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

top
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

top

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

top
  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. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.