×

zbMATH — the first resource for mathematics

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

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ralph Henstock, Theory of integration, Butterworths, London, 1963. · Zbl 0154.05001
[2] Ralph Henstock, Majorants in variational integration, Canad. J. Math. 18 (1966), 49 – 74. · Zbl 0138.27702
[3] Ralph Henstock, A Riemann-type integral of Lebesgue power, Canad. J. Math. 20 (1968), 79 – 87. · Zbl 0171.01804
[4] Ralph Henstock, A problem in two-dimensional integration, J. Austral. Math. Soc. Ser. A 35 (1983), no. 3, 386 – 404. · Zbl 0549.26007
[5] Jiří Jarník and Jaroslav Kurzweil, A nonabsolutely convergent integral which admits transformation and can be used for integration on manifolds, Czechoslovak Math. J. 35(110) (1985), no. 1, 116 – 139. · Zbl 0614.26007
[6] Jiří Jarník, Jaroslav Kurzweil, and Štefan Schwabik, On Mawhin’s approach to multiple nonabsolutely convergent integral, Časopis Pěst. Mat. 108 (1983), no. 4, 356 – 380 (English, with Russian summary). · Zbl 0555.26004
[7] Jaroslav Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J. 7 (82) (1957), 418 – 449 (Russian). · Zbl 0090.30002
[8] Jaroslav Kurzweil, Nichtabsolut konvergente Integrale, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 26, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1980 (German). With English, French and Russian summaries. · Zbl 0441.28001
[9] L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. · Zbl 0281.10001
[10] Peng Yee Lee and Naak In Wittaya, A direct proof that Henstock and Denjoy integrals are equivalent, Bull. Malaysian Math. Soc. (2) 5 (1982), no. 1, 43 – 47. · Zbl 0501.26006
[11] Jean Mawhin, Generalized Riemann integrals and the divergence theorem for differentiable vector fields, E. B. Christoffel (Aachen/Monschau, 1979) Birkhäuser, Basel-Boston, Mass., 1981, pp. 704 – 714.
[12] Jean Mawhin, Generalized multiple Perron integrals and the Green-Goursat theorem for differentiable vector fields, Czechoslovak Math. J. 31(106) (1981), no. 4, 614 – 632. · Zbl 0562.26004
[13] Robert M. McLeod, The generalized Riemann integral, Carus Mathematical Monographs, vol. 20, Mathematical Association of America, Washington, D.C., 1980. · Zbl 0486.26005
[14] James R. Munkres, Elementary differential topology, Lectures given at Massachusetts Institute of Technology, Fall, vol. 1961, Princeton University Press, Princeton, N.J., 1966. · Zbl 0161.20201
[15] Washek F. Pfeffer, Une intégrale riemannienne et le théorème de divergence, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 8, 299 – 301 (French, with English summary). · Zbl 0574.26009
[16] W. F. Pfeffer, The multidimensional fundamental theorem of calculus (to appear). · Zbl 0638.26011
[17] Walter Rudin, Principles of mathematical analysis, 3rd ed., McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976. International Series in Pure and Applied Mathematics. · Zbl 0346.26002
[18] Stanisław Saks, Theory of the integral, Second revised edition. English translation by L. C. Young. With two additional notes by Stefan Banach, Dover Publications, Inc., New York, 1964. · Zbl 1196.28001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.