# CALCULUS 2

## Learning outcomes of the course unit

Students must demonstrate knowledge and understanding of the basic results of multivariable calculus and ordinary differential equations. The course emphasizes concrete computations over more theoretical considerations with little emphasis on exceptional cases.

In particular, students must

1. exhibit thorough conceptual understanding and computational fluency in standard cases;

2. be able to evaluate coherence and correctness of results obtained by themselves or by others;

3. be able to communicate in a clear and precise way mathematical statements in the field of study.

## Prerequisites

Solid knowledge of single-variable calculus and linear algebra.

## Course contents summary

Multivariable calculus and ordinary differential equations.

## Course contents

1) Linear algebra and topology. Linear algebra and geometry: vector spaces, norm, scalar product and Cauchy-Schwarz inequality; matrices, eigenvalues and diagonal form of symmetric matrices, quadric forms; basic results of analytical geometry in space. Topology: interior, limit and bundary points; open and closed sets; compact sets and connected sets. 2) Multivariable differential calculus. Limits and continuity: limits for functions of several variables; continuous functions of several variables; Weierstrass' and intermediate values theorems. Multivariable calculus: directional and partial derivatives, differentiability of scalar and vector valued functions, gradient; tangent plane, tangent and normal vectors; chain rule;functions of class C^1; inverse function theorem, diffeomorphisms and change of variables. Functions of class C^2: Schwarz's theorem and hessian matrix; second order Taylor's formula; local and global maxima and minima, saddle points; necessary and sufficient conditions for optimality. Surfaces: implicit function theorem, Lagrange's multipliers. 3) Curves and vector fields. Curves: simple, closed and smooth curves, length of a smooth curve. Vector fields: line integral; potentials; irrotational vector fields. 4) Multiple integrals Integration: measure of sets; definition of integral and of integrable functions; dimensional reduction and Fubini's theorem. Change of variable formula: geometrical meaning of jacobian, spherical and cylindrical coordinates. 5) Ordinary Differential Equations Ordinary differential equations: definitions and examples; local existence and uniqueness of solutions; maximal and global solutions; solution methods for linear, separable and Bernoulli's equations. Second order linear differential equations: fundamental system of solutions, Lagrange's variation of parameters.

## Recommended readings

E. ACERBI - G. BUTTAZZO "Secondo corso di analisi matematica", Pitagora, Bologna 2016

## Teaching methods

Lectures and exercise sessions (4 + 2 hours per week).

## Assessment methods and criteria

The final exam consists of a written test supplemented by an oral examination. There are no intermediate tests.

The written test consists of exercises and multiple choices questions. The oral examination is subject to successful completion of the written test (grade 18/30 or higher). It consists of reviewing the written test and interviewing the candidate on program related arguments. The final grade is the average of the grades of the two parts (weights 1/3 and 2/3).

## Other informations

The course will be quite fast-paced and it is essential to work steadily throughout the semester.