87294 - Mathematical Analysis T-B

Learning outcomes

To give a good knowledge of the calculus concerning functions with several variables.

Course contents

THE EUCLIDEAN SPACE R^n. The vector space structure, the dot product and the euclidean norm. Open, closed, bounded, compact, connected subsets of R^n.

LIMITS, CONTINUITY AND DIFFERENTIAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES.

Generalities on real and vector functions of several real variables. Definition of limit of a function and continuous function and of . The Weierstrass, zeros and Heine-Cantor's theorem and the intermediate value theorem for functions of several variables. Partial and directional derivatives. Differentiable and C^1 functions; the differential and the Jacobian matrix. The chain rule. Partial derivatives of higher order. Hessian matrix. Taylor's formula of the second order for functions of several variables. Interior and constrained local extrema for real functions of several variables.

CURVE INTEGRALS.

Curves, length of a curve, orientation. Integral of a function over a curve.

The integral of a vector field over an oriented curve. Conservative vector fields and their potentials. Work of a vector field.

MULTIPLE INTEGRALS.

Normal domains. Double and triple integrals. The reduction formula. The change of variables theorem for a double integral.Gauss-Green's formulas and Stokes'Theorem in the plane.

SURFACE INTEGRALS.

Smooth surfaces. Tangent plane and normal vector. Area of a surface. Integral of a function over a surface. The divergence theorem and the Stokes theorem.

SERIES.

Numerical series: definition, convergence,  criteria of convergence.

Readings/Bibliography

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

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

E. Giusti: Analisi Matematica 2 (Boringhieri).

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

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

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

Teaching methods

Lessons in classroom. complimentary lecture notes will be available on IOL: https://iol.unibo.it/

Assessment methods

The grading is split in several written preparatory parts and ends with a final colloquium.

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 split in 4 parts: A,B,C,D.

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

The "theoretical part" is composed by test C and colloquium 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 obtains 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 the B test is sufficient.

The duration of test C is 30 minutes. The maximal score about C part is 5.

If the sums of tests, A,B,C is greater o equal to 15 the student is admitted to the final colloquium 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 in case commision agrees.

Example of the test A+B [https://campus.unibo.it/239076/1/A2_III_appello_160211.pdf]

Example of the test C [https://campus.unibo.it/239077/1/C_A2_16_02_16.pdf]

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.

 

Remote Evaluation (only in case the University orders to apply it)

The grading procedure will be decided with on line tests respectively named A,B and by an on line oral 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: 30 minutes;

B test duration: 90 minutes;

Colloquium examination: live oral, on line examination.

The admission to the B test depends on the grade obtained in A test. To be admitted to the B test the candidte has to obtain at least 5 over 10.

The admission to the C test depends on the grade obtained in B test. To be admitted to the C test the candidte has to obtain at least 10 over 20.

A,B tests will take place during the same day.  To be admitted to the A test, the candidate has to be enrolled in the AlmaEsami list associated with the exam.

In order to be admitted to the colloquium the student has to be obtained at least 5 in test A and at least 10 in test B. The colloquium will be done in a different day.

The final grade, during the remote evaluation, is a weighted average of A, B exams and the colloquium exam.

In case it will be permitted by the rules, students can face the colloquium in presence.

 

For further details, please read the Italian version and possibly

https://www.unibo.it/it/servizi-e-opportunita/servizi-online/servizi-online-per-studenti-1/lezioni-ed-esami-online

Teaching tools

Lecture notes concerning the lessons on IOL (only in Italian) IOL https://iol.unibo.it/ . Tutor (if assigned).

Links to further information

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

Office hours

See the website of Fausto Ferrari