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 . If *F* is a smooth vector field defined near , then *F* is integrable over provided has two-dimensional Lebesgue measure zero and *F* is sufficiently smooth. When actually computing the integral , one might hope that for PL or smooth approximators of . Several examples show that this is not the case. However, there are algorithms for choosing so that converges to exponentially quickly.

**[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)****[H1]**J. Harrison,*Box dimension vs. Hausdorff dimension in the theory of geometric integration*(in preparation).**[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)****[W]**H. Whitney,*A function not constant on a connected set of critical points*, Duke Math. J.**1**(1935), 514-517. MR**1545896**

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DOI:
https://doi.org/10.1090/S0002-9939-1994-1185264-9

Article copyright:
© Copyright 1994
American Mathematical Society