27213 - Mathematical Analysis 2

Learning outcomes

At the end of the course, any student will possess the knowledge of advanced Mathematical Analysis, as a useful and creative central science. He/she will know: the main theorems pertaining differentiability and integrability for the functions of several real variables, and of poitwise and uniform convergence of series of functions. He/she can apply this knowledge to the solution of problems typically appearing in pure and applied sciences. He/she can solve optimization and measure problems, and further has autonomy of judgment in reference to the mathematical formalization of simple problems of the applied sciences.

Course contents

Elements of metric spaces and their topology. Compactness. Contractions.

Differential calculus for functions of several real variables. Taylor's formula. Local maxima and minima. Local invertibility and implicit functions. Constrained extrema.

Local existence and prolongability of the solutions of the Cauchy problems for ordinary differential equations; solution methods for equations of a particular type. Linear equations and systems: general integral, resolution of equations and systems with constant coefficients.

Sequences and series of functions: pointwise and uniform convergence. Power series. Convergence criteria.

Path integrals. Vector fields, potentials.

Elements of measure theory and Lebesgue's integration in R^n. Passage to the limit under the integral sign, reduction and change of variable theorems.

Readings/Bibliography

Notes by the teachers will be available on the related web-sites.

In order to get an in-depth study of the topics of the course, students may want to consult:

E. Lanconelli: Lezioni di Analisi Matematica 2, prima parte, ed. Pitagora

N. Fusco, P. Marcellini, C. Sbordone: Analisi Matematica due, ed. Liguori

G.C. Barozzi, G. Dore, E. Obrecht: Elementi di Analisi Matematica, vol. 2, ed. Zanichelli

Exercise Textbooks:

M. Bramanti, Esercitazioni di Analisi Matematica 2, ed. Esculapio

P. Marcellini, C. Sbordone: Esercitazioni di Analisi Matematica due, parte I e parte II, ed. Zanichelli.

Teaching methods

Lectures and exercises in the classroom

Assessment methods

The examination consists of both a written and an oral exam.

The written exam consists of several exercises related to the arguments of the course. In order to participate the written test, any student must register at least three days before the test through AlmaEsami.

The written test remains valid for the oral exam in the same examination period.

The oral test follows the written test; it mainly concerns the theoretical aspects of the course. Any student must exhibit his/her knowledge of the concepts provided during the course (in particular: definitions, theorems and their proofs) and how to properly connect them.

In the period between the first and second cycle, students can take a partial examination on the topics of the first module of the course. The rules of partial examination are the same as for the total examination.

The students who have passed the examination relative to the first module can take the examination relative to the second module within July. After this date the first partial examination is canceled.

Teaching tools

Online pdf's and possibly some videos will be provided.

Office hours

See the website of Annalisa Baldi

See the website of Andrea Bonfiglioli