270 - guida - Facoltà di Ingegneria

GUIDA DELLO STUDENTE
Aims
(english version)
To know the techniques of integration: line integrals of functions and vector fields, surface and volume integrals. To know solving methods for
ordinary differential equations.
To know the basic elements and techniques of complex analysis, and of Laplace and Fourier transform.
To be able to apply them for solving scientific and technological problems
Topics
Smooth curves. Line integrals. Arc length. Vector fields, path integral along a curve. Conservative and irrotational fields. Differential forms.
Exact and closed forms. Poincaré's theorem. Multiple integrals. Reduction formulas. Change of variables. Green's theorem. Regular surfaces,
surface integrals. Measure theory and Lebesgue integral. Ordinary differential equations. Initial problem. Linear first and second order
equations. Separable differential equations.
Sequences, series, limits in the complex field. Continuous and differentiable functions in the complex field. Cauchy-Riemann equations.
Holomorphic and analytic functions. Properties of analytic functions. Integration in the complex field. 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. 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
Written and oral.
Textbooks
- Bramanti, Pagani, Salsa, "Analisi Matematica Due", Ed. Zanichelli.
- Fusco, Marcellini, Sbordone, "Analisi Matematica Due", Ed. Liguori.
- G. C. Barozzi, "Matematica per l'Ingegneria dell'Informazione", Ed. Zanichelli.
Tutorial session
Two hours per week.
29