Generalized spherical and simplicial coordinates

Elementary trigonometric quantities are defined in l2,p analogously to that in l2,2, the sine and cosine functions are generalized for eachp >0 as functions sinp and cosp such that they satisfy the basic equation |cosp(ϕ)|p + |sinp(ϕ)|p = 1. The p-generalized radius coordinate of a point ξ ∈ Rn is defined for each p >0 as rp = ( n i=1 |ξi |p)1/p. On combining these quantities, ln,p-spherical coordinates are defined. It is shown that these coordinates are nearly related to ln,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 ln,p-sphere and with the derivation of certain generalized statistical distributions. © 2007 Elsevier Inc. All rights reserved