Study-unit MATHEMATICS II

Course name Mechanical engineering
Study-unit Code GP004937
Curriculum Comune a tutti i curricula
Lecturer Anna Rita Sambucini
CFU 12
Course Regulation Coorte 2017
Supplied 2017/18
Supplied other course regulation
Type of study-unit Obbligatorio (Required)
Type of learning activities Attività formativa integrata
Partition

ANALYSIS

Code GP004944
CFU 6
Lecturer Anna Rita Sambucini
Lecturers
  • Anna Rita Sambucini - Didattica Ufficiale
Hours
  • 54 Hours - Didattica Ufficiale - Anna Rita Sambucini
Learning activities Base
Area Matematica, informatica e statistica
Sector MAT/05
Type of study-unit Obbligatorio (Required)
Language of instruction italian
Contents Partial differentiation, direction derivatives. Extreme of functions of two or three variables. Parametric curves and arc length. Polar, cylindrical and spherical coordinates. Line integrals, multiple integrals, Green formulas. Potential. Function series.
Reference texts Calogero Vinti - Lezioni di Analisi Matematica II - Galeno Ed.
Educational objectives the course goal is to enable students to develop the concepts acquired with the pourpose of being able to use them to interpret and describe problems of applied sciences and in particular of engineering.
Prerequisites In order to understand and know how to apply the topic of this course it is important to follows/ have followed the course Mathematics I.
Teaching methods The course is organized as follows:

-lectures on all subjects of the course

-exercises in classroom.
Other information The examination involves passing the exams of two modules: Geometry and Analysis.
Learning verification modality The method of verification of the learning results of the Analysis module is divided in two phases:
a written exam of 3 hours and an oral examination of more or less 40 minutes.

For both modules the written test includes the solution of three exercises on topics covered in the program and is designed to verify the ability to correctly apply the theoretical knowledge, the understanding of the exercices proposed and the ability to communicate in writing language. The oral test is designed to assess the level of knowledge and the understanding reached by the student on the contents listed in the program, this test is also used for verifying the presentation skills of the student.
The final score of Mathematics II will be given from the average of the scores of the two modules and it is necessary to have passed the exam of Mathematics I.
The two tests together allow to ensure the ability to:
- knowledge and understanding,
- apply the skills acquired,
- exposition,
- develop solutions.
Extended program Partial differentiation, direction derivatives. Extreme of functions of two or three variables. Parametric curves and arc length. Polar, cylindrical and spherical coordinates. Line integrals, multiple integrals, Green formulas. Potential. Function series.

GEOMETRY

Code GP004943
CFU 6
Lecturer Fernanda Pambianco
Lecturers
  • Fernanda Pambianco - Didattica Ufficiale
Hours
  • 54 Hours - Didattica Ufficiale - Fernanda Pambianco
Learning activities Base
Area Matematica, informatica e statistica
Sector MAT/03
Type of study-unit Obbligatorio (Required)
Language of instruction italian
Contents Vector spaces. Bases and dimension. Linear transformations and isomorphisms. Matrices. Determinants. Linear systems. Autovalues and Autovectors.
Geometric vectors. Affine space and parallelism. Euclidean space and orthogonality. Projective space. Conics. Quadrics (samples).
Reference texts A. BASILE , L. STRAMACCIA, ALGEBRA LINEARE E GEOMETRIA ED. COM s.r.l. - PERUGIA
Educational objectives Knowledge of basic mathematical language and of foundamental concepts of linear algebra and Cartesian geometry.
Prerequisites No prerequisites except the basic knowledge of arithmetic and algebra.
Teaching methods Lectures on all subjects of the course.
Learning verification modality The examination is divided in two phases:

a) A written exam of one hour. The written test includes the solution of two exercises on topics covered in the program and is designed to verify the ability to correctly apply the theoretical knowledge, the understanding of the exercises proposed and the ability to communicate in writing language. The student must justify explicitly the used techniques with theoretical recalls. Valuation: from 0 to 30.

b) Oral examination of more or less 20 minutes. The oral test is designed to assess the level of knowledge and the understanding reached by the student on the contents listed in the program, this test is also used for verifying the presentation skills of the student. In this oral examination, taking into account the outcome of the previous part, it establishes the final score of the module.
Extended program Vector spaces. Generator systems. Linear dependence. Bases and vector's coordinates. Bases in generator systems. Exchanging theorem and dimension. Linear transformations. The space Hom(V,W). Definition of a linear transformation on the vectors of a basis.
Kernel and Image of a linear transformation. Relation between their dimension. Isomorphic vector spaces and their dimension.
Vector spaces of matrices. Row-column product. Matrix of a linear transformation. Matrix of a composed linear transformation. Matrix of a bases exchange.
Calculus of a matrix determinant. Transpose of a matrix, product of matrices, their determinant. Invertible matrices, their determinant, linear dependence of the columns.
Linear systems. Cramer's systems. Rank of a matrix and its determination. Homogenehous linear systems and the space of solutions. General case and theorem of
Rouché-Capelli. Autovalues and autovectors.
Orientate lines and segments. Cartesian reference systems. The space of geometric vectors. Coordinates of a vector
and coordinates of the extreme points of its representative segments. Parallel vectors, complanar vectors and conditions on their coordinates.
Affine Space. Parametric representation of lines and planes. Cartesian equation of a plane. Bundles of planes and lines. Cartesian equations of a line.
Conditions of parallelism. Exchanges of affine reference systems.
Euclidean Space. Definitions of angles. Scalar product. Distance between two points and sphere. Orthogonality conditions.
Projective space. Homogeneous coordinates. Representation of planes and lines in homogeneous coordinates.
Algebraic curves (samples) Theorem of Bézout. Classification of conics. Bundles of conics. Configuration of basic points and reducible conics in a bundle.
Quadrics (samples).