# CALCULUS 2 PART. 1

## Learning outcomes of the course unit

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

## 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

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 and laboratories in the classroom

## Assessment methods and criteria

written and oral examinations