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
-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.
-HANDS-ON: CONSTRUCTION OF POLYHEDRA, RULED SURFACES, paper models
Programme
- COURSE SYLLABUS-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.
-HANDS-ON: CONSTRUCTION OF POLYHEDRA, RULED SURFACES, paper models
Core Documentation
ANY TEXT AT THE LEVEL OF THE SECOND AND THIRD SEMESTER OF A THREE-SEMESTER COLLEGE CALCULUS.Reference Bibliography
more reading suggestions on the course page at www.formulas.itType of delivery of the course
Classes and recitationsAttendance
Attendance is compulsory for at least 75% of the hoursType of evaluation
a written exam, followed by an oral arguing conversation teacher profile teaching materials
2- and 3-dimensional vector spaces. Vectors in the plane and in 3-space. Scalar, vector and mixed products and their geometric significance.
Equations of lines and places in parametric and cartesian form; parallel, intersecting and skewed lines; intersections of planes and lines; distance between points in 3-space, distances between lines and points, points and planes, lines and planes, parallel planes.
Matrices: sum and product operations, determinants. Geometric components, determinants as scaling factors for area.
Quadric surfaces:
Paraboloids, hyperboloids, cones, cylinders, ellipsoids. Equations, sections and level curves. Inverse operations: the reconstruction (sketches and equations) of quadrics given their sections. Ruled and double-ruled surfaces. Origami used to aid the study of surfaces.
Infinitessimal calculus in three-dimensional space:
(prerequisite: calculus of one variable)
Real-valued functions of one or more variable: domain of definition; planar representation of functions z=f(x,y): level curves, sections and their graphic representation.. Surfaces with free (or cylindrical) variables. Open and closed sets; internal, external, boundary and isolated points.
Limits and continuity for functions of several variables. Counterexamples. Partial derivatives. Differentiability. Tangent planes and normal lines. Gradient of a function, relation between the gradient and the other geometrical features of a surface: level curves, tangent planes, direction of maximal slope. Taylor's formula in more than one variable. Investigation of the nature of critical points: (relative) maxima and minima, saddle points for functions of two variables, Hessian determinant. Multiple integrals: simple vertical and horizontal domain of integration; double integrals as integrals iterated over simple domains; inversion of the order of derivation; application to the calculation of areas and volumes.
A topic to develop independently, which will be a part of the oral examination, chosen from:
• Two surfaces, one saddle-like and the other an ellipsoid, made by folding a single sheet of paper.
• A topic of your choice, extracted from one of the following books:
• “flussi e riflussi” by Lucio Russo
• “le curve celebri” by Luciano Cresci
• “Project origami : activities for exploring mathematics” Thomas Hull
• “How to fold it : the mathematics of linkages, origami and polyhedra”
by Joseph O’Rourke
• Courant, Robbins, “What is Mathematics?”, Oxford University Press
• L’America dimenticata. I rapporti tra le civiltà e un errore di Tolomeo, Lucio Russo
• Sulla coclea libri quattro: (facendo discendere l’acqua, la fa salire) by Del Monte, Guidobaldo
O qualunque altro testo di livello universitario, ad esempio:
Bramanti-Pagani-Salsa: “Calcolo infinitesimale e algebra lineare Seconda edizione “
G.B. Thomas, R.L. Finney “Analisi Matematica” ed. Zanichelli (comprende la maggior parte degli argomenti delle due annualità di Matematica, ed i necessari esercizi, lo trovate in biblioteca)
Salsa- Squellati: ESERCIZI DI MATEMATICA volume 1 e volume 2.
Programme
Linear algebra using geometry:2- and 3-dimensional vector spaces. Vectors in the plane and in 3-space. Scalar, vector and mixed products and their geometric significance.
Equations of lines and places in parametric and cartesian form; parallel, intersecting and skewed lines; intersections of planes and lines; distance between points in 3-space, distances between lines and points, points and planes, lines and planes, parallel planes.
Matrices: sum and product operations, determinants. Geometric components, determinants as scaling factors for area.
Quadric surfaces:
Paraboloids, hyperboloids, cones, cylinders, ellipsoids. Equations, sections and level curves. Inverse operations: the reconstruction (sketches and equations) of quadrics given their sections. Ruled and double-ruled surfaces. Origami used to aid the study of surfaces.
Infinitessimal calculus in three-dimensional space:
(prerequisite: calculus of one variable)
Real-valued functions of one or more variable: domain of definition; planar representation of functions z=f(x,y): level curves, sections and their graphic representation.. Surfaces with free (or cylindrical) variables. Open and closed sets; internal, external, boundary and isolated points.
Limits and continuity for functions of several variables. Counterexamples. Partial derivatives. Differentiability. Tangent planes and normal lines. Gradient of a function, relation between the gradient and the other geometrical features of a surface: level curves, tangent planes, direction of maximal slope. Taylor's formula in more than one variable. Investigation of the nature of critical points: (relative) maxima and minima, saddle points for functions of two variables, Hessian determinant. Multiple integrals: simple vertical and horizontal domain of integration; double integrals as integrals iterated over simple domains; inversion of the order of derivation; application to the calculation of areas and volumes.
A topic to develop independently, which will be a part of the oral examination, chosen from:
• Two surfaces, one saddle-like and the other an ellipsoid, made by folding a single sheet of paper.
• A topic of your choice, extracted from one of the following books:
• “flussi e riflussi” by Lucio Russo
• “le curve celebri” by Luciano Cresci
• “Project origami : activities for exploring mathematics” Thomas Hull
• “How to fold it : the mathematics of linkages, origami and polyhedra”
by Joseph O’Rourke
• Courant, Robbins, “What is Mathematics?”, Oxford University Press
• L’America dimenticata. I rapporti tra le civiltà e un errore di Tolomeo, Lucio Russo
• Sulla coclea libri quattro: (facendo discendere l’acqua, la fa salire) by Del Monte, Guidobaldo
Core Documentation
R. Adams “Calcolo Differenziale 2, (funzioni di più variabili)” , quarta edizione, casa editrice AmbrosianaO qualunque altro testo di livello universitario, ad esempio:
Bramanti-Pagani-Salsa: “Calcolo infinitesimale e algebra lineare Seconda edizione “
G.B. Thomas, R.L. Finney “Analisi Matematica” ed. Zanichelli (comprende la maggior parte degli argomenti delle due annualità di Matematica, ed i necessari esercizi, lo trovate in biblioteca)
Salsa- Squellati: ESERCIZI DI MATEMATICA volume 1 e volume 2.
Reference Bibliography
R. Adams “Calcolo Differenziale 2, (funzioni di più variabili)” , quarta edizione, casa editrice Ambrosiana O qualunque altro testo di livello universitario, ad esempio: Bramanti-Pagani-Salsa: “Calcolo infinitesimale e algebra lineare Seconda edizione “ G.B. Thomas, R.L. Finney “Analisi Matematica” ed. Zanichelli (comprende la maggior parte degli argomenti delle due annualità di Matematica, ed i necessari esercizi, lo trovate in biblioteca) Salsa- Squellati: ESERCIZI DI MATEMATICA volume 1 e volume 2.Type of delivery of the course
Taught class and exercisesAttendance
Attendance is compulsory for at least 75% of the hoursType of evaluation
written test and oral exam