# MATHEMATICAL ANALYSIS 2

## AIMS AND CONTENT

LEARNING OUTCOMES

In the Analysis module we provide the tools for the comprehension and computation of double and triple integrals, of curvilinear integrals of scalar and vector functions, and we introduce the related theorems (divergence, Gauss-Green).

We show how to deal with linear systems of differential equations (considering in particular the case of constant coefficients).

AIMS AND LEARNING OUTCOMES

The first goal is the understanding of integral calculus for functions of two or three real variables: double and triple integrals, line and surface integrals of scalar fields.

We will discuss the divergence theorem in two and three dimensions.

The second objective is a general understanding of systems of ordinary differential equations (with particular emphasis on linear systems in low dimension).

We will discuss the convergence properties of sequences and series of functions, in particular of power series expansions.

Teaching methods

Lectures and practice

SYLLABUS/CONTENT

Integration theory for functions of several variables. Double and triple integrals, changes of variables in multiple integrals. Polar, cylindrical, spherical coordinates. Parametric curves. Line integrals of scalar functions, length of a curve. Vector fields, line integrals of differential forms, closed and exact forms, potentials. Divergence theorem and Gauss Green formulas in the plane.

Parametric surfaces in space, area of a surface, surface integrals. Flow of a field through a surface. Divergence theorem in space.

Systems of ordinary differential equations. Existence and uniqueness for the Cauchy problem.

Linear systems, fundamental matrix. Solution of systems with constant coefficients. Stability and asymptotic behavior.

Sequences and series of functions. Pointwise and uniform convergence of sequences and series of functions. Power series.

RECOMMENDED READING/BIBLIOGRAPHY

C. Canuto e A. Tabacco, Analisi Matematica II, Springer-Verlag, 2008.

## TEACHERS AND EXAM BOARD

Exam Board

EDOARDO MAININI (President)

FRANCO BAMPI (President)

MANUEL MONTEVERDE

ROBERTO CIANCI

MAURIZIO CHICCO

## LESSONS

Teaching methods

Lectures and practice