# MATHEMATICS 2 AND EXERCISES

## Learning outcomes of the course unit

The course of Mathematics II and Exercises is designed to provide tools and mathematical methods useful for applications.

The theoretical treatment of the fundamental concepts will be followed by examples and exercises.

We expect that at the end of the course the students have the following abilities:

- Knowledge and understanding: students will know the main properties of multi-variable functions, multiple integrals, series of functions and other mathematical tools useful in applications, as Fourier/Laplace transforms;

- Applying knowledge and understanding: students will be able to apply their knowledge to solve exercises on the main topics of the course, and moreover they will be able to treat at a formal level through proper mathematical models some basic chemical and physical problems.

- Making judgements: students will be able to identify suitable relations among the various arguments of the course, and to choose the proper methods to face new exercises on the matter.

- Communication: students will have to express the contents of the course in a clear way, even in the oral exam, owing to a formally correct mathematical language;

- Lifelong learning skills: students will be able to use the acquired knowledge to solve problems even in different frames (Physics, Chemical Physics, and so on), so that at the end of First Level Degree they will be able to continue their studies at a Master Degree level with a good independence.

The course of Mathematics II and Exercises is designed to provide tools and mathematical methods useful for applications.

The theoretical treatment of the fundamental concepts will be followed by examples and exercises.

We expect that at the end of the course the students have the following abilities:

- Knowledge and understanding: students will know the main properties of multi-variable functions, multiple integrals, series of functions and other mathematical tools useful in applications, as Fourier/Laplace transforms;

- Applying knowledge and understanding: students will be able to apply their knowledge to solve exercises on the main topics of the course, and moreover they will be able to treat at a formal level through proper mathematical models some basic chemical and physical problems.

- Making judgements: students will be able to identify suitable relations among the various arguments of the course, and to choose the proper methods to face new exercises on the matter.

- Communication: students will have to express the contents of the course in a clear way, even in the oral exam, owing to a formally correct mathematical language;

- Lifelong learning skills: students will be able to use the acquired knowledge to solve problems even in different frames (Physics, Chemical Physics, and so on), so that at the end of First Level Degree they will be able to continue their studies at a Master Degree level with a good independence.

## Prerequisites

Basic tools on functions of a real variable, studied in the course Matematica I ed Esercitazioni (Mathematics I and Exercises)

Basic tools on functions of a real variable, studied in the course Matematica I ed Esercitazioni (Mathematics I and Exercises)

## Course contents summary

Multi-variable functions. Double or triple integrals. Functions series. Fourier and Laplace transforms.

Multi-variable functions. Double or triple integrals. Functions series. Fourier and Laplace transforms.

## Course contents

- Functions of several real variables: limits; continuity and differentiability; maxima and minima.

- Curves and surfaces: integrals.

- Series; functions series; Fourier series and Fourier coefficients; power series: convergence properties and sum of the series.

- Fourier and Laplace transforms: definitions, properties, basic rules and applications to differential problems.

- Functions with complex variable: examples, and Cauchy-Riemann conditions.

- Functions of several real variables: limits; continuity and differentiability; maxima and minima.

- Curves and surfaces: integrals.

- Series; functions series; Fourier series and Fourier coefficients; power series: convergence properties and sum of the series.

- Fourier and Laplace transforms: definitions, properties, basic rules and applications to differential problems.

- Functions with complex variable: examples, and Cauchy-Riemann conditions.

## 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.

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.

## Teaching methods

Lectures with theoretical explanations and several exercises

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.

- Written exam: exercises on the main arguments of the course (steady points of multi-variable functions, conservative vector fields, double or triple integrals, power series, differential problems solved owing to Fourier or Laplace transforms).

During the course there will be two written "intermediate exams" that, if both with a positive result, allow the students to do directly the oral exam.

- Oral exam: questions on the theoretical arguments of the course and on the methods used to solve the exercises.

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

- Written exam: exercises on the main arguments of the course (steady points of multi-variable functions, conservative vector fields, double or triple integrals, power series, differential problems solved owing to Fourier or Laplace transforms).

During the course there will be two written "intermediate exams" that, if both with a positive result, allow the students to do directly the oral exam.

- Oral exam: questions on the theoretical arguments of the course and on the methods used to solve the exercises.