27213 - Mathematical Analysis 2

Learning outcomes

The student acquires mathematical knowledge of more informative character, using a wide basic instrumentation to face the description of different physical phenomena. Specifically, at the end of the course, the student is able to: solve the problems of maximum or minimum constraints; Calculate simple integral functions of more variables; Calculate simple integral functions defined on surfaces. In particular, at the end of the course, the student is able to: - solve problems of maximum or minimum bounds; - to study the convergence of integrals in more variables; - surface integral calculations.

Course contents

Metric spaces, nomea spaces, spaces with inner product. Complete metric spaces. Banach spaces and Hilbert spaces. Banach-Caccioppoli theorem. Uniform and total convergence. Differential equations: existence and uniqueness of the Cauchy (Peano-Picard Theorem). Differential calculus for function with several variables: limits, directional derivative, partial differential derivative, differential of a function, chain's rule, gradient of a function, Taylor's formula. Research free maximum and minimum: necessary condition and sufficient conditions. Infinitesimal calculus for curves: regular curves and integral of fist type. Differential calculus for function with several variables with vector values: limit, continuity and differentiability of a function, Jacobian matrix, Hessian matrix, Taylor's formula. Implicit function Theorem (Dini). Inverse function Theorem. k-dimensional manifolds.
Maximum and minimum on manifold, integral on curves, integral calculus in several variables. Tangent space and space at a point in a manifold. Lagrange Theorem, maximum and minimum with constrains. Integral calculus: multiple integral, simple sets, regular and measurable sets. Fubini Theorem. Theorem on change of variables. Basics about measure theory, basics on abstract theory, Lebesgue integral,
Fubini's theorem and Lebesgue Theorem, L^p spaces. Vector fields Vector fields: integral of second type. Vector fields, potentials and differential forms. Integral on surfaces.
Gauss-Green Theorems, Gauss and Stokes Theorems.

Readings/Bibliography

The lessons of the teacher will be available on

Insegnamenti on line (IOL), only in Italian.

Below a list of some books where the students can find the subjects of the course:

C.D. Pagani, S. Salsa: Analisi Matematica 2 (Masson);

W. Rudin: Analisi Reale e Complessa (Boringhieri);

E. Giusti: Analisi Matematica 2 (Boringhieri).

S. Salsa, A. Squellati: Esercizi di Matematica volume 2 (Zanichelli);

T. Tao: An introduction to measure theory, GSM 126, AMS;

Marcellini Sbordone: Esercitazioni di Matematica, Secondo volume (Liguori Editore);

M. Bramanti: Esercitazioni di Analisi Matematica 2, Progetto Leonardo - Esculapio (2012).

Teaching methods

The course is taught by lessons in classroom. Some complementary material will be available on IOL (only in Italian).

Assessment methods

The grading is split of several written preparatory parts and ends with a final colloquium. The candidate must collect in an exercise book all the exercises that the teacher has assigned during all the lessons of the course. The exercise book has to be written by hand by the candidate itself (no papers written on computers or photocopies or scansions are accepted).

In order to partecipate to the tests the students have to book themselves on the lists on AlmaEsami [https://almaesami.unibo.it/almaesami/welcome.htm] . The students can not partecipate to the tests without the inscriptions to the lists on AlmaEsami. The exam is splitter in 4 parts: A,B,C,D.

The "exercises part" is composed by tests A and test B.

The "theoretical part" is composed by tests C and D

The duration of the parts A plus B is 2 hour overall. Tests A and B have to be solved together in the same day.

The maximal score of the part A is 9. The maximal score of the part B is 10. The candidate is admitted to the part A only obtaining at least 4 points. The part B is sufficient only if the candidate obtain at least 6 points.

Tests C and D have to be faced in the same day.

The students are admitted to the part C only if they have overcome the B test.

The C test is 45 minutes long during which the candidate has to reply to three theoretical questions. The maximal score of part C is 5.

If the sums of tests, A,B,C is greater o equal to 15 the student is admitted to the final test D at the blackboard, where two questions are asked to the student. The range of the score of the D part is from -6 to +6. Some queries could also be posed about the assigned exercises.

The final grade is the sum if the scores realized in the A,B,C and D tests, plus possibly 2 additional points if the candidate exercises book is correctly solved.

Example of the test A+B

Example of the test C

Further details about the final exam may be found in the parallel italian page of the course. In any case the candidate may ask to the teacher all the clarification about the structure of the tests and their grade.

The teacher plans to schedule also two tests during the lessons called "in itinere" tests (tests during the lessons). The "in itinere" tests are alternative to the A and B tests (in this way the student may skip A anB tests) in order to be admitted to the "theoretical part" (always composed by C and D tests) at the end of the lessons. The structure and the score of "in itinere" tests are very similar to the A, B tests. Further technical details can be found in the parallel italian version or asking to the teacher.

 

Due to COVID-19 desease sins the summer season 2020 the grading will be decided with on line tests respectively named A,B,C and the by an examination on line called D examination.

The exam will be delivered on line, using EOL, IOL (Esami OnLine, Insegnamenti OnLine), Zoom Meetings, Teams platforms.

The candidate has to have a computer endowed by a microphone and a webcam and a internet connection supporting a  audio/video datas transmission.

In order to be admitted to the exam the candidate has to be enrolled in AlmaEsami list opened to the exam.

A test duration: 20 minutes;

B test duration: 80 minutes;

C test duration: 30 minutes.

D examination: live oral, on line examination.

The admission to the B test depends on the grade obtaine in A test. To be admitted to the B test the candidte has to obtain at least 4 over 9.

The admission to the C test depends on the grade obtaine in B test. To be admitted to the C test the candidte has to obtain at least 6 over 10.

To be admitted to the D test, A+B+C has to be grater than 15.

A,B tests will be done during the same day and to be admetted to the A test the candidate has to be enrolled in the AlmaEsami list associated with the exam.

C,D evaluations will be done in a different day.

The examination D is done by the professor and either it may improves the grade until 6 point or downgrade the final evaluation by 6 point.

The final evaluation is given summing A,B,C,D evaluations.

In case the final grade is strictly less than 18, the candidate has to repeat the exam in another session repeating all the tests A,B,C.

For further details please read the Italian version.

Teaching tools

Lecture notes concerning the lessons on IOL (only in Italian). Tutor (if assigned).

Links to further information

https://www.unibo.it/sitoweb/fausto.ferrari

Office hours

See the website of Fausto Ferrari