In mathematicsthe exterior product or wedge product of vectors is an algebraic construction used in geometry to study areasvolumesand their higher-dimensional analogues. One way to visualize a bivector is as a family of parallelograms all lying in the same plane, having the same area, and with the same orientation —a choice of clockwise or counterclockwise. When regarded in this manner, the exterior bacheca incontri altedo of two vectors is called a 2-blade. More generally, the exterior product of any number k of vectors can be defined and is sometimes called a k -blade. It lives in a space known as the k th exterior power. The magnitude of the resulting k -blade is the volume of the k -dimensional parallelotope whose edges are the given vectors, just as the magnitude of incontri olimpici algebra scalar triple product of vectors in three dimensions gives the volume of the parallelepiped generated by those vectors. The exterior algebraor Grassmann algebra after Hermann Grassmann is the algebraic system whose product is the exterior product. The exterior algebra provides an algebraic setting in which to answer geometric questions. For instance, blades have a concrete geometric interpretation, and objects incontri olimpici algebra the exterior algebra can be manipulated according to a set of unambiguous rules. The exterior algebra contains objects that are not only k -blades, but sums of k -blades; such a sum is called a k -vector.Navigation menu
In terms of the coproduct, the exterior product on the dual space is just the graded dual of the coproduct:. Differential forms are mathematical objects that represent infinitesimal areas of infinitesimal parallelograms and higher-dimensional bodies , and so can be integrated over surfaces and higher dimensional manifolds in a way that generalizes the line integrals from calculus. In other words, the exterior product provides a basis-independent formulation of area. The fact that this coefficient is the signed area is not an accident. A short while later, Alfred North Whitehead , borrowing from the ideas of Peano and Grassmann, introduced his universal algebra. It is one of these more general constructions where the exterior algebra finds one of its most important applications, where it appears as the algebra of differential forms that is fundamental in areas that use differential geometry. In fact, in the presence of a positively oriented orthonormal basis , the exterior product generalizes these geometric notions to higher dimensions. Differential forms play a major role in diverse areas of differential geometry. These ideas can be extended not just to matrices but to linear transformations as well: All results obtained from other definitions of the determinant, trace and adjoint can be obtained from this definition since these definitions are equivalent.
Esercizi di Algebra Incontri Olimpici - Montecatini Terme Esercizio 1. Sia p(x) un polinomio a coe cienti interi tale che p(1) = 7 e p(7) = 1. Incontri Olimpici Stage per Insegnanti su argomenti di matematica olimpica Dipartimento di Matematica "wishwantwear.com" - Viale Morgagni 67/A Firenze, Dicembre ALGEBRA Prof. Paolo Gronchi (Università di Firenze) Video Alessandra Caraceni (SNS, Pisa) Video. Gli Incontri Olimpici sono rivolti a docenti della scuola secondaria. Le quattro giornate sono dedicate ai quattro argomenti in cui possono essere suddivisi gli argomenti tipici delle competizioni matematiche: algebra, aritmetica (teoria dei numeri), combinatoria e geometria. Incontri Olimpici Stage per insegnanti su argomenti di matematica olimpica Aemilia Hotel - Bologna Lunedì 14/10 – Tema della giornata: ALGEBRA – Prof. Emanuele Callegari (Univ. di Roma “Tor Vergata”) – Prof. Devit Abriani (Univ. di Urbino).