A descriptive, additive modification of Mawhin's integral and the Divergence Theorem with singularities

Dirk Jens F. Nonnenmacher

Annales Polonici Mathematici (1994)

  • Volume: 59, Issue: 1, page 85-98
  • ISSN: 0066-2216

Abstract

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Modifying Mawhin's definition of the GP-integral we define a well-behaved integral over n-dimensional compact intervals. While its starting definition is of Riemann type, we also establish an equivalent descriptive definition involving characteristic null conditions. This characterization is then used to obtain a quite general form of the divergence theorem.

How to cite

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Dirk Jens F. Nonnenmacher. "A descriptive, additive modification of Mawhin's integral and the Divergence Theorem with singularities." Annales Polonici Mathematici 59.1 (1994): 85-98. <http://eudml.org/doc/262524>.

@article{DirkJensF1994,
abstract = {Modifying Mawhin's definition of the GP-integral we define a well-behaved integral over n-dimensional compact intervals. While its starting definition is of Riemann type, we also establish an equivalent descriptive definition involving characteristic null conditions. This characterization is then used to obtain a quite general form of the divergence theorem.},
author = {Dirk Jens F. Nonnenmacher},
journal = {Annales Polonici Mathematici},
keywords = {Mawhin's integral; additive interval function; divergence theorem; divergence of vector fields},
language = {eng},
number = {1},
pages = {85-98},
title = {A descriptive, additive modification of Mawhin's integral and the Divergence Theorem with singularities},
url = {http://eudml.org/doc/262524},
volume = {59},
year = {1994},
}

TY - JOUR
AU - Dirk Jens F. Nonnenmacher
TI - A descriptive, additive modification of Mawhin's integral and the Divergence Theorem with singularities
JO - Annales Polonici Mathematici
PY - 1994
VL - 59
IS - 1
SP - 85
EP - 98
AB - Modifying Mawhin's definition of the GP-integral we define a well-behaved integral over n-dimensional compact intervals. While its starting definition is of Riemann type, we also establish an equivalent descriptive definition involving characteristic null conditions. This characterization is then used to obtain a quite general form of the divergence theorem.
LA - eng
KW - Mawhin's integral; additive interval function; divergence theorem; divergence of vector fields
UR - http://eudml.org/doc/262524
ER -

References

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  1. [JKS] J. Jarník, J. Kurzweil and S. Schwabik, On Mawhin's approach to multiple nonabsolutely convergent integral, Časopis Pěst. Mat. 108 (1983), 356-380. Zbl0555.26004
  2. [Ju-Kn] W. B. Jurkat and R. W. Knizia, A characterization of multi-dimensional Perron integrals and the fundamental theorem, Canad. J. Math. 43 (1991), 526-539. Zbl0733.26008
  3. [Ju-No] W. B. Jurkat and D. J. F. Nonnenmacher, A generalized n-dimensional Riemann integral and the Divergence Theorem with singularities, Acta Sci. Math. (Szeged), to appear. Zbl0810.26007
  4. [Ku-Jar1] J. Kurzweil and J. Jarník, Equivalent definitions of regular generalized Perron integral, Czechoslovak Math. J. 42 (117) (1992), 365-378. Zbl0782.26004
  5. [Ku-Jar2] J. Kurzweil and J. Jarník, Differentiability and integrability in n dimensions with respect to α-regular intervals, Results Math. 21 (1992), 138-151. Zbl0764.28005
  6. [Ku-Jar3] J. Kurzweil and J. Jarník, Equiintegrability and controlled convergence of Perron-type integrable functions, Real Anal. Exchange 17 (1991-92), 110-139. 
  7. [Maw] J. Mawhin, Generalized multiple Perron integrals and the Green-Goursat theorem for differentiable vector fields, Czechoslovak Math. J. 31 (106) (1981), 614-632. Zbl0562.26004
  8. [No] D. J. F. Nonnenmacher, Every M₁-integrable function is Pfeffer integrable, Czechoslovak Math. J. 43 (118) (1993), 327-330. Zbl0789.26006
  9. [Pf] W. F. Pfeffer, The divergence theorem, Trans. Amer. Math. Soc. 295 (1986), 665-685. Zbl0596.26007
  10. [Saks] S. Saks, Theory of the Integral, Dover, New York, 1964. 

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