Function parameterization guess

[Narasimham]

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

[djmpark]

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
***@comcast.net
http://home.comcast.net/~djmpark/index.html



From: Narasimham [mailto:***@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