Politecnico di Torino
Anno Accademico 2015/16
04KWYLI, 04KWYJM, 04KWYLM, 04KWYLP, 04KWYLS, 04KWYLX, 04KWYLZ, 04KWYMA, 04KWYMB, 04KWYMC, 04KWYMH, 04KWYMK, 04KWYMO, 04KWYMQ, 04KWYNX, 04KWYOD, 04KWYPC, 04KWYPI, 04KWYPL
Geometry
Corso di Laurea in Ingegneria Dell'Autoveicolo (Automotive Engineering) - Torino
Corso di Laurea in Ingegneria Meccanica (Mechanical Engineering) - Torino
Corso di Laurea in Ingegneria Informatica (Computer Engineering) - Torino
Espandi...
 Docente Qualifica Settore Lez Es Lab Tut Anni incarico Carlini Enrico O2 MAT/03 80 20 0 0 3 Vaccarino Francesco A2 MAT/03 80 20 0 0 5
 SSD CFU Attivita' formative Ambiti disciplinari MAT/03MAT/05 7.52.5 A - Di baseA - Di base Matematica, informatica e statisticaMatematica, informatica e statistica
 Presentazione The course Geometry has two principal aims. The first one is teaching students how to analyse a problem and solve it through a deductive path using a mathematical language. The second aim is to present an array of topics spread over Linear Algebra, Elementary, Differential Geometry and the Theory of Multivariable Functions, which thus build a bridge to the Analysis I and II courses. Part of the subject matter is in fact dedicated to extending differential calculus from one to several variables, and touches upon the basic applications. Moreover, the Geometry of Euclidean spaces is a necessary tool or integral calculus, as treated in Analysis II. Risultati di apprendimento attesi Ability to follow proofs; understanding of the essential properties of elementary geometry of space, the basics of vector spaces plus continuity and differenziability of multivariable functions. Prerequisiti / Conoscenze pregresse Real and complex numbers, trigonometry, equations and inequalities of degree one and two, differential calculus in one variable. Programma VECTORS The notion of a vector and operations. Dot product, distances, angles. Cross (wedge) product, triple (mixed) product, areas and volumes. Geometry of the plane, lines and conics. Equations of a plane and of a line in the space. MATRICES AND SYSTEM Matrices, sum and product of matrices. Powers and inverse of square matrices. Systems of equations, matrix form of a system. Rouchč-Capelli theorem. Linear independence in R^n. Linear combinations and subspaces spanned by vectors. Subspaces in R^n, bases. Spaces generated by rows or columns of matrices. LINEAR MAPS. The notion of vector space. Linear maps of vector spaces. Isomorphism. Matrices associated to linear maps. Kernel and image. Rank and nullity. Operations of subspaces. Sum and direct sum of subspaces. DIAGONALISATION. Endomorphism, change of bases. Eigenvalues, determinant of square matrix. Eigenvectors, subspaces, multiplicity. Symmetric matrices and quadratic form. Orthogonal matrices. Diagonalisability of square and symmetric matrices. DIFFERENTIAL CALCULUS. Curves in space. Tangent vector and tangent line. Length of a curve and arc length. Functions over R^n. Rudiments of topology. Limits and continuity. Partial derivatives and gradient. Vector-valued functions. Jacobian matrix. FUNCTION OF TWO VARIABLES. The graph of a map in two variables. Paraboloids. Differentiability and tangent plane. Second derivatives, Hessian matrix. Expansions in Taylor series. Relative maxima and minima. Critical points. SURFACES AND QUADRICS. Surfaces of revolution, spheres. Ellipsoids, hyperboloids. Rotations in the plane and in space. Quadrics, lines, planes. Parametric surfaces. Normal vector, tangent plane and normal line. Organizzazione dell'insegnamento The course consists of lectures and exercises classes in team. Testi richiesti o raccomandati: letture, dispense, altro materiale didattico Course materials will be communicated by professors in class. We will anyway provide a list of course materials that are generally used. Teoria 1) L. Gatto, Lezioni di Algebra lineare e Geometria, Clut 2013. 2) S.Greco, P. Valabrega, Lezioni di Geometria, Vol. 1 Algebra lineare, Vol. 2 Geometria Analitica, Ed. Levrotto e Bella, Torino 2009. 3) A. Sanini, Lezioni di Geometria, Ed. Levrotto e Bella, Torino 1993. 4) C. Canuto, A. Tabacco, Mathematical Analysis II, Springer-Verlag 2015. 5) Wilfred Kaplan; Donald J. Lewis, Calculus and Linear Algebra. Vol. 1: Vectors in the Plane and One-Variable Calculus; freely downloadable at the web address http://quod.lib.umich.edu/s/spobooks/5597602.0001.001/--calculus-and-linear-algebra-vol-1-vectors-in-the-plane 6) Wilfred Kaplan; Donald J. Lewis, Calculus and Linear Algebra. Vol. 2: Vector Spaces, Many-Variable Calculus, and Differential Equations; freely downloadable at the web address http://quod.lib.umich.edu/s/spobooks/5597602.0002.001/--calculus-and-linear-algebra-vol-2-vector-spaces-many Esercizi 1) E. Carlini, 50 quiz di Geometria, Celid 2011. 2) G. Casnati, M.L. Spreafico, Allenamenti di Geometria, Ed. Esculapio, Bologna 2013. 3) J. Cordovez, Chissą chi lo sa?, Clut 2013. 4) A. Sanini, Esercizi di Geometria, Ed. Levrotto e bella, Torino 1993. Useful course materials are available online on the website of Politecnico di Torino or directly on the webpages of single Courses of Study. Criteri, regole e procedure per l'esame The final evaluation will be made with a written examination (10 multiple choices questions and 2 exercises). However, an oral colloquium can be request by the teacher. Orario delle lezioni Statistiche superamento esami Programma definitivo per l'A.A.2015/16

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