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# MATHEMATICAL ANALISYS II & PROBABILITY CALCULATION

 Degree course Information and Communication Technologies (ICT) Engineering Curriculum Curriculum unico Learnings Orientamento unico Academic Year 2016/2017

## Module: MATHEMATICAL ANALISYS II

 Degree course Information and Communication Technologies (ICT) Engineering Curriculum Curriculum unico Learnings Orientamento unico Academic Year 2016/2017 ECTS 6 Scientific Disciplinary Sector MAT/05 Year First year Time unit Second semester Class hours 48 Educational activity Basic training activities

## Single group

 Professor LUISA ANGELA MARIA FATTORUSSO Objectives N.D. Programme Differential calculus in several variables. Sets of Rn. Measure of a bounded set. Functions of several variables. Limits of functions of several variables. Continuity of functions of several variables. Uniform continuity. First partial derivatives. Higher order partial derivatives. Gradient. Schwartz Theorem. Differentiability and differentials. Theorems and geometric interpretation of differential. Derivatives of composite functions. Directional derivative. Taylor formulas for functions in two variables. Relative minimum and maximum. Absolute minimums and maximums of a function over bounded domains with parameterized boundary.Integral calculus in several variables. Double integrals and geometric interpretation. Computing double integrals as iterated. Volume of solid of revolution. Changing variables in double integrals. Polar coordinates. Triple integrals. Computing triple integrals as iterated. Changing variables in triple integrals. Spherical coordinates.Ordinary differential equations. General definitions. Cauchy Problem. General, particular, singular solutions with geometric interpretation. Linear differential equations. Fundamental systems of solutions. Homogeneous and nonhomogeneous linear differential equations with constant coefficients. First order differential equations with separable variables. First order linear differential equations.Sequences and series of real valued functions. Sequences of real valued functions. Pointwise and uniform convergence of a sequence of real valued functions. The Uniform Limit of Continuous Functions. Integration of Uniformly Convergent Sequences. Differentiability of Uniformly Convergent Sequences. Series of real valued functions. Pointwise, Uniform, absolute convergence of a series of real valued functions. Weierstrass M-test. Power series in R. Radius and interval of convergence. Abel Theorem. Taylor series. Taylor expansions of more fundamental functions. Periodic functions. Integration using series expansion. Trigonometric polynomial. Fourier Series. Dirichlet Theorems. Curves and surfaces Differential forms. Parametric curves. Regular curves. Lenght of a curve. Line integrals and geometric interpretation. Curvilinear abscissa. Linear differential forms. Exact differential forms. Line integral of an exact differential form.Gaus-Green theorem. Potential function.Regular surfaces.Surface integrals.Stokes Theorem.Complex valued functions and Fourier transform. Complex Valued Functions. Limit of a Complex Valued Function. Complex logarithm, exponential and power functions. Complex Trigonometric Functions. Complex differentiation. Complex integration. Fourier Transform. Applications. Adjoint Fourier transform and applications. Books N.Fusco, P. Marcellini- C. Sbordone, Elementi di Analisi Matematica Due, Liguori Editore. R. Adams Calcolo differenziale 2. Edit. AmbrosianaJames Stewart. Calcolo Funzioni di piu variabili .Edit. ApogeoM.Bramanti, Pagani, S.Salsa, Analisi Matematica II, ZanichelliC. D. Pagani S. Salsa, Analisi Matematica 2 , Zanichelli, 2015 BolognaP. Marcellini, C. Sbordone, Esercizi di Matematica due(4 vol), Liguori Editore. Traditional teaching method Yes Distance teaching method No Mandatory attendance No Written examination evaluation Yes Oral examination evaluation No Aptitude test evaluation No Project evaluation No Internship evaluation No Evaluation in itinere No Practice Test No

## Further information

Description Document
Compito analisi II sett. 2016 (esercitazioni) No news posted
No class timetable posted

## Module: PROBABILITY CALCULATION

 Degree course Information and Communication Technologies (ICT) Engineering Curriculum Curriculum unico Learnings Orientamento unico Academic Year 2016/2017 ECTS 3 Scientific Disciplinary Sector MAT/05 Year First year Time unit Second semester Class hours 24 Educational activity Basic training activities

## Single group

 Professor SOFIA GIUFFRE' Objectives Knowledge of the fundamentals of Probability and its main random variables, knowledge of the joint distributions and of the main limit theorems. Programme Basic Probability Concepts. Sample space, events, Probability space. Axiomatic definition of probability. Independent events. Basic properties of probability. Conditional probability. Inclusion/exclusion Principle. Bayes Theorem. Total probability Theorem . Basic notions of Combinatorics.Random variables. Definition of a random variable. Distribution function and properties. Discrete random variables. Mass function. Absolutely continuous random variables. Density function. Functions of random variables. Expected value of a random variable and of a function of a random variable. Properties. Variance. Properties. Markov Inequality. Chebyshev inequality. Important discrete distributions: Bernoulli, binomial, geometric, Poisson distribution. Important absolutely continuous distributions: uniform, normal, exponential, chi-squared.Rn-valued Random variables. Discrete random vectors. Absolutely continuous random vectors. Joint cumulative distribution function and joint density. Marginal distributions and marginal densities. Functions of random vectors. Expected value of a function of a random vector. Independence of random variables. Covariance and correlations. The Weak Law of Large Numbers. Central limit Theorem. Books S.M.Ross, Calcolo delle Probabilità, Seconda Edizione, Apogeo.P.Baldi, Calcolo delle Probabilità, McGraw-Hill.L.M.Ricciardi, S.Rinaldi, Esercizi di Calcolo delle Probabilità, Liguori Editore. Traditional teaching method Yes Distance teaching method No Mandatory attendance No Written examination evaluation Yes Oral examination evaluation No Aptitude test evaluation No Project evaluation No Internship evaluation No Evaluation in itinere No Practice Test No

## Further information

No document in this course
No news posted
No class timetable posted
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