# MATHEMATICS 2 AND EXERCISES

## Learning outcomes of the course unit

Contents: the course of Mathematics II and Exercises is designed to provide tools and mathematical methods useful for applications. Particular emphasis is given to Fourier series and integral transforms. The theoretical treatment of the fundamental concepts is followed by examples and exercises.

Understanding and applications: attention is focused on the acquisition of a language formally correct, and on stimulating students ability to express the content in a clear but rigorous way; connections between different parts of the course are highlighted, also through several exercises.

## Prerequisites

Matematica I ed Esercitazioni (Mathematics I and Exercises)

## Course contents summary

Functions of several real variables. Mathematical methods and applications: Fourier and Laplace transform to solve simple problems involving (ordinary/partial) differential equations. Learning of a formally correct mathematical language, of methods to solve simple problems, and of instruments to find connections among the different topics of the course.

## Course contents

- Functions of several real variables: domain, limits, partial derivatives, characterization of the stationary points. Coordinate transformations. Integrals in two and three dimensions. Vector fields, differential forms.

- Short summary on numerical series. Sequences of functions. Series of functions. Power series, series of McLaurin. Trigonometric series and Fourier series, convergence in mean square norm, exponential form.

- Introduction to distributions, Dirac delta. Fourier transform and integral, and their applications. Laplace transform and applications. Functions of a complex variable.

## Recommended readings

M.Bramanti, C.D.Pagani, S.Salsa, Matematica (Calcolo Infinitesimale e Algebra lineare), Zanichelli Ed., in particular from Chapter 10 to Chapter 14

or, equivalently,

M.Bramanti, C.D.Pagani, S.Salsa, Analisi Matematica 2, Zanichelli Ed., in particular from Chapter 3 to Chapter 7.

Esercise book:

S.Salsa, A.Squellati, Esercizi di Matematica, Vol.2, Zanichelli, Bologna, 2005.

## Teaching methods

Lectures with theoretical explanations and several exercises

## Assessment methods and criteria

The knowledge and understanding of the topics of the course will be verified through a written and oral exam.

In particular, exam aims to check the learning of mathematical methods useful for some applications, the acquisition of a language formally correct, the ability to solve simple problems, the development of links between the different parts of the course. Students should be able to apply by themselves the mathematical tools to problems of interest in chemistry.