in the plane and in the space

4. Ability to use the tools they have studied in different framework.

Lesson period: First semester

Lessons timetable

Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi

Exams calendar

Single course

This course can be attended as a single course.

Take a single course

Course syllabus and organization

CORSO A

Responsible

Calanchi Marta

Lesson period

First semester

Syllabus
Course structure

Course syllabus

Implicit functions.
Maxima and minima with constraints; Lagrange multipliers.
Lebesgue measure and integral in R^n; multiple integrals.
Curves and curve integrals.
Linear differential forms.
Surfaces and surface integral.

Prerequisites for admission

1. Differential and integral calculus for real functions of a real variable.
2. Sequences and numerical series.
3. Basic knowledge of Analytical Geometry and Linear Algebra.
4. Differential calculus for vectorial functions of several real variables.
5. Free optimization for functions of several variables.
6. Differential equations: some techniques of resolution.
7. Cauchy problems: the main results for existence and / or uniqueness of the solution.
8. Sequences and series of functions.

Teaching methods

Attendance mode: strongly recommended;
Mode of delivery: lectures and exercise classes.

Teaching Resources

Textbook:
N. Fusco, P. Marcellini, C. Sbordone, Analisi Matematica due, Liguori ed.

Exercise book:
C. Maderna, G. Molteni, M. Vignati, Esercizi scelti di Analisi matematica 2 e 3, Città Studi ed.

Other textbooks:
G. Molteni, M. Vignati, Analisi matematica 3, Città Studi ed.
C. Maderna, P.M. Soardi, Lezioni di Analisi matematica II, Città Studi ed.

Assessment methods and Criteria

The exam consists of a written and an oral part. The written part consists of a written test
about the arguments discussed during the exercise classes. The duration of the written exam
will be proportional to the number of exercises assigned, also taking into account the nature
and complexity of the exercises themselves (however, the duration will not exceed three hours). The oral exam can be taken only if the written component has been successfully passed.
The oral part consists of an oral examination that focuses mainly on theoretical topics of the
course (knowledge of the major theoretical results, connections between different parts of
programs, demonstrations of proven results in class) and on the written examination.
The solution of some exercises may also be required during the oral part of the examination.
The complete final examination is passed if all the two parts (written and oral) are successfully
passed. Final marks are given using the numerical range 0-30, and will be communicated immediately
after the oral examination.

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6

Practicals: 24 hours
Lessons: 32 hours

Professors: Calanchi Marta, Calzi Mattia

CORSO B

Responsible

Stuvard Salvatore

Lesson period

First semester

Syllabus
(active tab)
Course structure

Course syllabus

Implicit functions theorem.
Constrained optimization; Lagrange multipliers.
Lebesgue measure and integral in R^n; multiple integrals.
Curves and curve integrals.
Linear differential forms.
Surfaces and surface integral.

Prerequisites for admission

1. Differential and integral calculus for real-valued functions of one real variable.
2. Sequences and numerical series.
3. Basic knowledge of Analytical Geometry and Linear Algebra.
4. Differential calculus for vector-valued functions of several real variables.
5. Free optimization for functions of several variables.
6. Ordinary differential equations: some techniques of resolution.
7. Cauchy problems: the main results for existence and / or uniqueness of the solution.
8. Sequences and series of functions.

Teaching methods

The course is delivered through lectures and exercise classes.
Attendance is strongly recommended.

Teaching Resources

Textbook:
N. Fusco, P. Marcellini, C. Sbordone, Analisi Matematica due, Liguori ed.

Exercise book:
C. Maderna, G. Molteni, M. Vignati, Esercizi scelti di Analisi matematica 2 e 3, Città Studi ed.

Other reference books:
G. Molteni, M. Vignati, Analisi matematica 3, Città Studi ed.
C. Maderna, P.M. Soardi, Lezioni di Analisi matematica II, Città Studi ed.

Assessment methods and Criteria

The exam consists of a written and an oral part. The written part consists of a written test about the arguments discussed during the exercise classes. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). Admission to the oral exam will be granted only upon passing the written exam.
The oral exam focuses mainly on theoretical topics of the course (knowledge of the major theoretical results, connections between different parts of the program, knowledge and understanding of the proofs presented in class) and on the written examination.
The solution of some exercises may also be required during the oral part of the examination.
The complete final examination is passed if the two parts (written and oral) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6

Practicals: 24 hours
Lessons: 32 hours

Professors: Cavalletti Fabio, Stuvard Salvatore

Educational website(s)

Analisi Matematica 3 CORSO A e CORSO B (a.a. 2024/25)

Professor(s)

Calanchi Marta

Web site

Reception:

on appointment every day

dipartimento di matematica

Reception:

Please, request an appointment via email

Room 1041, Department of Mathematics, Via Cesare Saldini 50, first floor or via Zoom conference call

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