Facoltà di Ingegneria - Guida degli insegnamenti (Syllabus)

Differential and integral calculus for functions of one real variable, linear algebra and analytic geometry in the plane and in the space.

The aim of the course is that of providing further mathematical tools commonly employed in the engineering sciences, by means of introducing the basic elements of the differential and integral calculus for real functions of several variables and of the ordinary differential equations

The many applications of the course topics in the applied sciences, for example in chemistry and in physics, will provide the student with the ability of modeling and solving practical engineering problems; they will also increase the ability of choosing independently the best solution techniques. The course will also provide the student with the ability to use mathematical laws in general scientific problems.

The expertise acquired in this course will be needed in order to study the material of later courses. Individual and collective problem-solving sessions will improve the ability to develop independent thought and learning capabilities. Oral presentations of the topics taught in the course, with the language proper of the basic disciplines of the degree course will help developing communication skills.

Smooth curves and elements of differential geometry of the curves in R2 and R3. Infinitesimal and differential calculus for functions of several real variables: limits, continuity, derivability, differentiability, Taylor's formula, local maxima and minima. Implicit function and Dini's Theorem. Constrained maxima and minima. Path and mulpliple integrals. Smooth surfaces and surface integrals. Conservative and irrotational vector fields. Flux and circulation of vector fields. Gauss-Green formulas, divergence and Stokes Theorem. Ordinary differential equations, existence and unicity results. Linear ordinary differential equations and resolution of some ordinary nonlinear differential equations.

The student will be assessed through on a written test and an oral test. The written test will assess the ability to solve problems by using the learned techniques. The oral test will assess the learning of the theory and the exposition skills.

In the exams the student have to prove that he understands the concepts presented in the course, that he knows the results and methods presented in the lectures, and finally that he is able to set a problem and solve it properly through the learned methods.

For each of the tests indicated above it is assigned a score between zero and thirty. The student will be admitted to the oral exam only if he passed the written test. The highest rating, equal to thirty of thirty, is achieved by demonstrating in-depth knowledge of the course contents and full autonomy in the performing the test. The minimum assessment, equal to eighteen of thirty, is assigned to students who manage to solve the proposed problems and who demonstrate sufficient knowledge of the topics of matter.

The overall grade, out of thirty, is derived from the comparative evaluation of both tests. The praise is reserved to the students who, having carried out the tests correctly and completely, has shown a special independence and excellence.

Teacher's lecture notes (available on teacher's web page) and/or M. Bramanti, C. Pagani, S. Salsa, "Analisi matematica 2", Zanichelli

- Ingegneria Meccanica (Corso di Laurea Triennale (DM 270/04))

**Università Politecnica delle Marche**

P.zza Roma 22, 60121 Ancona

Tel (+39) 071.220.1, Fax (+39) 071.220.2324

P.I. 00382520427