# MATHEMATICAL ANALYSIS II

OVERVIEW

The course is aimed at sophomore students and needs basic skills in Calculus, Linear Algebra and Geometry.

## AIMS AND CONTENT

LEARNING OUTCOMES

The course provides basic notions about multiple integrals, line integrals, surface integrals and vector fields. It provides also basic skills about holomorphoic functions, Laplace transforms together with some appplications to ODE's.

AIMS AND LEARNING OUTCOMES

At the end of the course students will be required to

-calculate double or triple integrals by using reduction formulae or by changing variables. In particular students will be required to calculate the area, the volume, the coordinates of the center of mass or the components of the tensor of inertia.

-calculate line and surface integrals by using the Divergence Theorem and the Gauss-Green formula.

-calculate the potentials of conservative vector fields;

-calculate the integral of functions of a complex variable by using the Residue theorem

-solve ODE's by using Laplace transform.

PREREQUISITES

Basic Calculus, Linear algebra and Geometry.

Teaching methods

Frontal lessons. Examination mode: written and oral examination.

SYLLABUS/CONTENT

Riemann integral in R^n. Fubini' s theorem in 2D and 3D: applications. Change of variables. Curves in R^n: lenght of a curve, line integrals. Parametric surfaces in R^3, area, surface integrals. Divergence Theorem. Vector fields: irrotational vector fields and conservative vector fields. Gauss- Green formula and Stokes Theorem.

Functions of a complex variable, holomorphic functions, Laplace transform. Applications

RECOMMENDED READING/BIBLIOGRAPHY

Analisi Matematica

M. Bertsch, R. Dal Passo, L. Giacomelli

Mc Graw Hill

## TEACHERS AND EXAM BOARD

**Ricevimento:** Besides the hours of exercises in the afternoon, every student can fix an appointment.

## LESSONS

Teaching methods

Frontal lessons. Examination mode: written and oral examination.