Multidimensional differential and integral calculus. Introduction to Lebesgue measure and Lebesgue integral. Combinatorial problems. Elementary probability. Discrete random variables. Gaussian distribution.
1. G.Giaquinta, G.Modica, Analisi Matematica IV, Funzioni di più variabili. Pitagora, 2005.
2. G.Modica , L.Poggiolini. Note di Calcolo delle Probabilità, Pitagora, 2011.
Learning Objectives
The course has a goal providing students with
KNOWLEDGE about elements of calculus for functions of several variables: partial derivatives, gradient, multiple integrals, Lebesgue measure, area and volumes. Enumeration problems, basic probability, discrete and absolutely continuous random variables.
EXPERTISE necessary for construction and analysis of mathematical models, which are based on multivariable calculus. One gains expertise in resolution of combiantorial problems. We teach fundamentals of probability theory so that students will be capable to develop and analize basic probabilistic models
Prerequisites
Calculus (differential and integral) of the functions of one variable
Teaching Methods
Lectures: presentation of the theory contained in the program of the course, with direct interaction between student and teacher, to ensure comprehension of the contents.
Moodle online learning platform: online teacher-student interaction, allocation of lists of exercises, bibliographic references, written tests from previous years.
Further information
Type of Assessment
Two intermediate written tests which contain 7-8 theoretic questions and exercises. Final oral examination which evaluates the capacity of understanding and presenting the theoretical contents, as well as originality and autonomy of reasoning.
Course program
1.1. Differential calculus of the functions of several variables. Directional and partial derivatives, gradient and differential. Jacobian matrix. Derivation of the composition.
1.2. Implicit function and inverse function theorems.
1.3. Curves in R^n; tangents, curvature. Surfaces in R^n ; tangent space and normal vector.
1.4. Maxima and minima of the functions of several variables without constraints; optimality conditions.
Constrained extrema; Lagrange principle.
1.5. Integral calculus: introduction onto measure theory and Lebesgue integral. Fubini theorem and computation of multiple integrals.
Change of variables in multiple integral.
Measure and area.
2.1. Enumeration problems: binomial coefficients. Summation and multiplication principle in combinatorics.
Permutations, permutations without fixed points, enumeration of subsets and words. Enumeration of placements and selections of the objects.
2.2. Elementary probability. Examples: throwing dice and coin(s), selection of numbers.
2.3. Axiomatic probability theory, σ-algebras, σ- additivity of probability measure, continuity axiom.
2.4. Inclusion-exclusion principle in probability. Conditional probability. Bayes formula.
2.5. Bernoulli trials, finite Bernoulli processes.
2.6. Random variable (discrete). Expectation and variance. Binomial. hypergeometric and Poisson distributions. Moivre theorem. Bernoulli theorem.
2.7. Absolutely-continuous (AC) random variables. Computation of the expectation and of the variance. Theorem of composition.
2.8. Chebyshev's inequality. Examples of AC random variables. Uniform distribution. Gaussian (normal) distribution. Standartization.