# A theory of non-absolutely convergent integrals in Rn with singularities on a regular boundary

Fundamenta Mathematicae (1994)

- Volume: 146, Issue: 1, page 69-84
- ISSN: 0016-2736

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topJurkat, W., and Nonnenmacher, D.. "A theory of non-absolutely convergent integrals in Rn with singularities on a regular boundary." Fundamenta Mathematicae 146.1 (1994): 69-84. <http://eudml.org/doc/212052>.

@article{Jurkat1994,

abstract = {Specializing a recently developed axiomatic theory of non-absolutely convergent integrals in $ℝ^n$, we are led to an integration process over quite general sets $A ⊆ q ℝ^n$ with a regular boundary. The integral enjoys all the usual properties and yields the divergence theorem for vector-valued functions with singularities in a most general form.},

author = {Jurkat, W., Nonnenmacher, D.},

journal = {Fundamenta Mathematicae},

keywords = {vector fields; non-absolutely convergent integrals; divergence theorem},

language = {eng},

number = {1},

pages = {69-84},

title = {A theory of non-absolutely convergent integrals in Rn with singularities on a regular boundary},

url = {http://eudml.org/doc/212052},

volume = {146},

year = {1994},

}

TY - JOUR

AU - Jurkat, W.

AU - Nonnenmacher, D.

TI - A theory of non-absolutely convergent integrals in Rn with singularities on a regular boundary

JO - Fundamenta Mathematicae

PY - 1994

VL - 146

IS - 1

SP - 69

EP - 84

AB - Specializing a recently developed axiomatic theory of non-absolutely convergent integrals in $ℝ^n$, we are led to an integration process over quite general sets $A ⊆ q ℝ^n$ with a regular boundary. The integral enjoys all the usual properties and yields the divergence theorem for vector-valued functions with singularities in a most general form.

LA - eng

KW - vector fields; non-absolutely convergent integrals; divergence theorem

UR - http://eudml.org/doc/212052

ER -

## References

top- [Fed] H. Federer, Geometric Measure Theory, Springer, New York, 1969.
- [Jar-Ku 1] J. Jarník and J. Kurzweil, A non-absolutely convergent integral which admits ${C}^{1}$-transformations, Časopis Pěst. Mat. 109 (1984), 157-167. Zbl0555.26005
- [Jar-Ku 2] J. Jarník and J. Kurzweil, A non-absolutely convergent integral which admits transformation and can be used for integration on manifolds, Czechoslovak Math. J. 35 (110) (1985), 116-139. Zbl0614.26007
- [Jar-Ku 3] J. Jarník and J. Kurzweil, A new and more powerful concept of the PU integral, ibid. 38 (113) (1988), 8-48. Zbl0669.26006
- [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
- [Ju] W. B. Jurkat, The Divergence Theorem and Perron integration with exceptional sets, Czechoslovak Math. J. 43 (118) (1993), 27-45. Zbl0789.26005
- [Ju-No 1] W. B. Jurkat and D. J. F. Nonnenmacher, An axiomatic theory of non-absolutely convergent integrals in ${\mathbb{R}}^{n}$, Fund. Math. 145 (1994), 221-242. Zbl0824.26007
- [Ju-No 2] W. B. Jurkat and D. J. F. Nonnenmacher, A generalized n-dimensional Riemann integral and the Divergence Theorem with singularities, Acta Sci. Math. (Szeged) 59 (1994), 241-256. Zbl0810.26007
- [Ju-No 3] W. B. Jurkat and D. J. F. Nonnenmacher, The Fundamental Theorem for the ${\nu}_{1}$-integral on more general sets and a corresponding Divergence Theorem with singularities, Czechoslovak Math. J., to appear. Zbl0832.26008
- [Maw] J. Mawhin, Generalized multiple Perron integrals and the Green-Goursat theorem for differentiable vector fields, ibid. 31 (106) (1981), 614-632. Zbl0562.26004
- [No 1] D. J. F. Nonnenmacher, Sets of finite perimeter and the Gauss-Green Theorem with singularities, J. London Math. Soc., to appear. Zbl0835.26008
- [No 2] D. J. F. Nonnenmacher, A constructive definition of the n-dimensional ν(S)-integral in terms of Riemann sums, preprint 1992, to appear.
- [Pf 1] W. F. Pfeffer, The divergence theorem, Trans. Amer. Math. Soc. 295 (1986), 665-685. Zbl0596.26007
- [Pf 2] W. F. Pfeffer, The Gauss-Green Theorem, Adv. in Math. 87 (1991), 93-147. Zbl0732.26013

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