27993 - Mathematical Analysis T-2

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.

DIFFERENTIAL EQUATIONS.  Linear equations and Equations with separable variables. The Cauchy problem for differential equations and systems. Theorems on existence, uniqueness and continuation of solutions.

Readings/Bibliography

One of the following text for the theory part:

Fusco-Marcellini-Sbordone: Analisi Matematica Due, Liguori Editore.

M. Bramanti, C. D. Pagani, S. Salsa, Analisi matematica 2. Ed. Zanichelli.

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

V. Barutello, M. Conti, D. Ferrario, S. Terracini, G. Verzina: Analisi Matematica vol. 2, ed. Apogeo

M. Bertsch, R. Dal Passo, L. Giacomelli: Analisi Matematica, seconda edizione (2011) Mc Graw Hill

An exercise book on functions of several real variables, such as, for example:

Bramanti M.: Esercitazioni di Analisi Matematica 2 , Ed. Esculapio

or

G. Catino, F. Punzo: ESERCIZI SVOLTI DI ANALISI MATEMATICA E GEOMETRIA 1/2, Ed. Esculapio

Teaching methods

The course consists of lessons describing the fundamental concepts of real and vector functions of several real variables. Lessons are completed with examples and counterexamples illuminating the theoretical content. Futhermore a lot of exercises are solved in the classroom.

Assessment methods

The assessment consists in a written part, lasting three hours, containing both the resolution of various exercises and theoretical questions (definitions and theorems, possibly with proofs).

The written test is passed if one gets a grade greater or equal than 18/30.

Students, that have already passed the T1 Mathematical Analysis exam and who pass the written test of Mathematical Analysis T2, have the possibility to face a further theoretical test by registering on a suitable Almaesami list. In any case the obtained result can not be modified more than three points (positively or negatively). Otherwise we proceed to verbalize the result of the written test by tacit assent, four days later from the publication on Almaesami of the results of the written test.

On the other hand, students, passing the aforementioned written part but that do NOT have yet passed the T1 Mathematical Analysis exam, must necessarily take a further theoretical test, enrolling on Almaesami in the appropriate lists.

The theoretical part of the exam dwells upon the comprehension of the relevant concepts and on the knowledge of definitions and the statements of fundamental theorems. Proofs of some theorems, clearly detailed, may be required.

Teaching tools

Tutorship (if appointed).

Upload on the IOL website https://iol.unibo.it/

of several sheets of exercises, very important for the preparation to the written examination.

Office hours

See the website of Annalisa Baldi