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Mathematical analysis 3

A.Y. 2021/2022

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.

· 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.

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

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### CORSO A

Responsible

Lesson period

First semester

Videos with lessons and exercises collections will be made available at the official web space of this course and they will cover the topic of each week. Live meetings during the classes hours may be scheduled by using Zoom.us. The details of these meetings will be available in the course web page of the Ariel platform, as well as the videos and every kind of material needed for the course.

The exams will be done by following the ways suggested on the web page of the university. The written exam will have the same structure as the one done in presence, possibly reduced in time and number of exercises.

The exams will be done by following the ways suggested on the web page of the university. The written exam will have the same structure as the one done in presence, possibly reduced in time and number of exercises.

**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.

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.

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.

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.

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 written test can be replaced by two ongoing tests . 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

Lessons: 32 hours

Professors:
Calanchi Marta, Ciraolo Giulio

### CORSO B

Responsible

Lesson period

First semester

The exams will be done by following the ways suggested on the web page of the university. The written exam will have the same structure as the one done in presence, possibly reduced in time and number of exercises.

**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.

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.

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.

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.

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 written test can be replaced by two ongoing tests . 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

Lessons: 32 hours

Professors:
Cozzi Matteo, Molteni Giuseppe

Professor(s)

Reception:

My office: Dipartimento di Matematica, via Saldini 50, first floor