# CALCULUS 2

## Learning outcomes of the course unit

To learn about the subjects of the course, and to be able to solve problems.

## Prerequisites

Good knowledge of the theory of functions of one real variable and of the geometry of Euclidean spaces.

## Course contents summary

Normed and metric spaces. Norms, distances, equivalent norms and equivalent distances.

Limits and continuity of functions of several real variables.

Differential calculus for functions of several real variables: directional derivatives and their geometric meaning, partial derivatives, gradients, differentiation rules, tangent hyperplanes and their geometric meanings, Schwarz Theorem, Taylor formula, quadratic forms, local maxima and minima.

Regular curves, simple curves, equivalences among curves, paths, unit tangent vector to regular paths, curve lenghts, integrals of continuous functions along paths.

Implicit Function Theorem, Inverse Function Theorem, Lagrange Theorem.

Linear differential forms, integrals along oriented paths, primitives, equivalent conditions for the existence of primitives, primitives on starshaped sets, simply connected sets.

Elementary notions about multiple integrals: definitions, reduction theorem, changes of variables, Gauss-Green formulae in dimension 2.

## Recommended readings

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

G. Prodi: Lezioni di Analisi Matematica II. ETS Pisa (1974);

M. Bramanti, C.D. Pagani, S. Salsa: Analisi Matematica 2. Zanichelli (2009);

N. Fusco, P. Marcellini, C. Sbordone: Analisi matematica due. Liguori (1996).

## Teaching methods

Lectures are held in the classroom, that concern both theoretical and applicative aspects. Moreover, exercises are solved by students with the guidance of the teacher, in such a way that it is possible to verify the degree of comprehension and knowledge of the students.

## Assessment methods and criteria

The examinations consist of a written and an oral session. The students are allowed to the oral sessions only if they pass the written examination. In the written examination 3 or 4 questions are asked. The students should exhibit calculus skills and master of different subjects taught in the course. Marks are given to each question.

The oral examination consists of a discussion about the written examination and of questions to verify the level of comprehension of the theoretical parts of the course.