# COMPLEMENTARY TOPICS IN CALCULUS

## Learning outcomes of the course unit

The aim of the course is to give students knowledge and understanding of sequences and series of functions, ordinary differential equations, linear differential forms and their path integrals, and the ability to use these tools to solve basic problems. A main aim is to give them the competence to treat and apply these instruments.

Objects of the course are also: judgement independence, strong written and oral communication skills, learning ability, in accordance with the specific objects of the mathematics degree.

The aim of the course is to give students knowledge and understanding of sequences and series of functions, ordinary differential equations, linear differential forms and their path integrals, and the ability to use these tools to solve basic problems. A main aim is to give them the competence to treat and apply these instruments.

Objects of the course are also: judgement independence, strong written and oral communication skills, learning ability, in accordance with the specific objects of the mathematics degree.

## Prerequisites

Mathematical Analysis I and Mathematical Analysis 2, 1° module.

Mathematical Analysis I and Mathematical Analysis II

## Course contents summary

Paths in R^n and linear differential forms.

Sequences and series of functions.

Ordinary differential equations and systems.

Paths in R^n and linear differential forms.

Sequences and series of functions.

Ordinary differential equations and systems.

## Course contents

Paths in R^n. Linear differential forms and their integrals over paths. Closed forms, exact forms and their primitives. Theorems about forms defined in star-shaped or simply connected open sets.

Sequences and series of functions. Pointwise and uniform convergence. Total convergence of series of functions. Power series and Taylor series. Fourier series.

Ordinary differential equations and systems of ODE. The Cauchy problem. Local existence and uniqueness theorem. Extensions of the solutions. Qualitative study of solutions and continuous dependence on the initial data. Solutions of some type of first order ODE. Linear equations and systems with constant coefficients.

Paths in R^n. Linear differential forms and their integrals over paths. Closed forms, exact forms and their primitives. Theorems about forms defined in star-shaped or simply connected open sets.

Sequences and series of functions. Pointwise and uniform convergence. Total convergence of series of functions. Power series and Taylor series. Fourier series.

Ordinary differential equations and systems of ODE. The Cauchy problem. Local existence and uniqueness theorem. Extensions of the solutions. Qualitative study of solutions and continuous dependence on the initial data. Solutions of some type of first order ODE. Linear equations and systems with constant coefficients.

## Recommended readings

M. Fusco, P:Marcellini, C. Sbordone, Analisi Matematica due, ed. Liguori.

M: Bramanti, C.D. Pagani, S.Salsa, Analisi matematica 2, ed. Zanichelli

G. Prodi, Lezioni di Analisi Matematica II, ed. Tecnico Scientifica.

M. Fusco, P:Marcellini, C. Sbordone, Analisi Matematica due, ed. Liguori.

M: Bramanti, C.D. Pagani, S.Salsa, Analisi matematica 2, ed. Zanichelli

G. Prodi, Lezioni di Analisi Matematica II, ed. Tecnico Scientifica.

## Teaching methods

The teaching consists in frontal lessons where both theoretical and applicable aspects are expounded. The exercises are worked out with the collaboration of the students and are programmed in order that students can solve autonomously the problems arising from the theoretical lessons.

The teaching consists in frontal lessons where both theoretical and applicable aspects are expounded. The exercises are worked out with the collaboration of the students and are programmed in order that students can solve autonomously the problems arising from the theoretical lessons.

## Assessment methods and criteria

The final exam consists in a written test followed by an oral discussion, provided the written test is successful.

The written test consists in some open questions. To succeed, the students should demonstrate to have sufficient knowledge of the subjects of the course as well as calculus skills.

The oral test consists of a discussion about the written test, and in the verification of the level of comprehension of the theoretical parts of the course.

The test will be considered sufficient if the average of the marks of the written and oral tests is greater or equal to 18/30.

The final exam consists in a written test followed by an oral discussion, provided the written test is successful.

The written test consists in some open questions. To succeed, the students should demonstrate to have sufficient knowledge of the subjects of the course as well as calculus skills.

The oral test consists of a discussion about the written test, and in the verification of the level of comprehension of the theoretical parts of the course.

The test will be considered sufficient if the average of the marks of the written and oral tests is greater or equal to 18/30.