27213 - Mathematical Analysis 2

Learning outcomes

At the end of the course the student will master the basic results as well as the basic tools of advanced calculus. He will master the notions of differentiability and integrability 'for functions of several real variables. He will be able to apply this knowledge to the solution of problems posed by the pure and applied sciences. He can solve practical problems of optimization and measure. He can formalize autonomously elementary problems raised by applied sciences.

Course contents

Function sequences and function series. Uniform convergence. Power series. Taylor Series. Differential calculus for functions of several real variables. Mean value theorem. Taylor formula. Convex functions. Local maxima and minima. Local invertibility, implicit function theorem. Lagrange multipliers theorem. Fixed point theorem for contraction. Ordinary differential equations and systems. Cauchy problem: existence of local solutions. Extension of solutions. Some special ode's. Linear ode's and linear systems of ode's. Elements of measure theory and Lebesgue integral in R^N. Vector fields and their potentials

Readings/Bibliography

Ermanno Lanconelli, Analisi Matematica 2 (prima e seconda parte), Ed. Pitagora.

Alternatively, the student can use:

Mariano Giaquinta, Giuseppe Modica, Analisi Matematica 3,4,5, Ed. Pitagora.  

Enrico Giusti, Analisi Matematica  2, Ed. Boringhieri.

R.Beals, Analysis: An introduction, ed. Cambridge University Press.

The student can also use any good text of Mathematical Analysis that contains the arguments to the program, since it is a standard program. Students are urged to check in advance with the teacher the appropriateness of the chosen text.

Teaching methods

Lectures and exercices given by the teacher.

Assessment methods

Students can choose between two different methods of exam:  1) A final exam on the whole program after the end of the lectures or later (in the dates decided by the CCdL); 2) A partial exam in January/February on the first part of the program, followed by a second exam as in 1) on the second part of the program before Sept. 30, 2016. Both exams will consist of a written part (4/5 standard exercises -  4 hours in the case 1), 2 and 1/2 h for both parts in case 2) that will receive 4 possible rates: poor, almost fair, fair, good, very good. If the rate is "poor" the student must repeat the written exam. Otherwise, he can proceed to the second part of the exam, the oral exam, within 6 months. During the written exam the student can use books or notes. Electronic devices of any kind are forbidden.

Office hours

See the website of Bruno Franchi

See the website of Giovanni Dore