27993 - Mathematical Analysis T-2

Course contents

SERIES. Series with real and complex terms. Definition of a convergent series. Absolute convergence of a series. Convergence criteria for numerical series. 
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. SERIES OF FUNCTIONS AND INTEGRALS DEPENDING ON A PARAMETER. Generalities on series of functions.  Power series in R and in C: Abel lemma, radius of convergence, propeties of the sum of a power series, real and complex analytic functions.  nuity and differentiability of integrals depending on a parameter.  DIFFERENTIAL EQUATIONS. The Cauchy problem for differential equations and systems. Theorems on existence, uniqueness and continuation of solutions.

Readings/Bibliography

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)

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 and an oral exam. In the written part, lasting three hours, the solution of various exercises is required. Access to the oral part is allowed only to the students passing the written part. The oral 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. The oral part of the exam must be passed in the same session of the written part. Only in the period june-july the oral part may be passed in the subsequent session. The written part may be replaced by two partial examinations, lasting two hours each, showing the student studies regularly.

Teaching tools

Tutorship (if appointed)

Office hours

See the website of Enrico Obrecht

See the website of Andrea Bonfiglioli