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

at the end of this activity and after clearing the corresponding exam, the student will be able to

- compute integrals of vector fields along curves in space

- compute multiple and surface integrals

- compute solutions of linear differential equations of the second order

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

SYLLABUS/CONTENT

ANALYSIS

• double integrals

  surface integrals

  integral along curves

  gauss' and stokes' theorems

  potential theory

  differential equations of the second order with constant coefficients

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

T. Apostol, vol 3: analisi 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

Anna Maria MASSONE (President)

ALESSIO CAMINATA

VICTOR LOZOVANU

ELEONORA ANNA ROMANO

FRANCESCO VENEZIANO

MICHELE PIANA (President Substitute)

FABIO TANTURRI (President Substitute)

LESSONS

LESSONS START

TBA

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

ANALYSIS:  Written exam followed by an oral exam

GEOMETRY: Written exam followed by an oral exam

Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the teacher at the beginning of the course to agree on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools

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