31586 - Integrational Elements of Mathematical Analysis and Elements of Probability Calculation T

Learning outcomes

At the end of the course, the student has a practical and theoretical knowledge of basic concepts and tools in differential and integral calculus in several variables, and in probability theory.

Course contents

Integrational Elements of Mathematical Analysis (60h)

1. Review of series and improper integrals.
2. Differential equations
   Separable differential equations; Linear equations of the first order; Linear equations with constant coefficients of the second order.
3. Curves
   Vector-valued functions: limits and continuity; Regular curves and vector differential calculus; Length of a curve; Line integrals of the first kind.
4. Functions of several variables
   Limits: Calculation of limits; Continuity and theorems on continuous functions.
   Differential calculus: Partial derivatives and directional derivatives; Tangent plane; Differentiability conditions; Second-order derivatives, Hessian matrix, and Taylor's formula.
5. Maxima and minima of functions in several variables.
   Free optimization: Fermat's theorem; Study of the nature of critical points: sufficient conditions of the second order.
   Constrained optimization: Lagrange multiplier theorem.
6. Double and triple integrals
   Coordinate transformations in the plane and in space.
   Simple domains; Reduction theorems; Change of variables.
7. Vector fields
   Line integrals of the second kind, work and line integrals along a closed curve; Irrotational fields and conservative fields.

Elements of Probability Calculation (30h)

1. Probability spaces
   Measure of probability; Conditional probability and Independence, Partition Equation, Bayes' formula; Combinatorics.
2. Discrete models
   Discrete random variables and main distributions: Bernoulli, binomial, geometric, and Poisson distribution; Cumulative distribution function; Expected value; Variance.
3. Continuous models
   Absolutely continuous random variables; Density and Cumulative distribution function; Expected value; Variance. Examples: uniform, normal, and exponential random variables.

Readings/Bibliography

• Analisi Matematica 2. Teoria con esercizi svolti
  Autrice: Francesca G. Alessio
  Editore: Esculapio
  Anno edizione: 2020
  
  
• Analisi matematica 2
  Autori: Marco Bramanti, Carlo D. Pagani, Sandro Salsa
  Editore: Zanichelli
  Anno edizione: 2009
  
  
• Esercizi di Analisi matematica 2
  Autori: Sandro Salsa, Annamaria Squellati
  Editore: Zanichelli
  Anno edizione: 2011
  

• Introduzione alla probabilità - con elementi di statistica, 2a edizione
  Autore: Paolo Baldi
  Editore: McGrawHill
  Anno edizione: 2012

Teaching methods

Lectures and exercise sessions. A third of the lectures will be delivered online, with the use of specific software.

Assessment methods

The exam has practical part, and a more theoretical part. both in the form of a written test. The theoretical part will be shortly discussed with the instructors.

Teaching tools

Supplementary material will be made available on Virtuale.

Office hours

See the website of Nicola Arcozzi

See the website of Francesca Colasuonno