Parametric representation
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
,
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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,
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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) |
Parametric representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parametric_representation&oldid=48125