# MATHEMATICAL ANALYSIS I

**Academic Year 2017/2018**- 1° Year

**Teaching Staff**

- MODULO 1:
**Ornella NASELLI** - MODULO 2:
**Raffaela Giovanna CILIA**

**Credit Value:**18

**Scientific field:**MAT/05 - Mathematical analysis

**Taught classes:**105 hours

**Exercise:**36 hours

**Term / Semester:**1° and 2°

## Learning Objectives

**MODULO 1**The student will learn the principal concepts and technics of Mathematical Analysis.

**MODULO 2**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 differential calculus for functions of one variable and its important applications, the theory of integration of a single variable function, the solving methods of ordinary differential equations, the theory of sequences and series of functions.

In particular, the course aims to:**Knowledge and understanding**: the student will familiarize himself with the differential calcul of the functions of one real variable and will know many of its important applications. He will learn the integration methods and the theory of Rienmann's integration linked to Peano's theory of measure. It will be able to move on to the concept of sequences and series of functions . He will learn the solving methods of differential equations that he willfind in other disciplines. The theory of differential equations will be postponed to the second year. Some studies will be entrusted to the most willing students who, either alone or in groups, will be able to present them in short seminars.

**Ability to apply knowledge and understanding:**the student will not only learn the individual concepts but will succeed in linking them and will be led, in particular, to reflect on the structure properties (eg topological) that are the basis of the various topics studied.They can also exercise the ability to use the acquired knowledgement in other settings: for example, the student will be invited to independently demonstrate results similar to those studied, and to carry out numerous exercises of application of the theories studied. This will be donethrough exercises - both manipulative and demonstrative - that will be proposed for individual study.

**Making judgements**: Students are invited to study some contents individually and to improve own knowledge also comparing different books. They can also verify the solutions of the problems that they have studied with the other students during tutoring hours to find the right solutions.

**Communication skills:**By listening to the lessons and reading the recommended books, the student will continue to improve in the use of mathematical language that will be more and more suitable to express correctly and elegantly mathematically sophisticated concepts.

**Learning Skills:**Students will be guided to acquire a method that will allow them to approach a new topic, recognizing immediately what are the prerequisites needed. It will also develop the skills of computing and manipulating the mathematical objects studied.

## Detailed Course Content

**MODULO 1**1. Real and complex numbers

2. Sequences and series

3. Continuous functions

**MODULO 2**MODULO 2

Differential Calcul for Functions of one variable

Integral calcul.

Improper integration.

Sequences and series of Functions

Solving methods of differential equations

## Textbook Information

**MODULO 1**1. G. Emmanuele, Analisi Matematica 1, Pitagora (nuova edizione)

2. C.D. Pagani, S. Salsa, Analisi Matematica 1, Zanichelli

3. J.P. Cecconi, G. Stampacchia, Analisi Matematica vol. 1, Liguori

4. G. De Marco, Analisi uno, Zanichelli

**MODULO 2**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