27993 - Mathematical Analysis T-2

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 OF NUMBERS AND OF FUNCTIONS.

Numerical series: definition, convergence, ansolute convergence. Criteria of convergence.

Power series, Taylor series, Fourier series: definition and main properties.

DIFFERENTIAL EQUATIONS. The Cauchy problem for differential equations and systems. Theorems on existence, uniqueness and continuation of solutions.

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 room. complimentary lecture notes will be available on AMS Campus.

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. 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. In case the sum is grater or equal 28 the The final grade is the sum if the scores realized in the A,B,C and D tests. In case the sum is grater or equal 28 the panel may decide to ask an extra question to the candidate, the examination board, in order to appoint the higher classification "cum Laude" may decide to ask an extra question to the candidate.

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.

Example of test A+B

Esempio di prova C

 

Teaching tools

Lecture notes concerning the lessons on AMS Campus. Tutor (if assigned).

Office hours

See the website of Fausto Ferrari

See the website of Alberto Parmeggiani