# CALCULUS 2 PART. 1

## Learning outcomes of the course unit

Knowledge and understanding.

Students must achive thorough conceptual understanding of the theoretical foundations of multivariable differential and integral calculus as well as computational fluency.

Applying knowledge and understanding.

Students must be able to apply the forementioned notions to solve medium level problems related to the field of study and to understand how the forementioned notions can be used for solving problems in a more applied context.

Making judgements.

Students must be able to evaluate coherence and correctness of results obtained by themselves or by others.

Communication skills.

Students must be able to communicate in a clear, precise and complete way mathematical statements in the field of study, also in a broader context than mere calculus.

## Prerequisites

Differential and integral calculus for functions of one real variable. Linear algebra.

## Course contents summary

Multivariable differential and integral calculus.

## Course contents

1) Preliminaries of linear algebra and topology.

Linear algebra and geometry.

Vectors spaces. Norms and scalar products. Cauchy--Schwarz inequality. Linear mappings and matrices. Eigenvalues and diagonalization of symmetric matrices. Quadratic forms.

Metric and euclidean spaces.

Interior, cluster and boundary points. Open and closed sets. Sequences and complete metric spaces. Compact and connected sets. Continuous functions. Bounded linear operators. Lipschitz functions and contraction principle.

2) Multivariable differential calculus.

Limits and continuity.

Limits of multivariable functions. Continuous functions.

Differentiable functions.

Directional and partial derivatives. Differentiable functions. The gradient and its geometrical meaning. Tangent planes, tangent and normal vectors. Chain rule. Functions of class C^1. The inverse function theorem. Diffeomorphisms and changes of variables.

Functions of class C^k.

Higher order differentiable functions. Functions of class C^k. Schwartz's theorem. Taylor's formula (Peano's and Lagrange's reminders).

Optimization of multivariable functions.

Local and global maxima and minima, saddle points. Necessary and sufficient conditions for optimality.

Manifolds in R^N.

The implicit function theorem. Manifolds in R^N. Lagrange's multiplier.

3) Multiple integrals

Measure theory.

Volume of N-rectangles. Measurable sets and Jordan measure. Negligible sets and characterization of measurable sets. Sufficient conditions for measurability.

Integration.

Integration of bounded functions over measurable sets. Properties of the integral. Iterated integrals. Change of variables formula for linear maps. Jacobian of linear maps. Change of variable formula for multiple integrals. Polar and cylindrical coordinates.

## Recommended readings

N. Fusco - P. Marcellini - C. Sbordone "Analisi matematica 2", Liguori, Napoli 2001

W. Fleming "Functions of several variables", Springer, New York 1977

W. Rudin "Principles of Mathematical Analysis", McGraw--Hill, New York 1976

J. L. Taylor "Foundations of analysis", American Mathematical Society, Providence RI 2012

## Teaching methods

Lectures in classroom and laboratory activities.

## Assessment methods and criteria

Exams test for thorough conceptual understanding of theoretical results and computational fluency. Consist of a written text followed by a colloquium.