Corso di laurea in matematica

Students who have taken the classes of "Geometria 2" and "Analisi Matematica 2" already have these prerequisites

The theoretical structure of the course consists in a series of theorems and their proofs, the study of which will enable the student to autonomously produce rigorous proofs of mathematical results not identical to those already known but inspired to them in a relevant manner and to solve problems of moderate difficulty in the field of differential geometry and multivariate calculus.

 In particular, the course will provide:

• theoretical training objectives: development of a rigorous mathematical language; assimilation of abstract concepts, theorems and their proofs related to differential geometry and multivariate calculus
• applied training objectives: the student will learn computing techniques to solve problems; the student will be able to solve standard exercises and new problems, in which it will be necessary to develop new strategies and apply the concepts learned or develop simple proofs similar to those seen in class.

1. will be familiar with abstract arguments;
2. will be able to generalize and apply ideas to specific examples;
3. will know some differential geometry and its role in mathematics;
4. will be familiar with results which require ideas connected with differential geometry for their proofs.

2. Differential geometry of surfaces in space: Smooth surfaces in R³. Tangent plane and normal vector; orientability. The first fundamental quadratic form. Surface integral, area. Isometries and local isometries. The Gauss map, the differential of the Gauss map, and the second fundamental quadratic form. Gaussian curvature, mean curvature, principal curvatures; local behaviour of the surface with respect to the tangent plane. The Theorema Egregium.

3. Differential forms on Rⁿ and Stokes theorem: Multilinear alternating forms on a vector space, wedge product. Tangent vectors to Rⁿ, vector fields. Differential forms on Rⁿ. Pull-back, wedge product, exterior differential. Closed forms and exact forms. Relationship with line integrals (recall from Analisi 2). Poincaré's Lemma. Integral of a k-form along a k-chain. The boundary of a k-chain. General form of Stokes theorem for integrals of k-forms defined on an open set in Rⁿ. Interpretation in terms of vector fields: curl, divergence, flux. The classical theorems of vector analysis: Green theorem, divergence theorem (Gauss), Stokes (or curl) theorem.

The written examination consists in exercises to solve, and usually lasts 2 hours and a half. The students can consult their own books and notes during the exam, but not in electronic form; a basic calculator is allowed.

For admission to the oral examination, it is necessary to get a grade of at least 18/30 at the written examination. The oral examination must be taken in the same exam session of the written examination.  If a student fails the oral examination, s/he must repeat also the written examination.

The oral examination consists of questions on the theory and the proofs treated in the course, and ofter includes a discussion of the written examination.

Foreign students can choose to take the exam in Italian, English, or French.

For more details, and for the written examinations of the previous years, please see the web page of the course on moodle.

Curves and surfaces
Autore: Marco ABATE, Francesca TOVENA
Casa editrice: Springer
This book has been published also in English, but there are no copies in English in the math library

For differential forms, vector fields, and Stokes theorem: 

Analisi Matematica 2
Autore: Carlo Domenico PAGANI, Sandro SALSA
Casa editrice: Zanichelli
ISBN: 9788808637086 (<-- link al catalogo biblioteca)