27993 - Mathematical Analysis T-2

Learning outcomes

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

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

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 of learning consists in an exam divided into two written tests: a first written test, lasting two hours, containing exercises, and a second written test lasting 40 minutes, which contains theory questions (definitions, statements of main theorems of which the proof may also be required, if seen in class). The first written test is passed with a grade score greater than or equal to 17/30. Only in case of passing the first written test, the second test will  be corrected. The evaluation of the two tests leads to a final grade, which is the final grade of the exam.

Students, who obtained a score greater than or equal to 25/30 as a final grade, have the opportunity to take an additional oral exam by registering on a suitable Almaesami list. In any case the obtained result can not be modified more than two points (positively or negatively). Otherwise we proceed to verbalize the result of the written test by tacit assent, three days later from the publication on Almaesami of the results of the written test.

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://virtuale.unibo.it/

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

Links to further information

https://virtuale.unibo.it/

Office hours

See the website of Annalisa Baldi

See the website of Eugenio Vecchi