The purpose of the course is to provide the students with the necessary background of differential and integral calculus for functions of several variables, and of differential equations. A further aim is to apply analytical techniques in other scientific disciplines.
Learning outcomes (declined compared with respect to the Dublin descriptors)
Knowledge and understanding. Knowledge of the differential and integral calculus for functions of several variables. The student will be able to state and prove basic theorems of Mathematical Analysis.
Applying knowledge and understanding. The ability to understand the problems proposed during the course, the ability concerning a correct application of the theoretical knowledge. The student will be able to study functions, to solve integration problems, to solve differential equations of first and second order, to discuss the behavior of series of functions.
Making judgments. Develop the ability to critically evaluate the problems and propose the most appropriate approach
Communication skills. Ability to report and present the results with a logical-deductive and synthetic exposition. He must be able to explain (even to non-expert people) the power of some applications of the mathematical tools described in the course, in the field of Engineering.
Ability to learn.
Ability to develop, outline, summarize the contents from several sources, in order to achieve a broad overview of the problems connected to the topics discussed in the course. The student will also develop the skill of learning more advances techniques of Mathematical Analysis.
It is necessary to acquire and assimilate the following knowledge provided by the course "Mathematical I": sequences of real numbers, differential and integral calculus for functions of one variable, Ordinary differential equation, Linear Systems
Differential Calculus for Functions of several real variables (20 Hours): Topology in R2; continuous functions; partial derivatives; differentiability and relative theorem; directional derivatives and gradient; higher order derivatives and Schwarz's theorem; Maxima and minima.
Line Integral and vector fields in 2-D and 3-D (16 hours) Regular curves; length of curve; line integral of a function; vector fields in 2-D and 3-D and its line integral; potential and conservative field; irrotational field; integrability criteria.
Double and triple integrals (12 hours) Double integrals on normal domains; integrability of continuous functions; reduction formulas ; change of variables ; Gauss-Green formulas, divergence theorem, Stokes formula; triple integrals.
Regular surfaces (10 hours) Regular surfaces, Rotation surfaces. Area of a surface. The divergence theorem and the Stokes formula.
Power Series (14) Power Series in Real Field and Complex Field. Convergence set. Taylor's series .
Differential calculus for functions of several real variables
Line Integral and differential forms. Double and triple integrals.
Regular surfaces Series
Lectures, exercise sessions
Robert A. Adams, Christopher Essex, Calculus: A Complete Course, Pearson Canada.
The objective of the exam is to check the level of achievement of the above-mentioned training objectives. In particular, the students attending the courses will be evaluated on a 3 hours written exam, containing 5 exercises and theoretical questions. The student will pass the exam of he/she answers correctly at least 3 exercises and 2 theoretical questions. In case of a positive evaluation, the student will be given the chance either to accept the final score or to integrate it with an oral exam, based on the theoretical aspects of the course, focusing basically on the proofs of the main results shown during the course. In the latter case, the final score will be as a result of the average of the two tests. In any case, the student will have a positive evaluation if he shows to have a sufficient mastery in the classical topics of multivariable calculus, power series, differential equations and two dimensional manifolds.
Lectures are in Italian. The professor is fluent in English and is available to interact with students in English, also during the examination.
Time table: Monday from 11.00 to 13.00. I am also available on other days by appointment by e-mail.