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Parametric representation

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of a function

The specification of a function , say defined on , by means of a pair of functions , say on , for which has a single-valued inverse such that , that is, for any ,

Example. The pair of functions , , , is a parametric representation of the function , .

If at a point a parametric representation of is differentiable, that is, and are differentiable, and if , then is differentiable at and . Furthermore, if and have at derivatives of order , then has a derivative of order at , which is a fractional-rational function of the derivatives of and of orders , , where in the denominator there stands the -th power of ; for example,


Comments

The functions need not be real, the same as above holds for complex functions (i.e. , ).

References

[a1] T.M. Apostol, "Calculus" , 1–2 , Blaisdell (1967)
How to Cite This Entry:
Parametric representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parametric_representation&oldid=48125
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article