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A numerical characteristic of a mapping connected with its differentiability properties. Defined by S. Banach [[#References|[1]]]. The definition given below applies to the two-dimensional case only. Consider the mapping
 
A numerical characteristic of a mapping connected with its differentiability properties. Defined by S. Banach [[#References|[1]]]. The definition given below applies to the two-dimensional case only. Consider the mapping
  
<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/v/v096/v096130/v0961301.png" /></td> </tr></table>
+
$$
 +
\alpha : = f( u, v),\  y  = \phi ( u, v),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v0961302.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v0961303.png" /> are continuous functions on the square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v0961304.png" />. One says that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v0961305.png" /> is of bounded variation if there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v0961306.png" /> such that for any sequences non-intersecting squares <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v0961307.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v0961308.png" />), with sides parallel to the coordinate axes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v0961309.png" />, the inequality
+
where $  f $
 +
and $  \phi $
 +
are continuous functions on the square $  D _ {0} = [ 0, 1] \times [ 0, 1] $.  
 +
One says that the mapping $  \alpha $
 +
is of bounded variation if there exists a number $  M > 0 $
 +
such that for any sequences non-intersecting squares $  D  ^ {i} \subset  D _ {0} $(
 +
$  i = 1, 2 , . . . $),  
 +
with sides parallel to the coordinate axes $  u , v $,  
 +
the inequality
  
<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/v/v096/v096130/v09613010.png" /></td> </tr></table>
+
$$
 +
\sum _ { i }  \mathop{\rm mes}  D _ {xy}  ^ {i}  \leq  M
 +
$$
  
is true. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v09613011.png" /> denotes the image of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v09613012.png" /> under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v09613013.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v09613014.png" /> is the plane [[Lebesgue measure|Lebesgue measure]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v09613015.png" />. The numerical value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v09613016.png" /> of the variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v09613017.png" /> may be determined in various ways. For instance, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v09613018.png" /> be of bounded variation. The variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v09613019.png" /> may then be determined by the formula
+
is true. Here $  E _ {xy} $
 +
denotes the image of a set $  E \subset  D _ {0} $
 +
under the mapping $  \alpha $,  
 +
and $  \mathop{\rm mes}  E $
 +
is the plane [[Lebesgue measure|Lebesgue measure]] of $  E $.  
 +
The numerical value $  V( \alpha ) $
 +
of the variation of $  \alpha $
 +
may be determined in various ways. For instance, let $  \alpha $
 +
be of bounded variation. The variation $  V ( \alpha ) $
 +
may then be determined by the formula
  
<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/v/v096/v096130/v09613020.png" /></td> </tr></table>
+
$$
 +
V( \alpha )  = \int\limits _ {- \infty } ^ { {+ }  \infty }
 +
\int\limits _ {- \infty } ^ { {+ }  \infty } N( s, t)  ds  dt ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v09613021.png" /> is the number of solutions of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v09613022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v09613023.png" /> (the Banach indicatrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v09613024.png" />).
+
where $  N( s, t) $
 +
is the number of solutions of the system $  f( u, v) = s $,
 +
$  \phi ( u, v) = t $(
 +
the Banach indicatrix of $  \alpha $).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v09613025.png" /> is of bounded variation, then, almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v09613026.png" />, the generalized Jacobian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v09613027.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v09613028.png" />) exists, and it is integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v09613029.png" />; also,
+
If $  \alpha $
 +
is of bounded variation, then, almost-everywhere on $  D _ {0} $,  
 +
the generalized Jacobian $  J( P) $(
 +
$  P \in {D _ {0} } $)  
 +
exists, and it is integrable on $  D _ {0} $;  
 +
also,
  
<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/v/v096/v096130/v09613030.png" /></td> </tr></table>
+
$$
 +
J( P)  = \lim\limits _ { \mathop{\rm mes}  K \rightarrow 0 }
 +
 +
\frac{ \mathop{\rm mes}  K _ {xy} }{ \mathop{\rm mes}  K }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v09613031.png" /> is a square containing the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v09613032.png" /> with sides parallel to the axes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096130/v09613033.png" /> [[#References|[2]]].
+
where $  K \subset  D _ {0} $
 +
is a square containing the point $  P \in D _ {0} $
 +
with sides parallel to the axes $  u , v $[[#References|[2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Banach,  "Sur les lignes rectifiables et les surfaces dont l'aire est finie"  ''Fund. Math.'' , '''7'''  (1925)  pp. 225–236</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.D. Kudryavtsev,  "The variation of mappings in regions" , ''Metric questions in the theory of functions and mappings'' , '''1''' , Kiev  (1969)  pp. 34–108  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Banach,  "Sur les lignes rectifiables et les surfaces dont l'aire est finie"  ''Fund. Math.'' , '''7'''  (1925)  pp. 225–236</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.D. Kudryavtsev,  "The variation of mappings in regions" , ''Metric questions in the theory of functions and mappings'' , '''1''' , Kiev  (1969)  pp. 34–108  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Saks,  "Theory of the integral" , Hafner  (1952)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Saks,  "Theory of the integral" , Hafner  (1952)  (Translated from French)</TD></TR></table>

Revision as of 08:27, 6 June 2020


A numerical characteristic of a mapping connected with its differentiability properties. Defined by S. Banach [1]. The definition given below applies to the two-dimensional case only. Consider the mapping

$$ \alpha : x = f( u, v),\ y = \phi ( u, v), $$

where $ f $ and $ \phi $ are continuous functions on the square $ D _ {0} = [ 0, 1] \times [ 0, 1] $. One says that the mapping $ \alpha $ is of bounded variation if there exists a number $ M > 0 $ such that for any sequences non-intersecting squares $ D ^ {i} \subset D _ {0} $( $ i = 1, 2 , . . . $), with sides parallel to the coordinate axes $ u , v $, the inequality

$$ \sum _ { i } \mathop{\rm mes} D _ {xy} ^ {i} \leq M $$

is true. Here $ E _ {xy} $ denotes the image of a set $ E \subset D _ {0} $ under the mapping $ \alpha $, and $ \mathop{\rm mes} E $ is the plane Lebesgue measure of $ E $. The numerical value $ V( \alpha ) $ of the variation of $ \alpha $ may be determined in various ways. For instance, let $ \alpha $ be of bounded variation. The variation $ V ( \alpha ) $ may then be determined by the formula

$$ V( \alpha ) = \int\limits _ {- \infty } ^ { {+ } \infty } \int\limits _ {- \infty } ^ { {+ } \infty } N( s, t) ds dt , $$

where $ N( s, t) $ is the number of solutions of the system $ f( u, v) = s $, $ \phi ( u, v) = t $( the Banach indicatrix of $ \alpha $).

If $ \alpha $ is of bounded variation, then, almost-everywhere on $ D _ {0} $, the generalized Jacobian $ J( P) $( $ P \in {D _ {0} } $) exists, and it is integrable on $ D _ {0} $; also,

$$ J( P) = \lim\limits _ { \mathop{\rm mes} K \rightarrow 0 } \frac{ \mathop{\rm mes} K _ {xy} }{ \mathop{\rm mes} K } , $$

where $ K \subset D _ {0} $ is a square containing the point $ P \in D _ {0} $ with sides parallel to the axes $ u , v $[2].

References

[1] S. Banach, "Sur les lignes rectifiables et les surfaces dont l'aire est finie" Fund. Math. , 7 (1925) pp. 225–236
[2] L.D. Kudryavtsev, "The variation of mappings in regions" , Metric questions in the theory of functions and mappings , 1 , Kiev (1969) pp. 34–108 (In Russian)

Comments

References

[a1] S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)
How to Cite This Entry:
Variation of a mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Variation_of_a_mapping&oldid=49115
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article