# 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/She will master the notions of differentiability and integrability 'for functions of several real variables. He/She will be able to apply this knowledge to the solution of problems posed by the pure and applied sciences. He/She can solve practical problems of optimization and measure. He/She can formalize autonomously elementary problems raised by applied sciences.

## Course contents

Topology of metric spaces: connectedness and compactness. Differential calculus for functions of several real variables. Mean value theorem. Taylor formula. Convex functions. Local maxima and minima. Local invertibility, implicit function theorem. Elements of differentiable manifolds. Lagrange multipliers theorem. Ordinary differential equations (ODEs) and systems of ODEs. Cauchy problem: existence of local solutions. Extension of solutions. Linear ODEs and linear systems of ODEs. Elements of measure theory and Lebesgue integral in R^N. Line integrals, vector fields and differential forms, and their potentials.

Enrico Giusti: Analisi Matematica 2, Ed. Boringhieri

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.

Carlo Domenico Pagani, Sandro Salsa, Analisi Matematica 2, Ed. Zanichelli 2015

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

As regards the construction of the Lebesue integral, the student may also use the book

Walter Rudin: Principles of Mathematical Analysis, 3rd Edition, MAc Graw-Hill 2015

and as regards ODEs the book

Cesare Parenti, Alberto Parmeggiani: Algebra Lineare ed Equazioni Differenziali Ordinarie, Unitext 48, Springer 2010.

Students can also use any good text of Mathematical Analysis that contains the arguments of the program (the program being standard). Students will check with the professors the validity of the chosen alternative textbook depending on the program.

## Teaching methods

Lectures and exercises given by the teacher in the classroom using the blackboard.

## Assessment methods

The final exam consists of a written exam and an oral exam. The written exam can be taken according to the following two possibilities:

1) a partial exam related to the program of the first semester during the winter session 2018/19 (2 dates, duration of the exam: 2 hours), which is followed by a partial exam related to the program of the second semester during the summer session (3 dates, duration of the exam: 2 hours);

2) a final exam related to the whole program (first and second semester) during the summer session, the fall session and the winter session after the end of the course (3+1+2 dates, duration of each written exam: 3 hours).

In either case, the student will receive 4 possible rates: poor, almost fair, fair, good, excellent. If the rate is "poor" the student must repeat the written exam. Otherwise, he/she can proceed to the second part of the exam, the oral exam, within the summer session in case 1), within the session during which the written exam was taken in case 2). During the written exam the student is not allowed to use books or notes. Electronic devices of any kind are not allowed.

## Office hours

See the website of Alberto Parmeggiani

See the website of Bruno Franchi