27993 - Mathematical Analysis T-2

Learning outcomes

At the end of the course, the student will have acquired a solid methodological and practical experience of the fundamental concepts and techniques of differential and integral calculus for functions of several variables.

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  exam is divided into two parts. The first is a written test that consists of solving exercises. If the written test obtains a grade greater than or equal to 18/30, the student is allowed to access the second part of the exam, which is devoted to evaluate whether the concepts explained have been assimilated (comprehension of the relevant concepts, knowledge of definitions, statements of main theorems of which the proof might also be required, if seen in class). The theory exam can also be taken in a subsequent session with respect to the one in which the exercise test is taken, as long as it is within the same exam session (June/July or January/February).

Electronic devices are not allowed.

 

 

 

Students with learning disorders and\or temporary or permanent disabilities: please, contact the office responsible (https://site.unibo.it/studenti-con-disabilita-e-dsa/en/for-students) as soon as possible so that they can propose acceptable adjustments. The request for adaptation must be submitted in advance (15 days before the exam date) to the lecturer, who will assess the appropriateness of the adjustments, taking into account the teaching objectives

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.

Office hours

See the website of Annalisa Baldi

See the website of Andrea Bonfiglioli