## Studying at the University of Verona

Here you can find information on the organisational aspects of the Programme, lecture timetables, learning activities and useful contact details for your time at the University, from enrolment to graduation.

## Academic calendar

The academic calendar shows the deadlines and scheduled events that are relevant to students, teaching and technical-administrative staff of the University. Public holidays and University closures are also indicated. The academic year normally begins on 1 October each year and ends on 30 September of the following year.

## Course calendar

The Academic Calendar sets out the degree programme lecture and exam timetables, as well as the relevant university closure dates..

Period | From | To |
---|---|---|

I semestre | Oct 1, 2018 | Jan 31, 2019 |

II semestre | Mar 4, 2019 | Jun 14, 2019 |

Session | From | To |
---|---|---|

Sessione invernale d'esame | Feb 1, 2019 | Feb 28, 2019 |

Sessione estiva d'esame | Jun 17, 2019 | Jul 31, 2019 |

Sessione autunnale d'esame | Sep 2, 2019 | Sep 30, 2019 |

Session | From | To |
---|---|---|

Sessione di laurea estiva | Jul 22, 2019 | Jul 22, 2019 |

Sessione di laurea autunnale | Oct 15, 2019 | Oct 15, 2019 |

Sessione di laurea autunnale straordinaria | Nov 21, 2019 | Nov 21, 2019 |

Sessione di laurea invernale | Mar 19, 2020 | Mar 19, 2020 |

Period | From | To |
---|---|---|

Sospensione attività didattica | Nov 2, 2018 | Nov 3, 2018 |

Vacanze di Natale | Dec 24, 2018 | Jan 6, 2019 |

Vacanze di Pasqua | Apr 19, 2019 | Apr 28, 2019 |

Vacanze estive | Aug 5, 2019 | Aug 18, 2019 |

## Exam calendar

Exam dates and rounds are managed by the relevant Science and Engineering Teaching and Student Services Unit.

To view all the exam sessions available, please use the Exam dashboard on ESSE3.

If you forgot your login details or have problems logging in, please contact the relevant IT HelpDesk, or check the login details recovery web page.

Should you have any doubts or questions, please check the Enrolment FAQs

## Academic staff

## Study Plan

The Study Plan includes all modules, teaching and learning activities that each student will need to undertake during their time at the University. Please select your Study Plan based on your enrolment year.

Modules | Credits | TAF | SSD |
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Modules | Credits | TAF | SSD |
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Modules | Credits | TAF | SSD |
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1° Year

Modules | Credits | TAF | SSD |
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2° Year

Modules | Credits | TAF | SSD |
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3° Year

Modules | Credits | TAF | SSD |
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Modules | Credits | TAF | SSD |
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#### Legend | Type of training activity (TTA)

TAF (Type of Educational Activity) All courses and activities are classified into different types of educational activities, indicated by a letter.

### Mathematical analysis 2 (2019/2020)

Teaching code

4S00031

Academic staff

Coordinatore

Credits

12

Language

Italian

Scientific Disciplinary Sector (SSD)

MAT/05 - MATHEMATICAL ANALYSIS

Period

I semestre dal Oct 1, 2019 al Jan 31, 2020.

## Learning outcomes

Topics treated in this course are: Calculus for functions of several variables, sequences and series of functions, ordinary differential equations, Lebesgue measure and integral. Emphasis will be given to examples and applications. At the end of the course, students must possess adequate skills of synthesis and abstraction. They must recognize and produce rigorous proofs. They must be able to formalize and solve moderately difficult problems on the arguments of the course.

## Program

i) Calculus in several variables. Neighborhoods in several variables, continuity in several variables, directional derivatives, differential of functions in several variables, Theorem of Total Differential, gradient of scalar functions, Jacobian matrix for vector-valued functions, level curves of scalar functions. Parametrized surfaces, tangent and normal vectors, changes of coordinates. Higher order derivatives and differentials, Hessian matrix, Schwarz's Theorem, Taylor's Series.

(ii) Optimization problems for functions in several variables. Critical points, free optimization, constrained optimization, Lagrange's Multiplier Theorem, Implicit and inverse function theorem, Contraction Principle.

(iii) Integral of functions in several variables. Fubini and Tonelli theorems, integral on curves, change of variables formula.

(iv) Integral of scalar function on surfaces, vector fields, conservatice vector fields, scalar potentials, curl and divergence of a vector fields, introduction to differential forms, closed and exact forms, Poincare lemma, Gauss-Green formulas.

(v) Flux through surfaces, Stokes' Theorem, Divergence Theorem

(vi) Introduction to metric spaces and normed spaces, spaces of functions, sequence of functions, uniform convergence, function series, total convergence, derivation and integration of a series of functions.

(vii) Introduction to Lebesgue's Measure Theory. Measurable sets and functions, stability of measurable functions, simple functions, approximation results, Lebesgue integral. Monotone Convergence Theorem, Fatou's Lemma, Dominated convergence Theorem and their consequences.

(viii) Ordinary differential equation, existence and uniqueness results, Cauchy-Lipschitz's Theorem. Extension of a solution, maximal solution, existence and uniqueness results for systems of ODE, linear ODE of order n, Variation of the constants method,

other resolutive formulas.

(ix) Fourier's series for periodic functions, convergence results, application to solutions of some PDE.

Author | Title | Publishing house | Year | ISBN | Notes |
---|---|---|---|---|---|

Giuseppe De Marco | Analisi 2. Secondo corso di analisi matematica per l'università | Lampi di Stampa (Decibel Zanichelli) | 1999 | 8848800378 | |

V. Barutello, M. Conti, D.L. Ferrario, S. Terracini, G. Verzini | Analisi matematica. Dal calcolo all'analisi Vol. 2 | Apogeo | 2007 | 88-503-242 | |

Adams, R. | Calcolo differenziale (vol. 2). Funzioni di più variabili. | Ambrosiana | 2003 | 8840812687 |

## Examination Methods

The final exam consists of a written test followed, in case of a positive result, by an oral test. The written test consists of some exercises on the program: students are exonerated from the first part of the test if they pass a mid-term test at the beginning of december. The written test evaluates the ability of students at solving problems pertaining to the syllabus of the course, and also their skills in the analysis, synthesis and abstraction of questions stated either in the natural language or in the specific language of mathematics. The written test is graded on a scale from 0 to 30 points (best), with a pass mark of 18/30..

The oral test will concentrate mainly but not exclusively on the theory. It aims at verifying the ability of students at constructing correct and rigorous proofs and their skills in analysis, synthesis and abstraction. The oral exam is graded on a scale from -5 to +5 point, which are added to the marks earned in the written test.

## Bibliography

## Type D and Type F activities

**Modules not yet included**

## Career prospects

## Module/Programme news

##### News for students

There you will find information, resources and services useful during your time at the University (Student’s exam record, your study plan on ESSE3, Distance Learning courses, university email account, office forms, administrative procedures, etc.). You can log into MyUnivr with your GIA login details.

## Further services

I servizi e le attività di orientamento sono pensati per fornire alle future matricole gli strumenti e le informazioni che consentano loro di compiere una scelta consapevole del corso di studi universitario.

## Graduation

## Attachments

Title | Info File |
---|---|

1. Come scrivere una tesi | 31 KB, 29/07/21 |

2. How to write a thesis | 31 KB, 29/07/21 |

4. Regolamento tesi (valido da luglio 2020) | 259 KB, 29/07/21 |

5. Regolamento tesi (valido da luglio 2022) | 171 KB, 17/02/22 |

## List of theses and work experience proposals

theses proposals | Research area |
---|---|

Formule di rappresentazione per gradienti generalizzati | Mathematics - Analysis |

Formule di rappresentazione per gradienti generalizzati | Mathematics - Mathematics |

Mathematics Bachelor and Master thesis titles | Various topics |

Stage | Research area |
---|---|

Internship proposals for students in mathematics | Various topics |

## Attendance

As stated in point 25 of the Teaching Regulations for the A.Y. 2021/2022, except for specific practical or lab activities, attendance is not mandatory. Regarding these activities, please see the web page of each module for information on the number of hours that must be attended on-site.Please refer to the Crisis Unit's latest updates for the mode of teaching.