Variation of a mapping

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)