Unit MATHEMATICS 1
- Course
- Optics and optometry
- Study-unit Code
- A002480
- Location
- TERNI
- Curriculum
- In all curricula
- Teacher
- Alessio Troiani
- CFU
- 10
- Course Regulation
- Coorte 2023
- Offered
- 2023/24
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa integrata
LINEAR ALGEBRA WITH ELEMENTS OF COMPUTER SCIENCE
| Code | A002481 |
|---|---|
| Location | TERNI |
| CFU | 5 |
| Teacher | Alessio Troiani |
| Teachers |
|
| Hours |
|
| Learning activities | Base |
| Area | Discipline matematiche e informatiche |
| Academic discipline | MAT/03 |
| Type of study-unit | 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 software tools for linear algebra and data analysis. |
| 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 | Lectures and tutorials |
| Other information | |
| Learning verification modality | The exam is based on a written test which consists of the solution of computational problems and a practical test with a spreadsheet. The test lasts no more than two hours and is aimed at verifying the ability to correctly apply theoretical knowledge, the ability to understand the proposed problems and the ability to learn and develop solutions independently. Info on the support service for students with disability and/or DSA can be found at http://www.unipg.it/disabilita-e-dsa |
| Extended program | ## Matrices and linear systems: Systems of linear equations. Matrices: definition and operations. Gauss Elimination. 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 Rouché-Capelli 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. ## Analytic Geometry of the plane: 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. ## Analytic geometry of the space: 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 line-plane. 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 Algebra and data analysis with dedicated software tools |
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 study-unit | 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 McGraw-Hill -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 1,5 hours. 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. In particular during the exam, the student will solve logarithm exponential irrational trigonometric inequalities and also a trigonometric problem. Info on the support service for students with disability and/or DSA can be found at http://www.unipg.it/disabilita-e-dsa |
| 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. |