The cross product in Euclidean space IR3 is an operation in which two vectors are associated to generate a third vector, also in space IR3. This product can be studied rewriting its basic equations in a matrix structure, more specifically in terms of determinants. Such a structure allows extending, for analogy, the ideas of the cross product for a type of the product of vectors in higher dimensions, through the systematic increase of the number of rows and columns in determinants that constitute the equations. So, in a n-dimensional space with Euclidean norm, we can associate n – 1 vectors and to obtain an n-th vector, with the same geometric characteristics of the product in three dimensions. This kind of operation is also a geometric interpretation of the product defined by Eckman . The same analogies are also useful in the verification of algebraic properties of such products, based on known properties of determinants.
Recently I published a
paper in the journal ALAMT (Advances in
Linear Algebra & Matrix Theory) and explored the possibility of
obtaining products of vectors in dimensions higher than three . In
continuation to this work, it is proposed to develop, through dimensional
analogy, a vector field with notation and properties analogous to the curl, in
this case applied to the space IR4.
One can see how the similarities are obvious in relation to the algebraic properties and the geometric structures,
if the rotations are compared in spaces of three and four dimensions.
A theoretical model for the
propagation of acoustic waves in dry granular media is presented within the
framework of the nonlinear granular elasticity. An essential ingredient is the
dependence of the elastic moduli on compression. For the purpose of
illustration, we analyze the case of a time-harmonic plane wave propagation
under isotropic compression. We derive explicit relations for the wave speed
dependence with the confining pressure. The present approach provides an
accurate description of acoustic wave propagation in granular packings and
represents a powerful tool to interpret the results of current experiments.