# COMPLEMENTARY TOPICS IN CALCULUS

## Learning outcomes of the course unit

The course object is to supply the students with knowledge and understanding of sequences and series of functions, ordinary differential equations (ODE), implicit functions and multivariable integral calculus.

Give to 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

Implicit functions. The inverse function Theorem. Extremum problems with side conditions.

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

Ordinary differential equations (ODE). The Cauchy problem. Local existence and uniqueness theorem. Extension of the solutions. Solutions of some type of first order ODE. Linear equations.

## Course contents

Multivariable integral calculus. Implicit functions. The inverse function Theorem. Extremum problems with side conditions.

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

Ordinary differential equations (ODE). The Cauchy problem. Local existence and uniqueness theorem. Extension of the solutions. Continuous dependence of the data. Solutions of some type of first order ODE. Linear equations with constant coefficients. Linear systems.

## 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

## Teaching methods

Academic method. The teaching consists in frontal lessons where both theoretical and applicable aspects are expounded. The exercises are developped with the collaboration of the students and are programmed in order to they can solve independently the problems arising from the theoretical lessons.

## Assessment methods and criteria

The understanding check consists in a final written test and, if it will be positive, in an oral discussion. In both the tests the student has to demonstrate knowledge, comprehension and to be able to connect knowledge and comprehension about sequences and series of functions, implicit functions and multivariable integral calculus.

The written test consists in 3 open questions regarding the above arguments.

The highest mark for each question will be 10, the total amount of the written test will be 30.

The total amount of the oral test is 30.

The marks 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.