# COMPLEMENTARY TOPICS IN CALCULUS

## Learning outcomes of the course unit

Knowledge and understanding: At the end of this course the student should know the essential definitions and results on sequences and series of functions, and on ordinary differential equations (ODE), and he should be able to grasp how these enter in the solution of problems.

Applying knowledge and understanding: The student should be able to apply the aforementioned notions to obtain rigorous proofs of mathematical results not identical but strictly related to those already encountered, to solve medium level problems, and to understand how they relate to concepts seen in different courses; finally, the student should be able to mathematize problems in order to analyze and solve them.

Making judgements: The student should be able to evaluate coherence and correctness of the proofs given during the written test, by constructing and developing logical arguments with a clear distinction of assumptions and conclusions; the student should be able to check correct proofs and spot wrong reasonings.

Communication skills: The student should be able to communicate in a clear and precise way, suitable for a scientist-to-be in an intermediate stage of his formation, also through group work.

## Prerequisites

Analysis for functions of one real variable; linear geometry; linear algebra; differential and integral calculus in several variables.

## Course contents summary

Sequences and function series. Ordinary Differential Equations.

## Course contents

- Sequences and series of functions

Function sequences. Power series: radius of convergence; uniform convergence, continuity and integration.

- Differential Equations

The Cauchy problem. Regularity. Existence and uniqueness of solutions. Continuous dependence on data. Integration of some ordinary differential equations. Systems of First Order Linear Differential Equations. Qualitative behavior of solutions of ODE’s.

## Recommended readings

E. Acerbi, G. Buttazzo: Secondo corso di Analisi Matematica. Pitagora, Bologna, 2016.

Also, one may use any other good book on Analysis in several variables, as e.g.,

W. Fleming: Functions of several variables. Second edition. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1977.

## Teaching methods

Lectures are held in the classroom, encompassing both theoretical and applied aspects. Moreover, exercises are solved by students with the guidance of the teacher, so as to verify the degree of comprehension and knowledge of the students.

## Assessment methods and criteria

The final exam consist of a written and an oral session. Books, lecture notes, calculators, etc... are not allowed.

Students are admitted to the oral sessions only if they pass the written examination (with a mark greater or equal than 15/30). In the written examination some open questions are asked (usually in 2 or 3 groups within 1,5/2 hours). The students should exhibit calculus skills and mastery of different subjects taught in the course. Marks are given to each question, according to theoretical correctness, precision of execution, precision of exposition.

The oral examination consists of a discussion (to be given during the same exams session) about the written examination and of questions to verify the level of comprehension of the theoretical parts of the course. The oral and written scores are averaged to give the student the overall result for the exam.