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 (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 usual way. Moreover some tutorial exercitations will be carried out during the semester.

SYLLABUS/CONTENT

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

Exam Board

ADA ARUFFO (President)

LAURA BURLANDO

TOMMASO BRUNO (President Substitute)

EMANUELA SASSO (President Substitute)

LESSONS

LESSONS START

The class will start according to the academic calendar.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

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 session of written test.

If the University introduces again the obligation to perform even partially online exams (as happened in part of the academic years 2019-20, 2020-21 and 2021-22), the exam will consist only of oral test.

Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the teacher at the beginning of the course to agree on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools.

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.

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

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.