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, linear algebra and in his relationship with geometrical thinking. From the forms to formulas, and vice versa: introduction to inverse problems and parametrical thinking.
Canali
teacher profile teaching materials
-LINEAR ALGEBRA FROM A GEOMETRIC VIEWPOINT, VECTORS,
- PLANES, LINES, SKEW LINES, DISTANCES.
-CONICS , QUADRIC SURFACES: IDENTIFICATION, CLASSIFICATION, CONSTRUCTABILITY
IDENTIFICATION AS RULED, AS DEVELOPPABLE, AS SECTIONS...
-DIFFERENTIAL AND INTEGRAL CALCULUS OF TWO AND THREE VARIABLES.
EXTREMA AND CRITICAL POINTS OF A SURFACE GIVEN BY A FUNCTION, TANGENT PLANE.
- PARAMETRIC CURVES, Frenet–Serret frame of a curve.
- SUPERFACES IN SPACE,PARAMETRIC AND IMPLICIT FORMULATION .
- DOUBLE INTEGRALS, VOLUMES OF REGIONS BOUNDED BY REGULAR SUPERFACES.
(several variables and curves)
Programme
-MATHEMATICAL MODELS FOR HANDLING 3d GEOMETRICAL SPACE:-LINEAR ALGEBRA FROM A GEOMETRIC VIEWPOINT, VECTORS,
- PLANES, LINES, SKEW LINES, DISTANCES.
-CONICS , QUADRIC SURFACES: IDENTIFICATION, CLASSIFICATION, CONSTRUCTABILITY
IDENTIFICATION AS RULED, AS DEVELOPPABLE, AS SECTIONS...
-DIFFERENTIAL AND INTEGRAL CALCULUS OF TWO AND THREE VARIABLES.
EXTREMA AND CRITICAL POINTS OF A SURFACE GIVEN BY A FUNCTION, TANGENT PLANE.
- PARAMETRIC CURVES, Frenet–Serret frame of a curve.
- SUPERFACES IN SPACE,PARAMETRIC AND IMPLICIT FORMULATION .
- DOUBLE INTEGRALS, VOLUMES OF REGIONS BOUNDED BY REGULAR SUPERFACES.
Core Documentation
ANY TEXT AT THE LEVEL OF THE SECOND AND THIRD SEMESTER OF A THREE-SEMESTER COLLEGE CALCULUS.(several variables and curves)
Type of delivery of the course
standard lectures and recitation sessions. In case of persistence of the Covid19 emergency, we will follow the rules of our University.Attendance
attendance mandatoryType of evaluation
written exam followed by oral exam. In case of perduring emergency due to COVID-19 we will follow all provisions geared at ruling the educational and evaluation activities. Specifically, there will be oral examinations on the programme teacher profile teaching materials
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 stright 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.
A mathematical topic at a choice, to be developed autonomously, from a given list of lectures.
or a choosen textbook at university level, for example:
Bramanti-Pagani-Salsa: "Calcolo infinitesimale e algebra lineare", Seconda edizione, ed. Zanichelli
G.B. Thomas, R.L. Finney “Analisi Matematica”, ed. Zanichelli
Programme
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 stright 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.
A mathematical topic at a choice, to be developed autonomously, from a given list of lectures.
Core Documentation
R. Adams “Calcolo Differenziale 2, (funzioni di più variabili)”, quarta edizione, ed. casa editrice Ambrosianaor a choosen textbook at university level, for example:
Bramanti-Pagani-Salsa: "Calcolo infinitesimale e algebra lineare", Seconda edizione, ed. Zanichelli
G.B. Thomas, R.L. Finney “Analisi Matematica”, ed. Zanichelli
Reference Bibliography
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 Some mathematical definitions in Enciclopedia Treccani L’America dimenticata. I rapporti tra le civiltà e un errore di Tolomeo, di Lucio Russo “La strada che porta alla realtà” di Roger PenroseType of delivery of the course
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. In case of an extension of the health emergency from COVID-19, all the provisions that regulate the methods of carrying out the teaching activities and student assessment will be implemented.Attendance
Attendance at the course is mandatory for its 75% .Type of evaluation
The student assessment involves a written and an oral exam. The written exam consists of some exercises to assess students understanding of concepts and their autonomous application. In case of an extension of the health emergency from COVID-19, all the provisions that regulate the methods of carrying out the teaching activities and student assessment will be implemented. In particular the student assessment will be organized through an oral exam concerning the program of the course.