OVERVIEW

This module deals with some issues of Mathematical Analysis with the aim to complete the basic learning and introduce some theoretical tools used in engineering science.The main topics concern with the integration for functions of two and three variables, the theory of vector fields, the Fourier's series and the functions of a complex variable.

LEARNING OUTCOMES

The module aims to introduce the basic methods for the following topics in Mathematical Analysis: 1) Integration on R2 and R3, path and surface integration for vector fields, differential operators. 2) Fourier series and applications. 3) Functions of one complex variable, integration, computation of residues and applications.

AIMS AND LEARNING OUTCOMES

This module aims to provide the student with the knowledge of the following mathematical conceps:

1) Integration on R2 and R3. Integration of differential forms. Vector flux, divergence, curl and their properties for vector fields.

2) Series of functions, Fourier series and applications.

3) Functions of one complex variable and integration on a path in the complex plane. Computation of residues and applications.

The main learning outcomes consist of technical skills about the following issues:

Evaluation of integrals on R2 and R3. Integration of differential forms. Vector calculus and use of differential operators. Fourier series expansions and applications to differential equations. Integration on the complex plane and evaluation of residues with application to improper integrals.

PREREQUISITES

Basic knowledge on linear algebra and differential calculus are required as prerequisites. In particular, the student must be familiar with the analysis of functions of one and more variables, numerical series, and ordinary differential equations.

Teaching methods

The  teaching activity consists of online lessons on a Teams channel. Lessons are based on descriptions of lectures, examples and exercises which are uploaded to the channel weekly

SYLLABUS/CONTENT

First part: Curves and surfaces in R3. Integrals on R2 and R3. Vector analysis: Differential forms and path integrals, vector fields, circulation, flux, divergence and Stokes theorems.

Second part: Series of functions, Fourier series and applications.

Third part: Functions of a complex variable. Analytic functions and McLaurin series. Residue theorem and application to improper integrals.

RECOMMENDED READING/BIBLIOGRAPHY

• Lecture notes by the teacher (available on Aulaweb)
• L. Recine, M. Romeo,  Esercizi di Analisi Matematica (vol 2, Funzioni di più variabili ed equazioni differenziali), 2^ edizione, Maggioli (2013)
• G.C. Barozzi, Matematica per l’Ingegneria dell’informazione, Zanichelli 2004.
• F. Bagarello, Metodi matematici per fisici e ingegneri, Zanichelli 2019
• G. B. Folland, Fourier Analysis and its applications, Wadsworth, Belmont, 1992.
• J.E. Marsden and M.J. Hoffman , Basic  Complex Analysis, Freeman and Co., New York, 1987. 

Exam Board

MAURIZIO ROMEO (President)

ANGELO MORRO

CLAUDIO ESTATICO (President Substitute)

Teaching methods

The  teaching activity consists of online lessons on a Teams channel. Lessons are based on descriptions of lectures, examples and exercises which are uploaded to the channel weekly

Exam description

A written test on  tecnical skills (the required result must be greater or equal to 18/30). This is a prerequisite for an oral exam concerning theoretical issues.

Assessment methods

The written test is intended to verify technical skills about the following points: 1) Computation of integrals for functions of two or three variables, 2) Computation of field lines or potentials for vector fields, 3) Fourier series expansion of a periodic function and its application to  numeric series, 4) Evaluation of improper integrals by residues.

The objective of the spoken exam is to verify the student's knowledge about theoretical concepts. The student is required to state and prove theorems, showing to be aware of formulas and notations adopted.

Exam schedule