27993 - Mathematical Analysis T-2

Learning outcomes

The student acquires the fundamental notions of differential and integral calculus for the functions of several variables. Moreover, he deals with the study of some differential equations of the first order and of the linear ones of the second order.

Course contents

Function in several variables

Real and vectorial functions of several variables: generalities, limits and continuity.
Weierstrass theorem and Intermediate value theorem.

Differential calculus on R^n
Partial derivatives and differential for functions of several variables; Jacobian matrix, gradient.
The chain rule. Partial derivatives of higher order. Hessian matrix.Taylor's formula at second order.
Local extrema : definitions, necessary conditions, sufficient conditions.Theorem of Lagrange multipliers for local extremas of a function subject to constraints.

Multiple integrals
Double integrals: definition, reduction and change of variable theorems.

Differential equations
First order linear differential equations, explicit solution formula.
Linear differential equations of order two; the case of constant coefficient.
Equations with separable variables.

Numerical Series
Definition, convergence and absolute convergence of a series; criteria for the convergence of series.

Regular curves and curvilinear integrals
Definition of a regular curve.  Length of a curve. Curvilinear integral of a scalar function along a regular curve: definition and properties.  The integral of a vector field on a curve.

Vector fields
Definition of vector field. Integral of a continuous vector field along a regular oriented curve (work). Conservative and potential vector field of a conservative field. Closed (or irrotational) vector fields. Search for a potential for a conservation field.

Complex numbers.

Definition of complex numbers and operations between complex numbers. Algebraic, trigonometric and exponential forms.The n-th roots of a complex number.

 

 

Readings/Bibliography

Fusco-Marcellini-Sbordone: Elementi di Analisi Matematica Due, Liguori Editore (versione semplificata) .
G.C. Barozzi, G. Dore, E. Obrecht: Elementi di Analisi Matematica, vol. 2, Zanichelli,

Esercizi:
Bramanti M.: Esercitazioni di Analisi Matematica 2 , Ed. Esculapio.

Teaching methods

The course provides theoretical lessons supported by exercises who aims to help the students to become familiar with and master the tools and mathematical methods introduced during the lectures.

Assessment methods

Achievements will be assessed by the means of a final exam.

The exam consists of a written part and an oral part.This is based on an analytical assessment of the "expected learning outcomes" described above. In order to properly assess such achievement the examination is composed of different sections; a written session, which consist of a test composed of exercises.

The oral session, consists of three questions at least about the theory.

Higher grades will be awarded to students who demonstrate an organic understanding of the subject, a high ability for critical application, and a clear and concise presentation of the contents . To obtain a passing grade, students are required to at least demonstrate a knowledge of the key concepts of the subject, some ability for critical application, and a comprehensible use of technical language.

A failing grade will be awarded if the student shows knowledge gaps in key-concepts of the subject, inappropriate use of language, and/or logic failures in the analysis of the subject.

Teaching tools

Exercises , notes and and other materials available online at  https://iol.unibo.it/

Office hours

See the website of Maria Manfredini