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)

References

[a1]	S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)