# COMPLEMENTARY TOPICS IN CALCULUS

## Learning outcomes of the course unit

The object is to supply the students with knowledge and understanding of linear differential forms, sequences and series of functions, ordinary differential equations and systems.

Amain 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.

## Course contents summary

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 (ODE) 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.

## 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.

## Assessment methods and criteria

The final exam consists in a written test followed by an oral discussion. In both the tests the student has to demonstrate knowledge, comprehension and to be able to connect knowledge and comprehension about the subjects of the course.

The written test consists in some open questions.

The marks, ranging from 0 to 30, will be attributed considering the accuracy of the exposition and the operating methods.

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