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Analisi Matematica 2 (INF)
Mathematics 2
Alessandro Calamai

Seat Ingegneria
A.A. 2015/2016
Credits 9
Hours 72
Period II
Language ENG

Prerequisites
Calculus in one real variable and in several real variables. Linear Algebra.

Learning outcomes
The course aims to provide students with knowledge and application skills of: - the methods for solving several variables problems with unconstrained and constrained extrema; - the tools and techniques for the integration in several variables and on differential varieties; - the functions of complex variables and their applications; - the Fourier and Laplace transform.

Program
Smooth curves. Line integrals. Arc length. Vector fields, path integral along a curve. Conservative and irrotational fields. Differential forms. Exact and closed forms. Poincare'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. Fourier series. 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.

Development of the examination
LEARNING EVALUATION METHODS
The learning evaluation is carried out by two exams: - a practical examination, which consists of solving exercises and problems related to the topics explained in the course. The test must be completed in 3 hours; - a theoretical examination, consisting in a discussion, written and oral,of the topics of the course. In particular the knowledge and the understanding of all definitions, theorems and proofs explained in the classes will be tested. The practical exam is preliminary to the theoretical one. It is necessary to pass the practical exam in order to do the theoretical one. The two exams must be passed in the same exam session. If the student fails the theoretical exam, he/she must repeat also the practical one.

LEARNING EVALUATION CRITERIA
In order to pass the learning evaluation, the student must demonstrate that he/she has understood the advanced concepts of mathematical analysis explained in the course. In particular in the practical test the student must show that he/she is able to apply independently the learned techniques in solving exercises and problems. In the theoretical exam the student must be able to expose the theoretical contents with the correct language and accuracy.

LEARNING MEASUREMENT CRITERIA
Each of the tests is graded on a scale from 0 to 30. The final grade will be decided starting from the two test grades.

FINAL MARK ALLOCATION CRITERIA
The final grade will be positive only if in both of the tests the students gets the passing grade (18/30). The maximal grade is reached if the student proves a knowledge and a thorough understanding of the course content. The maximal grade with honors is reserved to the students who passed both of the tests in a complet and correct way, showing special independence and excellence.

Recommended reading
- Bramanti, Pagani, Salsa, ”Analisi Matematica 2”, Ed. Zanichelli. - Fusco, Marcellini, Sbordone, ”Analisi Matematica Due”, Ed. Liguori. - G. C. Barozzi, ”Matematica per l'Ingegneria dell'Informazione”, Ed. Zanichelli.

Courses

• Ingegneria Informatica e dell'Automazione (Corso di Laurea Triennale (DM 270/04))