# MATHEMATICAL ANALYSIS B

## Learning outcomes of the course unit

The aim of the course is to provide knowledge and abilities related to improper integrals, ordinary differential equations theory, curves theory, differential and integral calculus in several real variables.

At the end of the course the student is expected to be able

D1 - Knowledge and understanding:

To know

- improper integral theory

- ordinary differential equations theory

- curves theory

- basic notions of functions of several real variables: law, domain, zeros and sign, level sets, graph

- general equations and features of all surfaces showed in the course: plane, paraboloid of revolution, cone of revolution, half-spherical surface

- topology theory (neighbourhoods, interior, exterior, boundary of a set, open, closed, bounded and compact sets)

- continuous functions and Weierstrass Theorem

- basic notions of differential calculus for functions of several real variables: partial derivatives, gradient, tangent plane, differentiability, directional derivatives, higher order derivatives

- Total Differential and Schwartz Theorems

- definitions of local and absolute extreme points and saddle point

- free and constrained extrema theory: critical points, Fermat Theorem, sufficient conditions, Lagrangian multipliers

- double integral Theory: definition, geometric meaning, reduction theorems, change of variables, center of mass.

D2 - Applying knowledge:

Being able to

- evaluate the convergence or divergence of an improper integral

- solve an ordinary differential equation or a Cauchy problem

- recognize and draw the support of a plane curve, determine and draw tangent and normal vectors and unit vectors, determine tangent and normal lines equations and the length of the curve

- determine for a curve in space the tangent line equation at a point, the plane perpendicular to this line and the length of the curve

- solve a two variables inequality

- write the parametric equations of a given curve and of the boundary of a given set

- determine and draw domain, zeros, sign and level sets of a function of two real variables

- write graph equation, recognize the surface given by the graph and draw it

- draw a solid in space

- determine the interior, the exterior and the boundary of a set, recognize an open, closed, bounded or compact set

- compute partial derivatives, gradient, tangent plane, directional derivatives and higher order derivatives of a function of several real variables

- prove the differentiability of a function

- determine critical points of a function and their nature

- apply Weierstrass Theorem to prove the existence of the extrema of a function

- determine the extrema of a function

- apply Lagrangian multipliers

- compute a double integral and the volume of a solid

- determine the center of mass.

D3 - Making judgments:

Being able to

- understand the mathematical machinery employed in non-mathematical courses

- check the credibility of the results

- deal with a new problem and plan its solution

- organize work in a precise way.

D4 - Communicating skills:

Being able to communicate mathematical contents, even outside of an exclusively applicative context.

D5 - Learning skills:

To have acquired a good grounding in mathematical analysis to face, in the future, an autonomous analysis of possible applications in a study or in a project.

## Prerequisites

Mandatory propedeuticities: Mathematical Analysis A and Geometry (of the first year courses).

## Course contents summary

Topics:

1. IMPROPER INTEGRALS

2. CURVES IN PLANE AND SPACE

3. ORDINARY DIFFERENTIAL EQUATIONS

4. FUNCTIONS OF SEVERAL REAL VARIABLES

DIFFERENTIAL CALCULUS

SURFACES AND SOLIDS IN SPACE

5. FREE AND CONSTRAINED EXTREMA

6. DOUBLE INTEGRALS

## Course contents

Topics:

1. IMPROPER INTEGRALS

2. CURVES IN PLANE AND SPACE

3. ORDINARY DIFFERENTIAL EQUATIONS

4. FUNCTIONS OF SEVERAL REAL VARIABLES

DIFFERENTIAL CALCULUS

SURFACES AND SOLIDS IN SPACE

5. FREE AND CONSTRAINED EXTREMA

6. DOUBLE INTEGRALS

## Recommended readings

Reference books:

A.Coscia, Appunti ed esercizi di Analisi Matematica 2, Libreria Santa Croce (Parma, 2018)

E.Acerbi, G.Buttazzo, Secondo corso di Analisi Matematica, Pitagora Editrice (Bologna, 2016)

Teaching stuff (on the platform ELLY https://elly2020.dia.unipr.it):

Lecture notes.

Exercises with solution.

Previous years' examinations with solution (from 2014-15 to 2020-21)

## Teaching methods

Teaching methods:

The course (9CFU) is organised into a series of lectures (72 hours, 6 hours a week) and practices (48 hours, 4 hours a week).

The teaching activities will be performed in the University classrooms; moreover the teacher will be available online or at the University (fixing an appointment) to discuss lectured topics and exercises.

The course is based on the concepts (given in a precise and rigorous way) and on the applications and calculus.

At the beginning of the course all the material is uploaded on the Elly platform.

In order to download the teaching stuff, the on-line registration is needed.

The uploaded lectures are an integral part of the teaching material.

Non-attending students are advised to check the available teaching material and the information given by the teacher via the Elly platform.

## Assessment methods and criteria

Method of testing learning:

The final test of the course consists of a theoretical and practical written test followed by a theoretical and practical oral test; books, notes and electronic devices are not allowed. The theoretical questions concern definitions, theorems and applications.

The exam will take place in presence in the lecture-hall.

The student must prove he/she has understood, and is able to apply, the basic concepts of every topic in the programme.

The written test is calculated on 34 points; after obtaining a positive result in the first part of the exam, the student will be admitted to the oral test, in which a maximum of 6 more points are assigned. The exam is passed with a final mark of minimum 18/30.

The results of the exam will be published on the Elly platform within two weeks from the written test.

The students can examine their written tests during the time specified by the teacher or by appointment.

Instead of the final test, the student will be allowed to substain three written tests in itinere, with questions both theoretical and practical.

## Other informations

This course (9CFU) is mandatory for all students in Management Engineering.

Attending lectures is strongly recommended.