# MATHEMATICAL ANALYSIS II

**Academic Year 2022/2023**- Teacher:

**Giuseppa Rita CIRMI**

## Expected Learning Outcomes

The course aims to provide the basic knowledge of differential and integral calculus for several variables functions and the ODE's theory. Students will be able to apply the mathematical tools to some problems arysing from Physics.

In particular the course objectives are:

**Knowledge and understanding**: students will learn the differential calculus for functions of more real variables and its applications to optimization problems, the theory and the resolutive methods for some ordinary differential equations, the integral calculus on domains, curves and surfaces.

**Applying knowledge and understanding**: by means of examples related to applied sciences, students will focus on the central role of Mathematics within science and not only as an abstract topic. Furthermore, they will be able to apply the mathematical tools to some problems arysing from Physics.

**Making judgements**: students will be stimulated, individually or in groups, to work on specific topics they have not studied during the class, developing exercises related on the field knowledge with greater independence. Seminars and lectures are scheduled to give students the chance to illustrate guided exercise on specific topics in order to share them with the other students and to find together the right solutions.

**Communication skills**: studying Mathematics and dedicating time to guided exercise and seminars, students will learn to communicate with clarity and rigour both, in the oral and written analysis. Moreover, students will learn that using a properly structured language means to find the key to a clear scientific and non-scientific communication.

**Learning skills**: students, in particular the more willing one, will be stimulated to examine in depth some topics, thanks to individual activities or working in group.

## Course Structure

The principal concepts and learning outcomes will be structured by planning frontal lectures ( 10 CFU) and exercise (2 CFU). Furthermore, to improve the making judgements and communication skills, students will dedicate time to guided exercises and they can work in groups or individually.

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 programme planned and outlined in the syllabus.

Learning assessment may also be carried out on line, should the conditions require it.

## Required Prerequisites

Knowledge of the main topics of Mathematical Analysis is essential.

Knowledge of the main topics of Geometry is useful.

## Attendance of Lessons

Attendance of the lessons is mandatory (see "Regolamento Didattico del Corso di Studi")## Detailed Course Content

1. Sequences and series of functions. Power series. Expansion of given functions in power series.

2. Metric spaces.

3. Functions of several variables: limits, continuity, derivation and differentiability. Local and global extrema.

4.Curves and surfaces.

5.Implicit functions.

6. Multiple integrals.

7.Differential forms. Conservative vector fields.

8. Ordinary differential equations.

## Textbook Information

- N.Fusco, P.Marcellini, C.Sbordone, Analisi Matematica 2, Zanichelli
- C.Pagani, S.Salsa, Analisi Matematica 2, Zanichelli
- M.Bramanti, Esercitazioni di Analisi Matematica 2, Esculapio
- P.Marcellini, C.Sbordone, Esercitazioni di Matematica, Vol.2, Parte I e II, Zanichelli

## Course Planning

Subjects | Text References | |

1 | Ordinary differential equations (12 hours) | |

2 | Metric spaces (6 ore) | |

3 | Real functions of several variables (14 hours) | |

4 | ODE: theory (2 hours) | |

5 | Implicit functions8 hours) | |

6 | Curves, integration on curves (10 hours) | |

7 | Vrctor fields (12 hourse) | |

8 | Multiple integrals | |

9 | Integration on surfaces | |

10 | Sequences and Series of functions (14 ore) |

## Learning Assessment Procedures

The** final exam** consists in a written test and an oral interview. Students who pass the written test with a grade of at least 18/30 will be allowed to take the oral interview.

On February/March 2023 there will be an** intermediate exam**, consisting in a written test, dealing with the topics already covered.

The written test is divided in two parts

A) Theoretical questions

B) Technical exercises.

Students who take the grade of at least 18/30 in each of the parts A) and B) will be allowed to do the final exam studying only the remaining topics, within June / July 2023.

## Examples of frequently asked questions and / or exercises

Examples of questions are listed below.- Relationship between punctual, uniformly and total convergence.
- Power series
- Expansions in series
- Relationship between continuity and differentiability
- Local extrema
- Curves
- Conservative and irrotational vector fields
- Linear differential equations

Examples of exercises are listed below.

- Find local and global extrema of a real function of a several variables.
- Solve an ODE
- Calculate a multiple integral
- Calculate the flow of a vector field
- Calculate the work of a vector field
- Study a function series

The above lists aren't exhaustive.