31586 - Integrational Elements of Mathematical Analysis and Elements of Probability Calculation T (L-Z)

Course contents

 

Mathematical  Analysis 60 hours

Limits and continuity for function in several variables

Definition of bounded, open and closed sets. Real and vectorial functions of several variables: generalities, limits and continuity.
Weierstrass's Theorem,  Bolzano's Theorems.

Differential calculus for functions of several variables

Partial derivatives and differential for functions of several variables. DEfinition of Jacobian matrix and gradient of a function. The chain rule. Partial derivatives of higher order.  Schwarz's Theorem.Taylor's formula of the second order for functions of several variables.
Relative extrema for real functions of several variables: definitions, necessary conditions (Fermat's Theorem), sufficient conditions. The theorem of  Lagrange's  multipliers.

Multiple integrals

Integral of a continuous function.  Properties of the integral. Reduction and change of variable theorems for multple integrals.

Differential equations

The Cauchy problem for differential equations. Theorems on existence and  uniqueness of solutions.
First order linear differential equations, explicit solution formula.
Homogeneous and non homogeneous order linear differential equations of second order; constant coefficient equations.
Equations with separable variables.

Regular curves  and  curve  integrals

Definition of regular curves, length of a curve and  Integral of a function over a curve.
 

Vector fields

The integral of a vector field over an oriented curve. Conservative vector fields and their potentials.

 

Probability Calculation  30 hours (Prof. Giovanni Cupini)

Probability models

Basic definitions and rules, conditional probability, independence, elements of combinatorial analysis.

Discrete random variables

Definitions and samples (binomial, geometric, hypergeometric, Poisson  distributions), multiple random variables.

Continuous random variables

Definitions and samples (uniform distribution, exponential distribution, Gaussian distribution), central limit theory.

Introductory statistics

Elementary descriptive statistics.

 

Readings/Bibliography

Elementi di  Analisi Matematica Due, Fusco,  Marcellini, Sbordone,  Liguori Editore.

Analisi Matematica, Michiel Bertsch, Roberta Dal Passo, Lorenzo Giacomelli Ed. McGraw-Hill.

Introduzione alla probabilità, P.Baldi, Ed. Mc Graw Hill.

Teaching methods

Lectures and exercises in the classroom.

Assessment methods

Preliminary written examination and oral examination.

Links to further information

http://www.dm.unibo.it/~manfredi

Office hours

See the website of Maria Manfredini

See the website of Giovanni Cupini