Re: Function parameterization guess

*To*: mathgroup at smc.vnet.net*Subject*: [mg132648] Re: Function parameterization guess*From*: "djmpark" <djmpark at comcast.net>*Date*: Sun, 27 Apr 2014 21:45:24 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-outx@smc.vnet.net*Delivered-to*: mathgroup-newsendx@smc.vnet.net*References*: <8720827.39330.1398579243598.JavaMail.root@m03>

Narasimham, I'm rather wading into the edge of my knowledge, but I think that in general the answer is no, or it is often difficult to find parametrizations. If you want algebraic or rational parameterizations of algebraic equation solutions then you can't always do it. If you want to do calculus on the solution set then you can make it into a manifold if the Jacobian is nowhere zero on the solution set. But then it will often require a number of charts in an atlas that will cover the manifold and this can be done in an infinite number of ways. For example, the {Cos[t], Sin[t}} parameterization is not adequate because it is not a 1-1 map of R^1 to the circle. t = 0 and t = 2 Pi give the same point on the circle and that ruins the calculus. You would need two overlapping angular domains, or four projections to the axes, or two stereographic projections. So generally it's a difficult or at least a nitty-gritty problem. David Park djmpark at comcast.net http://home.comcast.net/~djmpark/index.html From: Narasimham [mailto:mathma18 at gmail.com] Using excellent function capabilities of Mathematica is it not possible to generally guess or propose some standard parameterizations of components given functions? For two variables and single parameter. Given x^2 + y^2 =1 we have {x,y}= {Cos[t],Sin[t]} and its variants {Sech[t],Tanh[t]}among others are solutions. For three variables and two parameters. Given x^2 + y^2 - z^2 =1 we have Cosh[u] Cos[v], Cosh[u] Sin[v], Sinh[u] and variants.. The number of parametric set variations for component variables is not infinite, can be indicated with an arbitrary constant. A general or possible sub parameterization may be considered for each functional relationship. Regards Narasimham