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

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 (with a maximum score 25), 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) with a score between -2 and +8.  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.

Links to further information

https://virtuale.unibo.it/

Office hours

See the website of Annalisa Baldi

See the website of Andrea Bonfiglioli