2021/2022

2021/2022

DM270

MAT/05 (MATHEMATICAL ANALYSIS)

DEPARTMENT OF ELECTRICAL,COMPUTER AND BIOMEDICAL ENGINEERING

INDUSTRIAL ENGINEERING

PERCORSO COMUNE

1°

2nd semester (07/03/2022 - 17/06/2022)

6

56 lesson hours

Italian

WRITTEN AND ORAL TEST

FORNARO SIMONA (titolare) - 6 ECTS

Students are expected to master the notions presented in the courses Analisi Matematica I, and Geometria e Algebra.

This is a second course in calculus and it includes power series, vector analysis, multiple integrals, line and surface integrals, the integral theorems of vector calculus. In general, there will be much emphasis on the comprehension of the definitions and the principal results. Only few proofs will be treated in full details. There will be several examples and exercises. At the end of the course, the students should be able to do computations on power series, limits, partial and directional derivatives, multiple integrals, line and surface integrals and have a deep knowledge of the mail notions.

1. Power series; absolute and simple convergence; series with positive terms; special series. Convergence results. Power series; derivation and integration. Taylor expansion.
2. Calculus for functions of several variables. Limits, continuity, partial derivatives, gradient, differentiability, Hessian; stationary points and their classification. Taylor's formula. Calculus for vector functions; Jacobian.
3. Multiple integrals. Two dimensional integrals; change of coordinates, polar coordinates, techniques of integration. Three dimensional integrals: spherical or cylindrical coordinates; evaluating the integral by the slice method or the line method.
4. Line and surface integrals. Parametric equations of a line; tangent line; arc lenght. Parametric equations of a surface; tangent plane; surface area; surface of revolution. Line integrals of scalar fields and of vector fields. Conservative vector fields. The differential operators curl and div. Surface integrals. Green's theorem; Stokes' theorem; divergence theorem.

Lectures

1) M. Bramanti, C. D. Pagani, S. Salsa. Analisi Matematica 2. Zanichelli, 2009.
2) M. Bramanti. Esercitazioni di Analisi Matematica 2 Esculapio.

The exam consists in a written test and an oral examination, which is not compulsory, for all the students who passed the written test with a positive mark.