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
teacher profile teaching materials
Parametric and Cartesian equations of a plane. Parametric and Cartesian equations of a line in space. Distance from a point to a line. Distance between two lines. Intersections between lines and planes. Intersecting, parallel, and skew lines.
Quadric surfaces. Cylinders, cones, ellipsoids, paraboloids, and hyperboloids. Level curves and cross sections. Ruled surfaces.
Functions of two variables. Domain of definition. Graph. Level curves and cross sections. Limits and continuity for functions of two variables. Partial derivatives. Tangent plane at a point to the graph of a function of two variables. Directional derivative. Differentiability. Gradient of a function of two variables. Geometric properties. Direction of maximum slope. Higher-order derivatives.
Study of critical points of a function of two variables. The matrix of second derivatives and its Hessian determinant. Maxima, minima, and saddle points.
Integrals of functions of two variables as limits of Riemann sums. Iterated integrals and their use in the computation of integrals of functions of two variables.
Vector-valued functions and parametric curves. Examples of parametric curves: lines, conic sections, spirals, and cycloids. Tangent, normal, and binormal unit vectors to a curve. Frenet formulas. Curvature and torsion. Curves on surfaces. Cylindrical helix.
• Stewart, Calcolo, funzioni in più variabili. Apogeo.
• O qualunque altro testo di livello universitario, ad esempio:
• Bramanti-Pagani-Salsa, Calcolo infinitesimale e algebra lineare Seconda edizione
• Salsa- Squellati, Esercizi di matematica volume 1 e volume 2.
• Caliò-Lazzari, Elements of Mathematics with numerical applications.
Esculapio
• R. Adams, Calcolo Differenziale 2, (funzioni di più variabili), quarta edizione.
Ambrosiana
Programme
Sets of points in the plane and in three-dimensional space. Two- and three-dimensional vector spaces. Vectors and unit vectors. Scalar (dot), vector (cross), and triple products and their geometric meaning. Matrices and determinants.Parametric and Cartesian equations of a plane. Parametric and Cartesian equations of a line in space. Distance from a point to a line. Distance between two lines. Intersections between lines and planes. Intersecting, parallel, and skew lines.
Quadric surfaces. Cylinders, cones, ellipsoids, paraboloids, and hyperboloids. Level curves and cross sections. Ruled surfaces.
Functions of two variables. Domain of definition. Graph. Level curves and cross sections. Limits and continuity for functions of two variables. Partial derivatives. Tangent plane at a point to the graph of a function of two variables. Directional derivative. Differentiability. Gradient of a function of two variables. Geometric properties. Direction of maximum slope. Higher-order derivatives.
Study of critical points of a function of two variables. The matrix of second derivatives and its Hessian determinant. Maxima, minima, and saddle points.
Integrals of functions of two variables as limits of Riemann sums. Iterated integrals and their use in the computation of integrals of functions of two variables.
Vector-valued functions and parametric curves. Examples of parametric curves: lines, conic sections, spirals, and cycloids. Tangent, normal, and binormal unit vectors to a curve. Frenet formulas. Curvature and torsion. Curves on surfaces. Cylindrical helix.
Core Documentation
Bibliography:• Stewart, Calcolo, funzioni in più variabili. Apogeo.
• O qualunque altro testo di livello universitario, ad esempio:
• Bramanti-Pagani-Salsa, Calcolo infinitesimale e algebra lineare Seconda edizione
• Salsa- Squellati, Esercizi di matematica volume 1 e volume 2.
• Caliò-Lazzari, Elements of Mathematics with numerical applications.
Esculapio
• R. Adams, Calcolo Differenziale 2, (funzioni di più variabili), quarta edizione.
Ambrosiana
Attendance
Students must have attended at least 75% of the lectures in order to be admitted to the examination.Type of evaluation
At the student’s choice: two midterm tests plus an oral examination in which an in-depth topic is presented; a written examination plus an oral examination in which an in-depth topic is presented. teacher profile teaching materials
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.
or a choosen textbook at university level, for example:
R. Adams “Calcolo Differenziale 2, (funzioni di più variabili)”, quarta edizione, ed. Ambrosiana
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 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.
Core Documentation
James Stewart “Calcolo - Funzioni di più variabili”, Ed. Maggiolior a choosen textbook at university level, for example:
R. Adams “Calcolo Differenziale 2, (funzioni di più variabili)”, quarta edizione, ed. Ambrosiana
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 Alcune voci matematiche nell’Enciclopedia Treccani L’America dimenticata. I rapporti tra le civiltà e un errore di Tolomeo, di Lucio RussoAttendance
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 some exercises to assess students understanding of concepts and their autonomous application. teacher profile teaching materials
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
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 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.
Core Documentation
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
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 Alcune voci matematiche nell’Enciclopedia Treccani L’America dimenticata. I rapporti tra le civiltà e un errore di Tolomeo, di Lucio RussoType 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.Attendance
Attendance at the course is mandatory for its 75% .Type of evaluation
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. teacher profile teaching materials
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.
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:
R. Adams “Calcolo Differenziale 2, (funzioni di più variabili)”, quarta edizione, ed. Ambrosiana
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 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.
A mathematical topic at a choice, to be developed autonomously, from a given list of lectures.
Core Documentation
James Stewart “Calcolo - Funzioni di più variabili”, Ed. Maggiolior a choosen textbook at university level, for example:
R. Adams “Calcolo Differenziale 2, (funzioni di più variabili)”, quarta edizione, ed. Ambrosiana
Bramanti-Pagani-Salsa: "Calcolo infinitesimale e algebra lineare", Seconda edizione, ed. Zanichelli
G.B. Thomas, R.L. Finney “Analisi Matematica”, ed. Zanichelli
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 some exercises to assess students understanding of concepts and their autonomous application.