# MATHEMATICAL ANALYSIS II

**Academic Year 2016/2017**- 2° Year

**Teaching Staff:**

**Raffaela Giovanna CILIA**

**Credit Value:**15

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

**Taught classes:**120 hours

**Term / Semester:**2°

## Learning Objectives

The student will continue the acquisition of Mathematical concepts. The themes will be linked to concepts learned in other disciplines. In particular, the course has the following objectives:

Knowledge and understanding (knowledge and understanding): the student will be introduced in the study of metric spaces, normed spaces and Hilbert spaces. You will see how the results known from the course Analysis I are part of a much wider and more general context. The rigorous abstraction capacity and in the same time critical synthesis will be developed. The student will become familiar with differential calculus of several variables vector functions, with the study of differential equations of which already knows the solution methods but will be introduced to fundamental issues such as the theorems of existence, uniqueness of solutions. Expand the knowledge related to the measurability of sets in the plane and in space by studying the measure theory and Lebesgue Peano. So students will be introduced to the integral calculus of several variables and functions to the different applications. Some insights will be entrusted to the most willing students who, alone or in groups, will present them in short seminars.

Applying knowledge and understanding (applying knowledge and understanding): the student will not only learn the individual concepts but fail to connect them and will be conducted, in particular, to reflect on the structural properties (eg topology) that form the basis of various topics studied. It will be used to reflect on an issue and build a mathematical model to be studied analytically. Students can also exercise their ability to use their knowledge in situations other than those in which they were submitted, for example they will be invited to demonstrate independently similar results to those studied, and to perform many exercises of application of the studied theorems. This will be done through guided exercises in the classroom and through exercises - both manipulative that demonstration - that will be proposed for self-study.

Making judgments (Making judgments): students can study the topics of the lectures not to get used to deepen their own knowledge and will be encouraged to search for additional applications of the arguments. also he is able to critically engage with the other students during tutoring hours to identify the most appropriate solutions.

Communication skills (communication skills): by listening to lectures and reading books recommended, the student will be helped to structure so more complex mathematical thinking and to better treat the mathematical language which was already introduced in the first year of course. Through guided exercises and seminars, learn how to communicate clearly and rigorous both orally and in writing. You will learn to use a proper language is one of the most important means to acquire the mentality mathematics.

Learning skills (learning skills): the student will be led to acquire a study method that allows him to turn to a new topic now recognizing what are the necessary prerequisites. Develop, in addition, the computing capacity and handling of the studied mathematical objects.

## Detailed Course Content

*Metric spaces. Normed Banach and Hilbert spaces. Sequences and series of functions.. Didfferential calculus for functions of several variables. Implicit functions. Measure theory Integrals of functions of several variables. Curves. Integral of differential forms.Surfaces and surface Integrals. Systems of differential equations.*

## Textbook Information

Giovanni Emmanuele Analisi Matematica II Foxwell and Davies 2004