27993 - Mathematical Analysis T-2

Course Unit Page

Academic Year 2018/2019

Learning outcomes

Fornire una buona padronanza metodologica ed operativa degli aspetti istituzionali del calcolo differenziale ed integrale per le funzioni di più variabili.

Course contents

SERIES. Numerical series. Definition of a convergent series. Absolute convergence of a series. Convergence criteria for numerical series. Series of functions series. Power series, Taylor series, definition and main properties.
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 a continuous function and of limit of a function. The Weierstrass 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. Taylor's formula of the second order for functions of several variables. Interior and constrained local extrema for real functions of several variables. MULTIPLE INTEGRALS. Peano-Jordan measure. Definition of Riemann double integral for functions defined on a bounded and measurable set. Properties of the double integral. Double integrals on rectangular and normal domains computed by iterated integrals. The change of variables theorem for a double integral. Generalizations to triple integrals. Outline of double improper integrals. CURVE AND SURFACE INTEGRALS. Smooth and piecewise smooth curves, length of a curve, integral of a function over a curve. The integral of a vector field over an oriented curve. Conservative vector fields and their potentials. The Green-Gauss theorem. Smooth and piecewise smooth surfaces in R^3, area of a surface, integral of a function over a surface. The flux of a vector field through an oriented surface. The divergence theorem and the Stokes theorem. DIFFERENTIAL EQUATIONS. The Cauchy problem for differential equations and systems. Theorems on existence, uniqueness and continuation of solutions.


Bramanti-Pagani-Salsa, Analisi Matematica 2, Zanichelli.


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


M.Bertsch, R. Dal Passo, L. Giacomelli: Analisi Matematica, second edition (2011) Mc Graw Hill


G.C. Barozzi, G. Dore, E. Obrecht: Elementi di Analisi Matematica, vol. 2 - Zanichelli (2015).

An exercise book on functions of several real variables, such as, for example: M. Bramanti: Esercitazioni di Analisi Matematica 2, Progetto Leonardo - Esculapio (2012), or P.Marcellini, C. Sbordone: Esercitazioni di Analisi Matematica Due (prima e Seconda Parte) ed. Zanichelli.

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 consists in a written part, lasting three hours, containing both the resolution of various exercises and theoretical questions (definitions and theorems, possibly with proofs).

The written test is passed if one gets a grade greater  or equal than 18/30.

Students, that have already passed the T1 Mathematical Analysis exam and who pass the written test of Mathematical Analysis T2, have the possibility to face a further theoretical test 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, a week later from the publication on Almaesami of the results of the written test.

On the other hand, students, passing the aforementioned written part but that do NOT have yet  passed  the T1 Mathematical Analysis exam, must necessarily take a further theoretical test, enrolling on Almaesami in the appropriate lists.

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). On-line material (on IOL: INSEGNAMENTI ONLINE [https://iol.unibo.it/].).

Office hours

See the website of Annalisa Baldi

See the website of Maria Carla Tesi