27993 - Mathematical Analysis T-2

Learning outcomes

At the end of the course the student knows the basic definitions, their relationship and the main properties of the following topics: -numercal series - curves, surfaces and vector fields, integration of functions and of vector fields -real functions of more than 1 real variables (in particular of 2 real variables): continuity, differentiability, critical points, integration. The student is able to solve suitable exercises on these topics.

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

Theory:

M. Bertsch, A. Dall'Aglio, L. Giacomelli: Epsilon 2, Secondo corso di Analisi Matematica, seconda edizione, Mc Graw Hill

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

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

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

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

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

Teaching methods

Lectures and classroom exercises.

Assessment methods

The exam consists in a written test, which contains both exercises and theory questions.

Teaching tools

Teaching materials on Virtuale (only in Italian) https://virtuale.unibo.it/. Tutor (if assigned).

Office hours

See the website of Eleonora Cinti

See the website of Cataldo Grammatico

See the website of Cataldo Grammatico