


Politecnico di Torino  
Academic Year 2015/16  
18BCGLZ, 18BCGLN, 18BCGLP, 18BCGLS, 18BCGLX, 18BCGMA, 18BCGMB, 18BCGMC, 18BCGMH, 18BCGMK, 18BCGMN, 18BCGMO, 18BCGMQ, 18BCGNX, 18BCGOA, 18BCGOD, 18BCGPC, 18BCGPI, 18BCGPL Geometry 

1st degree and Bachelorlevel of the Bologna process in Aerospace Engineering  Torino 1st degree and Bachelorlevel of the Bologna process in Automotive Engineering  Torino 1st degree and Bachelorlevel of the Bologna process in Electronic And Communications Engineering  Torino Espandi... 





Subject fundamentals
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 Mathematical Analysis II.

Expected learning outcomes
Ability to follow proofs; understanding of the essential properties of elementary geometry of space, the basics of vector spaces plus continuity and differentiability of multivariable functions.

Prerequisites / Assumed knowledge
Real and complex numbers, trigonometry, equations and inequalities of degree one and two, differential calculus in one variable.

Contents
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. RouchčCapelli 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. Vectorvalued 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. 
Delivery modes
The course consists of lectures and exercises classes in team.

Texts, readings, handouts and other learning resources
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, SpringerVerlag 2015. 5) Wilfred Kaplan; Donald J. Lewis, Calculus and Linear Algebra. Vol. 1: Vectors in the Plane and OneVariable Calculus; freely downloadable at the web address http://quod.lib.umich.edu/s/spobooks/5597602.0001.001/calculusandlinearalgebravol1vectorsintheplane 6) Wilfred Kaplan; Donald J. Lewis, Calculus and Linear Algebra. Vol. 2: Vector Spaces, ManyVariable Calculus, and Differential Equations; freely downloadable at the web address http://quod.lib.umich.edu/s/spobooks/5597602.0002.001/calculusandlinearalgebravol2vectorspacesmany 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. 
Assessment and grading criteria
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.

