270 - guida - Facoltà di Ingegneria

GUIDA DELLO STUDENTE
Aims
(english version)
Integration of multivariable functions: integrals along a curve, surface integrals and volume integrals . Methods for solving differential
equations. Tools and techniques for complex analysis and operational calculus (Laplace and Fourier transforms). Applications of problem
solving in the fields of science and technology
Topics
Topics: Just for elettronica
Functions of two or more variables: hints on the topology in R^n.
Curves and line integrals. Smooth curves and length of a curve. Torsion and curvature. Fresnet coordinates. Abscissa on a curve. Line
integral of a function.
Vector fields: work along a curve, conservative and irrotational fields. Characterization of conservative fields by means of potentials. Poincare’
s Theorem. Green formulas and applications.
Surface integrals: evaluation of areas and of the flow of a vector field through a surface.
Volume integrals: normal domains, reduction formulas, change of variables.
Existence and uniqueness for Cauchy problem. Basic theory of linear differential equation. Resolution of O.D.E.
For both biomedica and elettronica
Sequences, series, limits in the complex field. Continuous and differentiable functions in C. Cauchy-Riemann equations. Olomorphic and
analytic functions. Properties of analytic functions. Integration in C. Jordan theorem. Cauchy theorem. Fresnel integrals. Cauchy integral
formula. Sequences and series of functions. Types of convergence. Liouville theorem. Fundamental theorem of algebra and of maximum
modulus. Laurent series. Residues and integration. Hermite theorem. Lebesgue spaces. Fubini and Tonelli theorems. Dominated convergence
theorem. Fourier transform and its properties. Inversion formula. Schwartz spaces. Plancherel identity. Laplace transform and its properties.
Relation with Fourier Transform. Initial and final value theorems. Solving differential equations by means of Laplace and Fourier transform.
Laplace transform of periodic functions. Convolution and Fourier and Laplace transform. Inversion formula for the Laplace transform.
Bromwich formula and use of residues. Special functions and their Laplace transform.
Exam
The exam consists of two written part and an oral part.
Textbooks
N. Fusco, P. Marcellini e C. Sbordone, Analisi matematica 2, edizioni Liguori
N. Fusco, P. Marcellini e C. Sbordone, Esercitazioni di matematica 2, edizioni Liguori (vol 1 e 2)
G. C. Barozzi, "Matematica per l'Ingegneria dell'Informazione", Zanichelli, Bologna, 2001.
Tutorial session
At least two hours per week.
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