# 27993 - Mathematical Analysis T-2

• Moduli: Annalisa Baldi (Modulo 1) Gregorio Chinni (Modulo 2)
• Campus: Bologna
• Corso: First cycle degree programme (L) in Chemical and Biochemical Engineering (cod. 8887)
• from Feb 21, 2024 to Mar 22, 2024

• from Mar 27, 2024 to Jun 07, 2024

## 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.

Theory:

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, 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 and half, containing exercises, and a second written test lasting 60 minutes, which contains theory questions (comprehension of the relevant concepts, knowledge of definitions, statements of main theorems of which the proof is also  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. More details are given in class.

## Teaching tools

Tutorship (if appointed).

Upload on the ''VIRTUALE'' website https://virtuale.unibo.it/

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