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

Learning outcomes

At the end of the course, after passing the final exam, students will have acquired a basic understanding of the calculus of real functions of several variables (properties, maxima and minima), curves, potentials, multiple integrals, and their significance, as well as the solution of certain simple types of differential equations. In addition, they will have acquired a basic understanding of probability, with particular reference to certain continuous distributions (uniform and normal distributions).

Course contents

The course consists of two modules.

• Module 1 (Eugenio Vecchi): INTEGRATIONALE ELEMENTS OF MATHEMATICAL ANALYSIS
  
• Module 2 (Nicola Arcozzi): ELEMENTS OF PROBABILITY CALCULATION T

 

Course content of Module 1: INTEGRATIONALE ELEMENTS OF MATHEMATICAL ANALYSIS

THE N-DIMENSIONAL EUCLIDEAN SPACE AND COMPLEX NUMBERS. The structure of vector space, scalar product and Euclidean norm. Elements of topology. An overview of complex numbers.

LIMITS, CONTINUITY AND DIFFERENTIAL CALCULUS FOR FUNCTIONS OF SEVERAL REAL VARIABLES. Real and vector functions of several real variables: generalities. Definition of limit and continuous function. Weierstrass theorems, intermediate values for functions of several variables. Definition of partial derivative and directional derivative. Differentiable functions and functions of class C^1; the differential and the Jacobian matrix. The theorem on the differentiability of a compound function. Higher-order partial derivatives. Second-order Taylor formula for functions of several variables. Relative extrema for free real functions of several real variables.

MULTIPLE INTEGRALS. Definition of Riemann double integral on finite and measurable sets. Properties of the double integral. Reduction theorems on rectangles and on simple sets. The theorem of change of variables. Triple integrals: extension of definitions and theorems on double integrals.

CURVILINEAR AND SURFACE INTEGRALS. Regular and regular piecewise curves, length of a curve, integral of a function on a curve. The integral of a vector field on an oriented curve. Conservative vector fields and their potentials. Green-Gauss theorem. Regular and regular surfaces at strokes in R^3, area of a surface, integral of a function on a surface. Flow of a vector field through an oriented surface. Divergence and Stokes theorems. Curl fields and potential vector.

ORDINARY DIFFERENTIAL EQUATIONS. Linear differential equations of first order: general integral for homogeneous and non homogeneous equations, the Cauchy problem. Linear differential equations of second order with constant coefficients: general integral for homogeneous and non homogeneous equations, the Cauchy problem. Generalization to variable coefficients and arbitrary order equations. Separable ODEs.

 

 

Course content of Module 2: ELEMENTS OF PROBABILITY CALCULATION T

1. Probability Spaces
   Measure of probability; conditional probability and independence; total probability formula; Bayes’ theorem; combinatorial calculus.
2. Discrete Models
   Discrete random variables and main distributions: Bernoulli, binomial, geometric, and Poisson; distribution function; expected value; variance.
3. Continuous Models
   Absolutely continuous random variables; density and distribution function; expected value; variance. Examples: uniform, normal, and exponential random variables. Central Limit Theorem.

 

 

 

Readings/Bibliography

Module 1: INTEGRATIONALE ELEMENTS OF MATHEMATICAL ANALYSIS

THEORY.

1. M. Bertsch, R. Dal Passo, L. Giacomelli: Analisi Matematica (seconda edizione), McGraw-Hill (2011)
2. G.C. Barozzi, G. Dore, E. Obrecht: Elementi di Analisi Matematica, vol. 2, Zanichelli (2015)

EXERCISES. M. Bramanti: Esercitazioni di Analisi Matematica 2, Progetto Leonardo - Esculapio (2012)

 

Module 2: ELEMENTS OF PROBABILITY CALCULATION T

1. Appunti di Probabilità e Statistica, Fabio Zucca, 2010.
   Available for download at https://zucca.faculty.polimi.it/ita/didattica/2006-07/dispense/corsoSMA.pdf
2. INTRODUZIONE ALLA TEORIA DELLA PROBABILITÀ E DEI PROCESSI ALEATORI, Luigi Carassale 2011.             Available for download at:
   http://www.dicat.unige.it/carassale/AQFA/AQFA_Probabilita.pdf

Teaching methods

Module 1: INTEGRATIONALE ELEMENTS OF MATHEMATICAL ANALYSIS

Lecture at the blackboard and weekly student reception.

 

Module 2: ELEMENTS OF PROBABILITY CALCULATION T

Assessment methods

The exam consists of two parts. To pass the exam, students must pass both the exams of Module 1: INTEGRATIONALE ELEMENTS OF MATHEMATICAL ANALYSIS and Module 2: ELEMENTS OF PROBABILITY CALCULATION T in the same session (summer session = 4 exam dates between June and September; winter session = 2 exam dates between January and February). The final grade will be the weighted average of the two exams.

Module 1: INTEGRATIONALE ELEMENTS OF MATHEMATICAL ANALYSIS

At the end of the course there will be a written and an oral test, both of which are compulsory.

The written test will consist of exercises. It will last 2 hours. It is forbidden to use notes nor books.

Access to the subsequent oral test is allowed only to those who have passed the written test with a score greater than or equal to 18/31.

The oral test will focus on verifying understanding of the fundamental concepts and knowledge of the definitions and statements of the main results, examples and counterexamples.

The oral test is to be taken within the same roll call as the written test passed. Failure or absence in the oral will result in forfeiture of the validity of the score obtained in the written.

There will be 6 rounds (each of which will have both written and oral test): 4 in the summer session (3 between June-July + 1 in September) and 2 in the winter session (January-February).

 

Module 2: ELEMENTS OF PROBABILITY CALCULATION T

At the end of the course there will be a written test lasting 1 hour.

 

Students with Specific Learning Disabilities (SLD) or temporary/permanent disabilities are advised to contact the University Office responsible in a timely manner (https://site.unibo.it/studenti-con-disabilita-e-dsa/en ). The office will be responsible for proposing any necessary accommodations to the students concerned. These accommodations must be submitted to the instructor for approval at least 15 days in advance, and will be evaluated in light of the learning objectives of the course.

Teaching tools

Sheets of exercises will be made available, uploaded on the UniBo "VIRTUAL" website.

Office hours

See the website of Eugenio Vecchi

See the website of Nicola Arcozzi