# 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.

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

Topology of R^N.

Interior, cluster and boundary points. Open and closed sets. Compact sets and Heine-Borel theorem. Connected stes.

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. Schwarz's theorem. Taylor's formula. Lagrange's reminder.

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) Curves and vector fields.

Curves.

Oriented curves. Length of a curve. Homotopy of curves and simply connected stes.

Vector fields.

Integral along a curve. Conservative vector fields and potentials.

4) Multiple integrals

Integration.

Integration of bounded functions over compact rectangles. Properties of the integral. Jordan regions. Volume of a Jordan region. Integration over Jordan regions. Iterated integrals.

Change of variables.

Change of variables formula for linear maps. Jacobian of linear maps. Change of variable formula for multiple integrals. Polar and cylindrical coordinates.

## Recommended readings

Lecture notes and material from the following textbooks:

1) G. Prodi "Lezioni di Analisi Matematica 2", Bollati Boringhieri, Torino 2011

2) A. Browder "Mathematical Analysis. An introduction", Springer, New York 1996

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

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

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

## Teaching methods

Lectures and discussion section.

## 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.