Facoltà di Ingegneria - Guida degli insegnamenti (Syllabus)

Calculus in one real variable. Linear Algebra.

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.

Functions of two or more variables. Topology in R^n, continuity, derivability, differentiability. Taylors Formula of first and second order and classification of critical points in open sets. Implicit functions. Dinis Theorem. Maxima and minima for constrained functions. Lagrange multipliers. 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. Ordinary differential equations. Initial problem. Linear first and second order equations. Separable differential equations.

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.

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.

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.

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.

- Bramanti, Pagani, Salsa, Analisi Matematica 2, Ed. Zanichelli. - Fusco, Marcellini, Sbordone, Analisi Matematica Due, Ed. Liguori.

- Ingegneria Civile e Ambientale (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