# MATHEMATICAL ANALYSIS I

**Academic Year 2020/2021**- 1° Year

**Teaching Staff**

- MODULO 1:
**Raffaela Giovanna CILIA** - MODULO 2:
**Giuseppe DI FAZIO**

**Credit Value:**18

**Taught classes:**105 hours

**Exercise:**36 hours

**Term / Semester:**1° and 2°

## Learning Objectives

**MODULO 1**The student will study the principal concepts and technics of Mathematical Analysis. In particular, the course

**objectives**are:**Knowledge and understanding**: the student will acquire the main concepts of Mathematical Analysis and will be guided to link them to the concepts learned in Algebra, in General Topology or the concepts that he will study in Physics. He will learn the main techniques of Mathematical Analysis. In particular, he will study the concepts of inf and sup of a number set, of sequences and series of real numbers. He will be introduced into the study of infinitesimal calculus for a real function of one variable. Some tasks will be entrusted to the most willing students who, thanks to group work or individual work, will be able to present them in short seminars.

**Ability to apply knowledge and understanding****:**the students will be led not only to learn pure concepts, but also to link them and to reflect on the structure properties (e.g. topological) that are the basis of the various topics studied. They can also improve their ability to use the acquired knowledge in other settings: for instance, the students will be invited to prove results that are similar to those studied, and to carry out numerous exercises of application of the theories they have studied. This aspect**will be assessed through**exercises - both manipulative and demonstrative - that will be suggested for their individual study.

**Making judgements**: The students are invited to study some contents individually and to improve their own knowledge comparing different books.

**Communication skills:**By listening to the lessons and reading the recommended books, the students**will improve**their ability to use mathematical language, to express properly and elegantly sophisticated mathematically math concepts.

**Learning skills:**Students will be guided to acquire a method that will allow them to approach a new topic, recognizing immediately the prerequisites needed.**The course aims also at developing students' skills**of computing and manipulating the mathematical objects studied**MODULO 2**Primary goal of the course is to master the techniques and methods that are peculiar to Mathematical Analysis.

## Course Structure

**MODULO 1**The course is organized by lectures. There will be some team practices.

Periodical meetings will be organized in teamwork with the professor of Algebra and Geometry, in order to give students useful suggestions on how to use and apply the acquired knowledge to the other disciplines.

If necessary, the telematic way will be adopted. 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 online, should the conditions require it.

**MODULO 2**Almost all lectures are given in the traditional way. This is because the student must acquire the rigorous methods of Mathematical Analysis. This is achieved showing examples and results. Most of them are then discussed in detail with the students.

**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.**

## Detailed Course Content

**MODULO 1**Set of real numbers

Set of complex numbers

Structure of number sets

Infinitesimal Calcul for Functions of one variable

Series of real numbers.

**MODULO 2**Differential Calculus. Integral Calculus. Ordinary differential equations.

## Textbook Information

**MODULO 1**1) G. Emmanuele Analisi Matematica I Pitagora

2) G. Emmanuele Analisi Matematica II Foxwell Davies

Other books that the student is invited to use

3) Pagani Salsa Analisi Matematica 1, Zanichelli

4) Cecconi Stampacchia, Analisi Matematica , Liguori

5) DeMarco Analisi uno, Zanichelli

**MODULO 2**- M.Giaquinta - G. Modica
*Mathematical Analysis: Functions of one variable*Birkhäuser (2013) - W. Rudin -
*Principles of Mathematical Analysis*3 ed Mc Graw Hill

- M.Giaquinta - G. Modica