# CALCULUS 2

## Learning outcomes of the course unit

Knowledge and understanding.

At the end of the lectures, students should have acquired knowledge and understanding of the basic results of multivariable calculus, ordinary differential equations, curves in Euclidean n-dimensional spaces and of Peano-Jordan measure theory and integration.

Applying knowledge and understanding.

By means of the classroom exercises students learn how to apply the theoretical knowledges to solve concrete problems, such as optimization problems. models from the applied sciences which lead to solving ordinary differential equation or to solving particular integrals for functions of several variables.

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 and precise way mathematical statements in the field of study, also in a context broader than mere calculus. Through the front lectures and the assistance of the teacher, the students acquire the specific and appropriate scientific vocabulary.

Learning skills.

The student, who has attended the course, is able to deepen autonomously his/her knowledge of multivariate calculus, of the measure theory and integration for multivariate functions, starting from the basic and fundamental knowledges provided by the course. He/She will be also able to consult specialized textbooks, even outside the topics illustrated during the lectures. This to facilitate the entry in the labour market as well as a second-level study in a field which requires good mathematical skills.

## Prerequisites

Students may take the exams only after they pass the exams of the courses Analisi Matematica 1 and Geometria.

## Course contents summary

The course aims at providing students with the fundamental concepts of multivariable calculus, integration for multivariate functions, curves and (explicitly solvable) ordinary differential equations.

## Course contents

1) Curves.

Oriented curves; simple, closed and smooth curves; tangent vector and tangent straight line; length of a smooth curve, equivalent curves, curvilinear abscissa, line integral.

2) Elements of topology in R^n.

Interior, limit and bundary points; open and closed sets; compact sets, connected sets and convex sets.

3) Multivariable differential calculus.

Limits and continuity: limits for functions of several variables; continuous functions of several variables; Weierstrass' and intermediate value theorems. Multivariable calculus: directional and partial derivatives, differentiability of scalar and vector valued functions, gradient; tangent plane, tangent and normal vectors; chain rule; differentiability of a composite function; functions of class C^1, functions with null gradient. 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 optimization; Lagrange's multipliers; vector fields and line integral;

potentials; irrotational vector fields.

4) Multiple integrals (mainly for functions depending on two or three variables).

Integration: Peano-Jordan measure of sets; definition of integral and of integrable functions; dimensional reduction theorem. Change of variable formula: polar, spherical and cylindrical change of variables.

5) Ordinary Differential Equations.

First order linear differential equations with continuous coefficients, separable equations, linear equations of order n with constant coefficients. Variation-of-the-constants formula.

## Recommended readings

Marino Belloni, Luca Lorenzi: Analisi Matematica 2 - Teoria. Ed. Santa Croce.

Marino Belloni, Luca Lorenzi: Analisi Matematica 2 - Esercizi. Ed. Santa Croce.

## Teaching methods

The course schedules 4 hours of lectures per week plus other two hours per week for additional exercises.

The didactic activities consist of frontal lectures alternating with exercises.

During the lectures, the topics of the course will be presented and discussed in a rigorous way. Much emphasis will be given to the application of the abstract results presented. To this aim, particular importance will be given to exercises, which are the most useful way to make students understand the relevance of the results presented in the theoretical lectures and to learn how they can be applied.

## Assessment methods and criteria

The knowledges are verified in a traditional way, through the evalutation of a written test. Students should show that they have achieved knowledges on the subjects illustrated during the lectures. They should also be able to apply the knowledges to concrete situations, solving both multiple choices exercises and more elaborate exercises.

The test contains also a theoretical question which could be the statement of a theorem or the correct definition of a concept illustrated during the lectures.

The exam is passed if the score of the written test is at least 18 points.

The maximum score of the test is 33 points. Students who gain more than 30 points pass the exam with the grade "30 cum laude".

If interested, students who pass the written test (i.e., students with at least 18 points in the written test), may ask the teacher to take also an oral exam. The request should be formulated before the deadline for accepting the grade of the written test on ESSE3. The oral test will focus on all the program of the course and it will consist of both theoretical questions and practical exercises.

## Other informations

Even if not mandatory, it is strongly recommended to attend the lessons.