Teacher
|
Munday Sara Ann
(syllabus)
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
(reference books)
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.
|