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Transactions of the American Mathematical Society

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The divergence theorem


Author: W. F. Pfeffer
Journal: Trans. Amer. Math. Soc. 295 (1986), 665-685
MSC: Primary 26B20; Secondary 26A42, 26B15
MathSciNet review: 833702
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Abstract: We define a well-behaved multidimensional Riemann type integral such that the divergence of any vector field continuous in a compact interval and differentiable in its interior is integrable, and the integral equals the flux of the vector field out of the interval.


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  • [H $ _{\mathbf{1}}$] Ralph Henstock, Theory of integration, Butterworths, London, 1963. MR 0158047
  • [H $ _{\mathbf{2}}$] Ralph Henstock, Majorants in variational integration, Canad. J. Math. 18 (1966), 49–74. MR 0185075
  • [H $ _{\mathbf{3}}$] Ralph Henstock, A Riemann-type integral of Lebesgue power, Canad. J. Math. 20 (1968), 79–87. MR 0219675
  • [H $ _{\mathbf{4}}$] Ralph Henstock, A problem in two-dimensional integration, J. Austral. Math. Soc. Ser. A 35 (1983), no. 3, 386–404. MR 712816
  • [JK] 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. MR 779340
  • [JKS] 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). MR 727536
  • [K $ _{\mathbf{1}}$] Jaroslav Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J. 7 (82) (1957), 418–449 (Russian). MR 0111875
  • [K $ _{\mathbf{2}}$] 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. MR 597703
  • [KN] L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0419394
  • [LW] 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. MR 683810
  • [M $ _{\mathbf{1}}$] 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. MR 661109
  • [M $ _{\mathbf{2}}$] 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. MR 631606
  • [Mc] Robert M. McLeod, The generalized Riemann integral, Carus Mathematical Monographs, vol. 20, Mathematical Association of America, Washington, D.C., 1980. MR 588510
  • [Mu] James R. Munkres, Elementary differential topology, Lectures given at Massachusetts Institute of Technology, Fall, vol. 1961, Princeton University Press, Princeton, N.J., 1966. MR 0198479
  • [P $ _{\mathbf{1}}$] 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). MR 761251
  • [P $ _{\mathbf{2}}$] W. F. Pfeffer, The multidimensional fundamental theorem of calculus (to appear).
  • [R] Walter Rudin, Principles of mathematical analysis, 3rd ed., McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976. International Series in Pure and Applied Mathematics. MR 0385023
  • [S] 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. MR 0167578

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DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0833702-0
Article copyright: © Copyright 1986 American Mathematical Society