Difference between revisions of "Variation of a mapping"
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| + | $#C+1 = 33 : ~/encyclopedia/old_files/data/V096/V.0906130 Variation of a mapping | ||
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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 | |
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| − | + | $$ | |
| + | \alpha : x = f( u, v),\ y = \phi ( u, v), | ||
| + | $$ | ||
| − | where | + | 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|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 $[[#References|[2]]]. | ||
====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">[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> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)</TD></TR></table> | ||
Latest revision as of 13:07, 6 January 2024
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) |
| [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=15325
Variation of a mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Variation_of_a_mapping&oldid=15325
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article