To provide the algebraic and analytical tools that enable the treatment of three-dimensional space, and beyond. In particular, to introduce differential and integral calculus in several variables and linear algebra in its relationship with geometrical thinking. From forms to formulas, and vice versa: introduction to inverse problems and parametrical thinking.
Canali
scheda docente
materiale didattico
Parametric and cartesian equation of a plane. Parametric and cartesian equation of a straight line in space. Point-line distance. Line-line distance. Intersections between straight lines and planes. Intersecting, parallel and skew lines.
Quadric surfaces. cylinders, cones, ellipsoids, paraboloids and hyperboloids. Contour lines and sections. Ruled surfaces.
Vector functions and parametric curves. Examples of parametric curves: lines, conics, spirals and cycloids. Tangent, normal and binormal unit vectors. Frenet formulas. Curvature and torsion. Curves on surfaces. Cylindrical helix.
Functions of two variables. Domain of definition. Graph of a function. Contour lines and sections. Limits and continuity for two variables functions. Partial derivatives. Tangent plane on a point to the function graph. Directional derivative. Differentiability. Gradient of a two variables function. Geometric properties. Maximal slope direction. Higher order derivatives. Critical points of a two variables function. Second order derivatives matrix and its Hessian determinant. Maxima, minima and saddle points.
Visualization of curves and surfaces using the software Mathematica or Python.
A mathematical topic at a choice, to be developed autonomously, from a given list of lectures.
oppure un testo universitario a scelta, ad esempio:
Bramanti-Pagani-Salsa: "Calcolo infinitesimale e algebra lineare", Seconda edizione, ed. Zanichelli
Programma
Sets of points in the plane or in the three-dimensional space. Vector space in two and three dimensions. Vectors and unit vectors. Scalar product, vector product and scalar triple product with their geometrical interpretation. Matrices and determinants.Parametric and cartesian equation of a plane. Parametric and cartesian equation of a straight line in space. Point-line distance. Line-line distance. Intersections between straight lines and planes. Intersecting, parallel and skew lines.
Quadric surfaces. cylinders, cones, ellipsoids, paraboloids and hyperboloids. Contour lines and sections. Ruled surfaces.
Vector functions and parametric curves. Examples of parametric curves: lines, conics, spirals and cycloids. Tangent, normal and binormal unit vectors. Frenet formulas. Curvature and torsion. Curves on surfaces. Cylindrical helix.
Functions of two variables. Domain of definition. Graph of a function. Contour lines and sections. Limits and continuity for two variables functions. Partial derivatives. Tangent plane on a point to the function graph. Directional derivative. Differentiability. Gradient of a two variables function. Geometric properties. Maximal slope direction. Higher order derivatives. Critical points of a two variables function. Second order derivatives matrix and its Hessian determinant. Maxima, minima and saddle points.
Visualization of curves and surfaces using the software Mathematica or Python.
A mathematical topic at a choice, to be developed autonomously, from a given list of lectures.
Testi Adottati
R. Adams “Calcolo Differenziale 2, (funzioni di più variabili)”, quarta edizione, ed. casa editrice Ambrosianaoppure un testo universitario a scelta, ad esempio:
Bramanti-Pagani-Salsa: "Calcolo infinitesimale e algebra lineare", Seconda edizione, ed. Zanichelli
Bibliografia Di Riferimento
Courant, Robbins, “Che cos’è la matematica”, Bollati Boringhieri, 2000 “Le curve celebri” (almeno l’introduzione e un paragrafo tratto dai capitoli 1-6) di Luciano Cresci “Flussi e riflussi” di Lucio Russo Alcune voci matematiche nell’Enciclopedia Treccani L’America dimenticata. I rapporti tra le civiltà e un errore di Tolomeo, di Lucio RussoModalità Erogazione
The course is organized in lectures and exercise class, sometimes using a computer. During exercise classes we give some exercises and problems, let the students try to solve them then we discuss the solution and, if necessary, we give the full solution at the blackboard. In some of the exercise classes the students are requested to use a computer for curves and surfaces visualization.Modalità Frequenza
Attendance at the course is mandatory for its 75% .Modalità Valutazione
The student assessment involves a written and an oral exam. Some tests during the course are also planned. The written exam consists of some questions to assess students’ understanding of concepts and their autonomous application.