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Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(online) ISSN 0273-0979(print)


Stokes' theorem for nonsmooth chains

Author: Jenny Harrison
Journal: Bull. Amer. Math. Soc. 29 (1993), 235-242
MSC (2000): Primary 58C35; Secondary 28C99, 49Q15, 55N05
MathSciNet review: 1215309
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Abstract: Much of the vast literature on the integral during the last two centuries concerns extending the class of integrable functions. In contrast, our viewpoint is akin to that taken by Hassler Whitney [Geometric integration theory, Princeton Univ. Press, Princeton, NJ, 1957] and by geometric measure theorists because we extend the class of integrable domains. Let $ \omega $ be an n-form defined on $ {\mathbb{R}^m}$. We show that if $ \omega $ is sufficiently smooth, it may be integrated over sufficiently controlled, but nonsmooth, domains $ \gamma $. The smoother is $ \omega $, the rougher may be $ \gamma $. Allowable domains include a large class of nonsmooth chains and topological n-manifolds immersed in $ {\mathbb{R}^m}$. We show that our integral extends the Lebesgue integral and satisfies a generalized Stokes' theorem.

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PII: S 0273-0979(1993)00429-4
Article copyright: © Copyright 1993 American Mathematical Society

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