27213 - Mathematical Analysis 2

Learning outcomes

At the end of the course, the student has the knowledge of advanced mathematical analysis, identifying it as a useful and creative central science. He has the knowledge of differentiability and integrability for functions of several real variables and of punctual and uniform convergence of series of functions. He knows how to apply this knowledge to the solution of problems posed by pure and applied sciences. He knows how to solve practical optimization and measurement problems. He has autonomy of judgment about the mathematical formalization of simple problems of applied sciences.

Course contents

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

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 variables theorems.

Sequences and series of functions: punctual and uniform convergence, convergence criteria. Power series, Taylor series.

Local existence and extendability of the solutions of 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.

Readings/Bibliography

Notes of the teachers will be available on Virtuale [http://virtuale.unibo.it/] .

To study in depth the topics of the course, students can consult:

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

N. Fusco, P. Marcellini, C. Sbordone: Lezioni di Analisi Matematica due, ed. Zanichelli

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

Textbooks about exercises:

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

Mixed modality lessons.

Assessment methods

The examination consists of a preliminary written test and an oral one.

The written test consists of six exercises related to the arguments of the course. In order to participate to the written test the student must register at least three days before the test through AlmaEsami [https://almaesami.unibo.it/] .

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. The student must show to know the concepts explained during the course (in particular definitions, theorems and their proofs) and how to connect them.

In the period between the first and second cycle the student 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 student 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.

Office hours

See the website of Giovanni Dore

See the website of Giovanni Cupini