The divergence theorem. (English) Zbl 0596.26007

In 1981, the reviewer [Czech. Math. J. 31(106), 614-632 (1981; Zbl 0562.26004)] has introduced a non-uniformity with respect to the irregularity of the partitions in the Kurzweil-Henstock definition of integral over an interval of \(R^ n\). He obtained in this way an integral allowing a divergence theorem for merely differentiable vector fields. Unfortunately, integrability in this sense over two abutting intervals does not imply integrability over their union.
The present paper proposes a clever modification of the concept of irregularity (presented here in terms of an inverse concept of regularity) which overcomes the above difficulty without loosing any of the other interesting properties of the above integral. Moreover, this modification even allows to prove a divergence theorem when the vector field lacks differentiability on some ”sufficiently small” subsets of the interval. The paper also discusses the relationship of the new integral to Lebesgue’s one and a change of variable formula under some restricted class of transformations. It also contains a number of interesting remarks and open questions.
Reviewer: J.Mawhin


26B20 Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.)
26B15 Integration of real functions of several variables: length, area, volume
26A39 Denjoy and Perron integrals, other special integrals


Zbl 0562.26004
Full Text: DOI


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