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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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}}$] R. Henstock, Theory of integration, Butterworths, London, 1963. MR 0158047 (28:1274)
  • [H $ _{\mathbf{2}}$] -, Majorants in variational integration, Canad. J. Math. 18 (1966), 49-74. MR 0185075 (32:2545)
  • [H $ _{\mathbf{3}}$] -, A Riemann-type integral of Lebesgue power, Canad. J. Math. 20 (1968), 79-87. MR 0219675 (36:2754)
  • [H $ _{\mathbf{4}}$] -, A problem in two dimensional integration, J. Austral. Math. Soc. (Ser. A) 35 (1983), 386-404. MR 712816 (84k:26010)
  • [JK] J. Jarník and J. Kurzweil, A nonabsolutely convergent integral which admits transformation and can be used for integration on manifolds, Czechoslovak Math. J. 35 (1985), 116-139. MR 779340 (86e:26011)
  • [JKS] J. Jarník, J. Kurzweil and Š. Schwabik, On Mawhin's approach to multiple nonabsolutely convergent integral, Časopis Pěest. Mat. 108 (1983), 356-380. MR 727536 (85h:26011)
  • [K $ _{\mathbf{1}}$] J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J. 7 (1957), 418-446. MR 0111875 (22:2735)
  • [K $ _{\mathbf{2}}$] -, Nichtabsolut Konvergente Integrale, Teubner, Leipzig, 1980. MR 597703 (82m:26007)
  • [KN] L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley, New York, 1974. MR 0419394 (54:7415)
  • [LW] Lee Peng Yee and Wittaya Naak-In, A direct proof that Henstock and Denjoy integrals are equivalent, Bull. Malaysian Math. Soc. (2) 5 (1982), 43-47. MR 683810 (84f:26011)
  • [M $ _{\mathbf{1}}$] J. Mawhin, Generalized Riemann integrals and the divergence theorem for differentiable vector fields, in E. B. Christoffel, Birkhäuser-Verlag, Basel, 1981, pp. 704-714. MR 661109 (83m:26016)
  • [M $ _{\mathbf{2}}$] -, Generalized multiple Perron integrals and the Green-Goursat theorem for differentiable vector fields, Czechoslovak Math. J. 31 (1981), 614-632. MR 631606 (82m:26010)
  • [Mc] R. M. McLeod, The generalized Riemann integral, Carus Math. Monographs, 20, Math. Assoc. Amer., 1980. MR 588510 (82h:26015)
  • [Mu] J. R. Munkres, Elementary differential topology, Princeton Univ. Press, Princeton, N.J., 1966. MR 0198479 (33:6637)
  • [P $ _{\mathbf{1}}$] W. F. Pfeffer, Une intégrale riemannienne et le théorème de divergence, Analyse Math., C. R. Acad. Sci. Paris Sér. I 299 (1984). MR 761251 (85k:26021)
  • [P $ _{\mathbf{2}}$] W. F. Pfeffer, The multidimensional fundamental theorem of calculus (to appear).
  • [R] W. Rudin, Principles of mathematical analysis, McGraw-Hill, New York, 1976. MR 0385023 (52:5893)
  • [S] S. Saks, Theory of the integral, Dover, New York, 1964. MR 0167578 (29:4850)

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Article copyright: © Copyright 1986 American Mathematical Society

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