Unit MATHEMATICS 1
 Course
 Optics and optometry
 Studyunit Code
 A002480
 Location
 TERNI
 Curriculum
 In all curricula
 Teacher
 Massimo Giulietti
 CFU
 10
 Course Regulation
 Coorte 2021
 Offered
 2021/22
 Type of studyunit
 Obbligatorio (Required)
 Type of learning activities
 Attività formativa integrata
LINEAR ALGEBRA WITH ELEMENTS OF COMPUTER SCIENCE
Code  A002481 

Location  TERNI 
CFU  5 
Teacher  Massimo Giulietti 
Teachers 

Hours 

Learning activities  Base 
Area  Discipline matematiche e informatiche 
Academic discipline  MAT/03 
Type of studyunit  Obbligatorio (Required) 
Language of instruction  Italian 
Contents  Vector spaces in basic sciences. Notable examples. Polynomials as a vector space. Review of analytical geometry. Euclidean Geometry in dimensions 2 and 3. The notion of linearity. An algorithmic approach to classical problems of algebra and linear algebra and analytic geometry. Basics of computer science and coding. Illustration of at least one Computer Algebra package. 
Reference texts  Algebra lineare e geometria, E. Schlesinger, Zanichelli 
Educational objectives  The main objective of the course is to provide on the one hand the basics of the language of linear algebra and, on the other hand, the methods of solving some simple problems in an operational perspective that also includes the use of the computer tools. The main knowledge acquired will concern the basic notions of linear algebra related to vector spaces and linear applications and the algorithmic methods of solving classical mathematical problems. The main skills that the course aims to transmit are:  be able to describe simple problems related to basic sciences using the language of mathematics, and in particular linear algebra  knowing how to solve, even with the aid of a computer, simple numerical problems related to linear systems, polynomial equations, analytical geometry. 
Prerequisites  A good knowledge of high school math subjects. 
Teaching methods  The course consists of classroom lectures on all topics of the course. In each lesson about half of the time will be devoted to solving problems and exercises 
Learning verification modality  The exam consists of a written test (or two progress assessments) and a final oral exam. The written exam requires the solution of 4 problems and it has a duration of 2 hours.Its objectives are to evaluate the resolutive capacity of the problems and the proper use of acquired knowledge. The oral exam consists of a talk of about 20 minutes. It is aimed at testing the degree of comprehension the students have reached, expositive skills and capacity of finding connections between the topics studied. If it is required, the exam can be taken in English 
Extended program  Matrices. Determinants. Systems of linear equations. Matrices: definition and operations. Determinant of a square matrix: definition and properties. The Laplace rule. The Binet's theorem. Invertible matrices. Rank of a matrix. Systems of Linear equations. The RouchéCapelli theorem. The Cramer's Rule. Geometric Vectors. Definition and operations. Product of a scalar and a vector. Linear independence. Bases. Orientation. Scalar product. Vector product. Mixed product. Plane Analytic Geometry. Cartesian frame of reference. Cartesian coordinates. The equation of line incident with two points. Cartesian equation and parametric equations of a line in the plane. Mutual position of two lines. Angle between lines. Bundle of lines. Distance between two points, distance between a point and a line. The Circumference. The Conics. The conics as sections of a cone. The conics as geometric places. The canonical form of a conic. Center, axes, vertices, asymptotes, fires and directives. Eccentricity of a conic. Conics as algebraic curves: general equation of a conic. Invariants of a conic. Reduction to the canonical form of a conic. Space Analytic geometry. Cartesian frame of reference. Cartesian coordinates. Cartesian equation and parametric equations of a plane. Mutual position of two planes. Angles between planes. Bundles of planes. The line in the space: Cartesian equations and parametric equations. Mutual position lineplane. Angles between the line and plane. Mutual position of two lines. Skew lines and coplanar lines. Distance between a point and a line, distance between a point and a plane. Distance between parallel lines, distance between a line and parallel plane, distance between parallel planes, distance between skew lines. Spheres and circumferences in the space. Surfaces and curves. Planar curves. Ruled surfaces. Cones and cylinders. Quadrics. Solving problems in Linear Algera with the Computer Algebra System MAGMA. 
BASIC MATHEMATICS FOR OPTICS
Code  A002482 

Location  TERNI 
CFU  5 
Teacher  Ilaria Mantellini 
Teachers 

Hours 

Learning activities  Base 
Area  Discipline matematiche e informatiche 
Academic discipline  MAT/05 
Type of studyunit  Obbligatorio (Required) 
Language of instruction  Italian 
Contents  Set theory, rational, irrational, logarithmic and exponential equations. Trigonometry. System of equations. Definition of function. Graphics of the main functions. 
Reference texts  Matematica per i precorsi. Giovanni Malafarina McGrawHill Lecture notes by the teacher (if necessary) 
Educational objectives  The main objective of the module is to provide on the one hand the basics of mathematical language and, on the other hand, the methods of solving simple equations and inequalities which will then be used in the study of a function. The main knowledge acquired will concern the basic notions of set theory and the solution of inequalities. The main skills that the module aims to convey are:  be able to describe in analytical terms the conditions of existence of mathematical laws  knowing how to solve simple systems of equations and inequalities. 
Prerequisites  Operations with polynomials. Algebraic fractions. Radicals Decomposition and factoring 
Teaching methods  Lectures and tutorials 
Other information  Students are strongly advised to follow lessons 
Learning verification modality  The examination involves an individual written test lasting two hours and an individual oral examination lasting an average of half an hour . The written test is aimed at verifying the correct application of the rules and all theoretical skills reached by the student in manipulating the calculation methods . The oral test is instead aimed at the verification of theoretical knowledge and the awareness acquired by the student in dealing with the topics of study , in addition to the establishment of presentation skills and language reached . The written test is prevalent with respect to the oral examination, since this examination can be taken only if the written test has been passed. 
Extended program  Set theory. Rational. irrational, logarithmic and exponential equations. Trigonometry, trigonometric equations and inequalities. Systems of equations and inequalities. Definition of function, domain, codomain and properties (injectivity, surjectivity, biietivity, monotonicity) and inverse functions. Overview and graphics of the main functions. 