# Mathematical Analysis II

**Academic Year 2023/2024**- Teacher:

**Salvatore LEONARDI**

## Expected Learning Outcomes

1. Knowledge and understanding:The student will be able to understand and assimilate the definitions and main results of the basic mathematical analysis for functions of several real variables, necessary for the treatment and modeling of problems deriving from science applied.

2. Applying Knowledge and understaning:The student will be able to acquire an appropriate level of autonomy in theoretical knowledge and in the use of basic analytical tools. The course prepares for the study of Fourier series and Fourier and Laplace transforms.

3. Making judgments:Ability to reflect and calculate. Ability to apply the notions learned to solving problems and exercises.

4. Communication skills:Ability to communicate the notions acquired through an adequate scientific language.

5. Learning skills: Ability to deepen and develop acquired knowledge. Ability to critically use tables and analytical and computer tools of symbolic computation.

PLEASE NOTE: Information for students with disabilities and / or SLI To guarantee equal opportunities and in compliance with the laws in force, the interested students can ask for a personal interview so to program any compensatory and / or dispensative measures, according to the didactic objectives and specific needs.

It is also possible to contact the referent of CInAP (Centro per l’integrazione Attiva e Partecipata - Servizi per le Disabilità e/o i DSA) of the Department.

## Course Structure

Lectures in classroom.

###### Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programm planned and outlined in the syllabus.

**PLEASE NOTE: Information for students with disabilities and / or SLI**

**To guarantee equal opportunities and in compliance with the laws in force, the interested students can ask for a personal interview so to program any compensatory and / or dispensative measures, according to the didactic objectives and specific needs.**

**It is also possible to contact the referent of CInAP (Centro per l’integrazione Attiva e Partecipata - Servizi per le Disabilità e/o i DSA) of the Department.**

## Required Prerequisites

## Attendance of Lessons

## Detailed Course Content

Struttura del corso della prima parte:

8 CFU - 61 total hours

49 hours lectures

12 hours exercises

**1. Sequences and Series of Functions**.

Sequences of real functions of real variable. Series of functions. Pointwise, uniform and total convergence. Theorems of continuity, integration by series. Real power series. Pointwise onvergence. Theorem of D'Alembert and Cauchy - Hadamard. Radius of convergence of the derivatives series. Theorems of differentiation and integration of power series. Taylor series. Criterion for the expansion in a Taylor series. Major series.

2. Real functions of two or more real variables.

Elements of topology in R^2 and R^3.

Bounded sets. Open connected sets.

Limits and continuity. Weierstrass theorem.

Partial derivatives. Successive derivatives. Schwartz theorem. Gradient. Differentiability.

Differentiability and continuity. Differential Theorem.

Composition of functions. Theorem of derivation of composite functions. Functions with zero gradient in a connected set.

Extremals. Necessary and sufficient conditions for an extremal.

Conditioned extremals. Lagrange multipliers.

** 3. Curvilinear integrals and differential forms in R^n.**

Regular curves. tangent vector and the normal vector of a smooth curve at a point.

Rectificability . Length of a smooth curve. Oriented curves. Arc length.

Curvilinear integral of a function. Differential forms.

Curvilinear integral of a differential form.

Exact differential forms. Integration theorem of exact differential forms. Characterization of the exact differential forms. Potential of a differential form. Closed differential forms. Differential forms in a rectangle. Differential forms in a simply connected open set of R2 and R3

**3. Basics on Fourier series. **Trigonometric polynomials.** **Trigonometric series. Convergence in L^2 of Fourier series.

## Textbook Information

[1] Bramanti, C. Pagani, S. Salsa, Analisi Matematica due, Zanichelli.

[2] G. Di Fazio - P. Zamboni, Analisi Matematica Due, seconda edizione, Ed. Monduzzi.

[3] N. Fusco, P. Marcellini, C. Sbordone, Analisi Matematica due, Liguori Editore.

[4] M.Bramanti, Esercitazioni di Analisi Matematica 2,

[5] G. De Marco - C. Mariconda, Esercizi di calcolo in più variabili, Ed. Zanichelli - Decibel.

## Course Planning

Subjects | Text References | |
---|---|---|

1 | Topic 1 | [1,2] |

2 | Topic 2 | [1,2] |

3 | Topic 3 | [1,2] |

4 | Topic 4 | [1,2] |

## Learning Assessment

### Learning Assessment Procedures

1. A single in itinere written test is given (called test or section A) consisting of theoretical and practical questions concerning the part of the program treated up to an agreed date

2. The final exam consists of a written paper divided into two sections: A (with the topics covered up to the ongoing test) and B containing practical and theoretical questions concerning the part of the program treated after test A

3. Passing the in itinere test allows the student to be exempted from completing the questions of section A in the final exam (thus increasing the time available to him in the sessions of the current Academic Year)

4. Those who have not passed the in itinere test can also access the final exam, but in this case they will have to complete both the questions of section A and the questions of section B of the final exam.

5. The benefits of passing the in itinere test remain valid until the end of the third exam session of the current Academic Year.

Lectures in classroom.

As a rule, marks will be assigned according to the following scheme

- not approved: the student has not acquired the basic concepts and is not able to carry out the exercises.

- 18-23: the student demonstrates minimal mastery of the basic concepts, his skills in exposition and connection of contents are modest, she/he is able to solve simple exercises

- 24-27: the student demonstrates good mastery of the course contents, her/his presentation and content connection skills are good, she/he solves the exercises with few errors

- 28-30 cum laude: the student has acquired all the contents of the course and is able to explain them fully and connect them with a critical spirit; she/he solves the exercises completely and without errors.

Verification of learning can also be carried out electronically, should the conditions require it. In this case, the duration of the written test may be subject to change.

**PLEASE NOTE: Information for students with disabilities and / or SLI**

**To guarantee equal opportunities and in compliance with the laws in force, the interested students can ask for a personal interview so to program any compensatory and / or dispensative measures, according to the didactic objectives and specific needs.**

**It is also possible to contact the referent of CInAP (Centro per l’integrazione Attiva e Partecipata - Servizi per le Disabilità e/o i DSA) of the Department.**

### Examples of frequently asked questions and / or exercises

Differential forms (knowledge and understanding, applying knowledge and understanding)

Relation between existence of the gradient and differentiability for a function of two variables (knowledge and understanding, applying knowledge and understanding).

Conditional extremes of a function (knowledge and understanding, applying knowledge and understanding).