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Stokes’ theorem for nonsmooth chains. (English) Zbl 0863.58008

Summary: 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 H. Whitney [‘Geometric integration theory’, Princeton, Univ. Pr. (1957; Zbl 0083.28204)] 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 \(\omega\) is, the rougher \(\gamma\) may be. 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.

MSC:

58C35 Integration on manifolds; measures on manifolds
28C99 Set functions and measures on spaces with additional structure
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
55N05 Čech types

Citations:

Zbl 0083.28204
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References:

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