Namespaces
Variants
Actions

Difference between revisions of "Parametric representation"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
''of a function''
+
<!--
 +
p0715301.png
 +
$#A+1 = 39 n = 0
 +
$#C+1 = 39 : ~/encyclopedia/old_files/data/P071/P.0701530 Parametric representation
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
The specification of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p0715301.png" />, say defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p0715302.png" />, by means of a pair of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p0715303.png" />, say on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p0715304.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p0715305.png" /> has a single-valued inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p0715306.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p0715307.png" />, that is, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p0715308.png" />,
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p0715309.png" /></td> </tr></table>
+
''of a function''
  
Example. The pair of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153012.png" />, is a parametric representation of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153014.png" />.
+
The specification of a function  $  f $,
 +
say defined on  $  [ a, b] $,
 +
by means of a pair of functions $  \phi , \psi $,
 +
say on  $  [ \alpha , \beta ] $,
 +
for which  $  \phi : [ \alpha , \beta ] \rightarrow [ a, b] $
 +
has a single-valued inverse  $  \phi  ^ {-} 1 : [ a, b] \rightarrow [ \alpha , \beta ] $
 +
such that  $  f = \psi \circ \phi  ^ {-} 1 $,  
 +
that is, for any  $  x \in [ a, b] $,
  
If at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153015.png" /> a parametric representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153016.png" /> is differentiable, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153018.png" /> are differentiable, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153019.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153020.png" /> is differentiable at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153022.png" />. Furthermore, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153024.png" /> have at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153025.png" /> derivatives of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153027.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153028.png" /> has a derivative of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153029.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153030.png" />, which is a fractional-rational function of the derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153032.png" /> of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153034.png" />, where in the denominator there stands the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153035.png" />-th power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153036.png" />; for example,
+
$$
 +
f( x)  = \psi [ \phi  ^ {-} 1 ( x)].
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153037.png" /></td> </tr></table>
+
Example. The pair of functions  $  x = \cos  t $,
 +
$  y = \sin  t $,
 +
$  0 \leq  t \leq  \pi $,
 +
is a parametric representation of the function  $  y = \sqrt {1- x  ^ {2} } $,
 +
- 1 \leq  x \leq  1 $.
  
 +
If at a point  $  t _ {0} \in [ \alpha , \beta ] $
 +
a parametric representation of  $  f $
 +
is differentiable, that is,  $  \phi $
 +
and  $  \psi $
 +
are differentiable, and if  $  \phi  ^  \prime  ( t _ {0} ) \neq 0 $,
 +
then  $  f $
 +
is differentiable at  $  x _ {0} = \phi ( t _ {0} ) $
 +
and  $  f ^ { \prime } ( x _ {0} ) = \psi  ^  \prime  ( t _ {0} )/ \phi  ^  \prime  ( t _ {0} ) $.
 +
Furthermore, if  $  \phi $
 +
and  $  \psi $
 +
have at  $  t _ {0} $
 +
derivatives of order  $  n $,
 +
$  n = 2, 3 \dots $
 +
then  $  f $
 +
has a derivative of order  $  n $
 +
at  $  x _ {0} $,
 +
which is a fractional-rational function of the derivatives of  $  \phi $
 +
and  $  \psi $
 +
of orders  $  k $,
 +
$  k = 1 \dots n $,
 +
where in the denominator there stands the  $  ( 2n- 1) $-
 +
th power of  $  \phi  ^  \prime  ( t _ {0} ) $;
 +
for example,
  
 +
$$
 +
f ^ { \prime\prime } ( x _ {0} )  = 
 +
\frac{\psi  ^ {\prime\prime} ( t _ {0} ) \phi  ^  \prime  ( t _ {0} ) - \psi
 +
^  \prime  ( t _ {0} ) \phi  ^ {\prime\prime} ( t _ {0} ) }{[ \phi  ^  \prime  ( t _ {0} )]  ^ {3} }
 +
.
 +
$$
  
 
====Comments====
 
====Comments====
The functions need not be real, the same as above holds for complex functions (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071530/p07153039.png" />).
+
The functions need not be real, the same as above holds for complex functions (i.e. $  f: D \rightarrow \mathbf C $,  
 +
$  D \subset  \mathbf C $).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,  "Calculus" , '''1–2''' , Blaisdell  (1967)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,  "Calculus" , '''1–2''' , Blaisdell  (1967)</TD></TR></table>

Revision as of 08:05, 6 June 2020


of a function

The specification of a function $ f $, say defined on $ [ a, b] $, by means of a pair of functions $ \phi , \psi $, say on $ [ \alpha , \beta ] $, for which $ \phi : [ \alpha , \beta ] \rightarrow [ a, b] $ has a single-valued inverse $ \phi ^ {-} 1 : [ a, b] \rightarrow [ \alpha , \beta ] $ such that $ f = \psi \circ \phi ^ {-} 1 $, that is, for any $ x \in [ a, b] $,

$$ f( x) = \psi [ \phi ^ {-} 1 ( x)]. $$

Example. The pair of functions $ x = \cos t $, $ y = \sin t $, $ 0 \leq t \leq \pi $, is a parametric representation of the function $ y = \sqrt {1- x ^ {2} } $, $ - 1 \leq x \leq 1 $.

If at a point $ t _ {0} \in [ \alpha , \beta ] $ a parametric representation of $ f $ is differentiable, that is, $ \phi $ and $ \psi $ are differentiable, and if $ \phi ^ \prime ( t _ {0} ) \neq 0 $, then $ f $ is differentiable at $ x _ {0} = \phi ( t _ {0} ) $ and $ f ^ { \prime } ( x _ {0} ) = \psi ^ \prime ( t _ {0} )/ \phi ^ \prime ( t _ {0} ) $. Furthermore, if $ \phi $ and $ \psi $ have at $ t _ {0} $ derivatives of order $ n $, $ n = 2, 3 \dots $ then $ f $ has a derivative of order $ n $ at $ x _ {0} $, which is a fractional-rational function of the derivatives of $ \phi $ and $ \psi $ of orders $ k $, $ k = 1 \dots n $, where in the denominator there stands the $ ( 2n- 1) $- th power of $ \phi ^ \prime ( t _ {0} ) $; for example,

$$ f ^ { \prime\prime } ( x _ {0} ) = \frac{\psi ^ {\prime\prime} ( t _ {0} ) \phi ^ \prime ( t _ {0} ) - \psi ^ \prime ( t _ {0} ) \phi ^ {\prime\prime} ( t _ {0} ) }{[ \phi ^ \prime ( t _ {0} )] ^ {3} } . $$

Comments

The functions need not be real, the same as above holds for complex functions (i.e. $ f: D \rightarrow \mathbf C $, $ D \subset \mathbf C $).

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=18063
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article