Generalized spherical and simplicial coordinates

[摘要] Elementary trigonometric quantities are defined in l(2),p analogously to that in l(2,2), the sine and cosine functions are generalized for each p > 0 as functions sin(p) and cos(p) such that they satisfy the basic equation vertical bar cos(p) (phi)vertical bar(p) + vertical bar sin(p) (phi)vertical bar(p) = 1. The p-generalized radius coordinate of a point xi is an element of R-n is defined for each p > 0 as r(p) = (Sigma(n)(i)= 1 vertical bar xi(i)vertical bar(p))(1/p) . On combining these quantities, l(n, p)-spherical coordinates are defined. It is shown that these coordinates are nearly related to l(n, p)-simplicial coordinates. The Jacobians of these generalized coordinate transformations are derived. Applications and interpretations from analysis deal especially with the definition of a generalized surface content on ln,p-spheres which is nearly related to a modified co-area formula and an extension of Cavalieri's and Torricelli's indivisibeln method, and with differential equations. Applications from probability theory deal especially with a geometric interpretation of the uniform probability distribution on the l(n, p)- sphere and with the derivation of certain generalized statistical distributions. (c) 2007 Elsevier Inc. All rights reserved.

[发布日期] 2007-12-15

[关键词] generalized radius coordinate;generalized trigonometric functions;l(2,p)-trigonometry;p-generalized pythagoras type equation;l(2,p)-generalized polar coordinates;l(n,p)-spherical coordinates;l(2,p) -generalized triangle coordinates;l(n,p)-simplicial coordinates;Jacobians;l(n,p)-ball volume;l(n,p)-generalized indivisibeln method;generalized surface content;l(n,p)-generalized uniform distribution on the sphere;modified co-area formula;disintegration of Lebesgue measure;p-generalized chi(2)- and Student-distributions;l(n,p)-norm symmetric distributions