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Proceedings of the American Mathematical Society

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Numerical integration of vector fields over curves with zero area


Author: Jenny C. Harrison
Journal: Proc. Amer. Math. Soc. 121 (1994), 715-723
MSC: Primary 65D30
DOI: https://doi.org/10.1090/S0002-9939-1994-1185264-9
MathSciNet review: 1185264
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Abstract: The boundary of a Jordan domain A may be a nonsmooth curve $ \gamma $. If F is a smooth vector field defined near $ \gamma $, then F is integrable over $ \gamma $ provided $ \gamma $ has two-dimensional Lebesgue measure zero and F is sufficiently smooth. When actually computing the integral $ {\smallint _\gamma }F \bullet ds$, one might hope that $ {\lim _{k \to \infty }}{\smallint _{{\gamma _k}}}F \bullet ds = {\smallint _\gamma }F \bullet ds$ for PL or smooth approximators $ {\gamma _k}$ of $ \gamma $. Several examples show that this is not the case. However, there are algorithms for choosing $ {\gamma _k}$ so that $ {\smallint _{{\gamma _k}}}F \bullet ds$ converges to $ {\smallint _\gamma }F \bullet ds$ exponentially quickly.


References [Enhancements On Off] (What's this?)

  • [H-N] J. Harrison and A. Norton, Geometric integration on fractal curves in the plane, Indiana J. Math. 40 (1991), 567-594. MR 1119189 (92m:28011)
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  • [N] A. Norton, Functions not constant on fractal quasi-arcs of critical points, Proc. Amer. Math. Soc. 106 (1989), 397-405. MR 969524 (89m:28013)
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DOI: https://doi.org/10.1090/S0002-9939-1994-1185264-9
Article copyright: © Copyright 1994 American Mathematical Society

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