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

where and are continuous functions on the square . One says that the mapping is of bounded variation if there exists a number such that for any sequences non-intersecting squares (), with sides parallel to the coordinate axes , the inequality

is true. Here denotes the image of a set under the mapping , and is the plane Lebesgue measure of . The numerical value of the variation of may be determined in various ways. For instance, let be of bounded variation. The variation may then be determined by the formula

where is the number of solutions of the system , (the Banach indicatrix of ).

If is of bounded variation, then, almost-everywhere on , the generalized Jacobian () exists, and it is integrable on ; also,

where is a square containing the point with sides parallel to the axes [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)