# MATHEMATICAL ANALYSIS 3

OVERVIEW

The lectures are delivered in Italian.

Some basic topics in Mathematical Analysis are covered, with the aim to complete the ones already covered in the previous courses of Mathematical Analysis I and 2.

## AIMS AND CONTENT

LEARNING OUTCOMES

Introduction to Lebesgue's Integration Theory and to integration along curves and surfaces.

AIMS AND LEARNING OUTCOMES

To continue the study of Classical Mathematical Analysis (implicit functions, curves, surfaces and 1-differential forms) and to introduce the study of Lebesgue's Integration Theory: these are fundamental instruments in Mathematical Analysis, essential to get a well grounded knowledge in the basic branches of Mathematics and for the understanding of simultaneous and next courses.

**Expected learning outcomes:**

The students will become acquainted with the concepts and proofs carried out in class and how they are used in practice to solve exercises; moreover they will know how to produce easy variants of demonstrations seen and construct examples on topics covered in this course.

PREREQUISITES

Mathematical Analysis I and 2, Linear Algebra and Analitic Geometry, Geometry 1.

Teaching methods

Both theory and exercises are presented by the teacher in the classroom on the blackboard. Moreover some tutorial exercitations will be carried out during the semester.

SYLLABUS/CONTENT

Limsup and liminf. Implicit functions, Dini theorem, local invertibility. Notion of sigma-algebra and measure. Lebesgue integral and theorems of convergence under sign of integral. Riesz extension of Riemann integral for continuous functions with compact support. Lebesgue measurable sets and their measure. Fubini theorem. Integrability criteria. Integrals depending by a parameter. Lusin theorem. Curves and surfaces; length and area; integration on curves and surfaces. Differential forms of degree 1; integration of 1-differential forms on oriented curves; closed and exact 1-differential forms.

RECOMMENDED READING/BIBLIOGRAPHY

W. Rudin - Real and Complex Analysis - McGraw-Hill 1970

## TEACHERS AND EXAM BOARD

**Ricevimento:** At the end of lectures or by appointment.

Exam Board

ADA ARUFFO (President)

EMANUELA SASSO (President)

LAURA BURLANDO

## LESSONS

Teaching methods

Both theory and exercises are presented by the teacher in the classroom on the blackboard. Moreover some tutorial exercitations will be carried out during the semester.

LESSONS START

The class will start according to the academic calendar.

## EXAMS

Exam description

Written and oral tests. Oral test, which students can access whatever the outcome of written test, has to be taken in the same exam appeal of written test.

Furthermore, during the year, there will be two partial written tests. The students that will pass them (that is, obtaining an average greater than or equal to 15/30), will be enabled to access directly the oral exam; this is valid for all the exam appeals for the academic year, hence until February included.

Assessment methods

The written examination consists in some exercises about the topics covered in this course. In this test the ability to apply theoretical results in concrete situations is evaluated.

The test lasts two hours and it is possible to consult the notes and textbooks; the same procedures apply to each of the two partial written tests.

In the oral exam a discussion of the written examination is done; moreover some questions are asked on the course content and/or about the solution of some exercises about the topics covered in this course. In such way they are assessed the understanding, the knowledge of the concepts, and the skills in using them, acquired by the students.

Exam schedule

Date | Time | Location | Type | Notes |
---|---|---|---|---|

22/06/2020 | 10:00 | GENOVA | Scritto | |

29/06/2020 | 09:00 | GENOVA | Orale | |

20/07/2020 | 10:00 | GENOVA | Scritto | |

23/07/2020 | 09:00 | GENOVA | Orale | |

07/09/2020 | 10:00 | GENOVA | Scritto | |

09/09/2020 | 09:00 | GENOVA | Orale |

## FURTHER INFORMATION

Attendance is recommended.

Students are suggested to enroll in AulaWeb, in order to be able to get further information by the teachers about the course.