MATHEMATICAL ANALYSIS II

Student will acquire the basic notions of differential calculus and integral calculus for the real functions of several real variables as well as the ability to apply them to solving problems arising from other sciences such as Physics and Economics.

In particular, the course has the following goals

Knowledge and understanding: the student will first see how concepts and results known from the course of Mathematical Analysis I can be extended, with appropriate modifications when needed, to more general and abstract situations; In this way we will try to develop the abstraction abilities of the learner. Then we will try to apply the definitions, the results and the techniques thus obtained to particular cases, so as to illustrate how the general case can be passed to the particular case, demonstrating that the abstractions made are not merely a theoretical exercise but have always significant practical implications that can also solve problems apparently "far and different". This will try to stimulate the learner to develop capacity of abstraction as well as capacity of critical synthesis.

Ability to apply knowledge and understanding: the student will not only learn the individual concepts but will be led to reflect on the notions considered so as to isolate the peculiar aspects of a problem also in view of application to other issues that are analogous to the problem under consideration. We will try to accustom the learner to build mathematical models of various concrete situations and to apply the notions studied for their analytical study. Learner will be stimulated to also utilize his knowledge in situations other than the original one: for example, he will be invited to independently demonstrate results similar to those studied and to carry out numerous exercises of application of the theories studied.

Making judgments: Student will be able to study non-lectured topics to get used to deepen his own knowledge and will be invited to look for further applications of the topics. He can also critically confront with other learners during tutoring hours to find the most appropriate solutions.

Communication skills: By attending lessons and reading the text book and other books possibly indicated by the teacher, the student will familiarize himself with mathematical language. Through guided exercises and seminars, he will learn to communicate his knowledge clearly and rigorously. He will learn that using a correct language is one of the most important means of communicating science.

Learning skills: Student will be guided to perfect the correct method of study that he should have learned in the first year courses. This will allow him to approach a new topic, recognizing immediately what the necessary prerequisites are. It will also continue to develop the skills of computing and manipulating mathematical objects studied.

Concepts and techniques of the Approximation Theory of Functions by power series and Fourier series will be studied. Metric spaces will be introduced and studied deeply, sometimes generalizing notions (limit of sequences and functions, continuity and uniform continuity of functions) and results already met in the Mathematical Analysis 1 course. Main notions of both differential and integral calculus will be extended from one variable functions to multivariable ones. Systems of ordinary differential equations, differential geometry of curves and surfaces and the main results of vector calculus, interesting in themselves and extremely useful in applications to other sciences, will be studied in quite large generality

G. Emmanuele, Analisi Matematica 2, Foxwell and Davies Italia 2004 (ask teacher how to get the book)

It is possible to consult the teacher's web site at http://www.dmi.unict.it/~emmanuele/ for an extensive list of texts of exercises and official tests of previous academic years