OVERVIEW

The course is aimed at second year students who have acquired the fundamental knowledge related to the functions of one and two variables, to plane analytical geometry and linear algebra.

AIMS AND CONTENT

LEARNING OUTCOMES

The course aims to provide the basics on  Integration of functions of multiple variables, Integration on curves and surfaces, Vector fields. Provide algebraic calculation tools and knowledge of 3D analytical geometry 

AIMS AND LEARNING OUTCOMES

ANALYSIS

Students are expected to master the main techniques for calculating double and triple integrals, line integrals of scalar and vector fields, the basic properties of vector fields and the classical theorems of the differential calculus in Euclidean space (divergence, curl, Stokes, Gauss-Green).

At the end of the course students will be able to recall and present the theoretical notions that have been presnted during the lectures. Applying the various techniques introducted in the course and justifying every step, they will be able to:

• compute a double or triple integral using the main integration techniques over regions with significant geometry (portions of cones, cylinders, spheres, elipsoids);
• compute line and surface integrals and use tclassical theorems such as the divergence or the Gauss-Green theorems;
• sestablish if a vector field is conservative, and in the affirmative case, compute its potentials;
• use the main properties of functions of a complex variable to compute line integrals (the residue theorem)
• know and use the basic properties of the Laplace transform

GEOMETRY

The course aims at providing the necessary theoretical and operative knowledge for the understanding and solving of the problems in the following topics: change of coordinates in the 3-dimensional space, symmetric matrices and their signature, conics and quadrics, curves and surfaces from a differential point of view.

At the end of the course the student will be able to recall and present the theoretical notions she will have faced during the lectures. Moreover, by applying the algorithms and the solving technique introducted in the course and justifying every step, she will be able to:
- write down and analyse a change of coordinates in the 3-dimensional space given by a rotation and a translation;
- evaluate the signature and the positive/negative definition of a symmetric matrix;
- identify and study an assigned conic or quadric;
- study and characterise the geometry of a parametric curve or surface.

ANALYSIS

The student will have to know the tools for the calculation of double, triple, curvilinear and surface integrals and the fundamental properties of vector fields in view of the applications.

PREREQUISITES

Differential calculus for functions of one and two variables. Integration of the functions of one variable.

Plane analytic geometry,  linear algebra.

TEACHING METHODS

Lectures and exercise sessions for about 60h (Mathematical Analysis) and 30h (Geometry), with teaching methods to be announced depending on the evolution of the pandemic situation.

SYLLABUS/CONTENT

ANALYSIS

• Riemann integral for  functions of  2/3 variables. Measure in  R^2 andR^3. Reduction formulae for double and triple integrals.Change of variables. Polar, cylindrical and spherical coordinates.
• Curves in R^n. Length of a curve. Line integrals w.r.t. arclength.
• Parametric surfaces in R^3. Area of a surface, surface integrals. 
• Vector fields. Irrotational vector fields, conservative vector fields. Divergence theorem, Gauss-Green formulae.
• Functions of one complex variable. Holomorphic functions,  Cauchy Riemann equations. Cauchy’s theorem and  intergal formula. Isolated singularities and residue theorem.  Definition and basic properties of the Laplace transform, simple applications.

GEOMETRY

The following topics will be dealt with from both a theorical and operative point of view, withouth a clear subdivision between lectures and exercise sessions.
Elements of linear algebra.
Change of coordinates, rotations and translations in the 3-dimensional space.
Quadratic forms and symmetric matrices.
Classification of conics and quadrics.
Differential geometry of parametric curves.
Differential geometry of parametric surfaces.

RECOMMENDED READING/BIBLIOGRAPHY

ANALYSIS

Canuto-Tabacco, Analisi Matematica II

Pagani-Salsa, Analisi Matematica 2

GEOMETRY

Main references:

- A.Bernardi, A.Gimigliano, Algebra Lineare e Geometria Analitica, Città Studi Edizioni. 

- M.V. Catalisano, A. Perelli, Appunti di geometria e calcolo numerico, Lectures notes available on the aulaweb page of the course

- M.E. Rossi, Algebra lineare, Lectures notes available on the aulaweb page of the course

Further readings:

- Silvio Greco, Paolo Valabrega – GEOMETRIA ANALITICA – Levrotto e Bella

- Silvana Abeasis - ALGEBRA LINEARE E GEOMETRIA - ZANICHELLI 

- Marco Abate, Algebra Lineare , ed. McGraw-Hill

- E. Sernesi, Geometria vol 1, ed Bollati-Boringhieri

TEACHERS AND EXAM BOARD

Exam Board

FILIPPO DE MARI CASARETO DAL VERME (President)

ALESSIO CAMINATA

ANNA ONETO

ELEONORA ANNA ROMANO

FRANCESCO VENEZIANO

MATTEO SANTACESARIA (President Substitute)

FABIO TANTURRI (President Substitute)

LESSONS

LESSONS START

https://corsi.unige.it/8722/p/studenti-orario

EXAMS

EXAM DESCRIPTION

ANALYSIS:  Written exam followed by an oral exam

GEOMETRY: Written exam followed by an oral exam

ASSESSMENT METHODS

The written part will consist in the resolution of exercises on the topics of the teaching programme (see also the Aims and Learning Outcomes section). Through the resolution of the exercises the student will be evaluted in:
- her skills in identifying the theoretical and practical results which are necessary to approach and solve the problems, and general knowledge of the very same results;
- her skills in applying the suitable algorithms and procedures to solve the exercises;
- her skills in providing the right arguments and justifications for the involved steps she follows.

During the oral part the teacher will evaluated the student's knowledge which has not been positively emerged during the written part. At the same time, the teacher will evaluate her knowledge on the topics which have not been covered by the written part.

Exam schedule