MATHEMATICAL ANALYSIS II - channel 1

The 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 solve problems arising from other sciences. The student will first see how the concepts and results already known in the course of Mathematical Analysis I can be extended, with appropriate modifications when necessary, to more general and abstract situations; In this way we will try to develop the abstraction abilities of the learner. Then it will be shown how 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 extremely useful to also solve problems apparently "far" from the original one. We will thus seek to stimulate the learner to develop rigorous abstraction abilities and at the same time critical synthesis. 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 the 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 apply the notions studied for his analytical study. By listening to lessons and reading the textbook (and possibly other books as indicated by the teacher) students will familiarize themselves with mathematical language by learning how to use a correct language is one of the most important means of communicating science. The student will be guided to improve and make more precise the correct method of study that he should have learned in the first year courses. This will allow him to approach a new topic even independently. He 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