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Function parameterization guess

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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

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Function parameterization guess Narasimham <mathma18@gmail.com> - 2014-04-27 06:07 +0000

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