非光滑链的Stokes定理
Stokes' theorem for nonsmooth chains
摘要：
在过去的两个世纪里，关于积分的大量文献都是关于扩展可积函数类的。相比之下，我们的观点类似于哈斯勒惠特尼[{ \it几何积分理论}，普林斯顿大学出版社，普林斯顿，新泽西州，1957年]和几何测度理论家，因为我们扩展了可积{ \it域}的类。令$ω$是定义在$\Bbb R^m$上的$n$ -形式.我们表明，如果$ω$是足够光滑，它可以被集成在充分控制，但非光滑，域$γ$。较平滑的是$ ω$，较粗糙的可以是$γ$。允许域包括一个大类的非光滑链和拓扑$n$ -流形沉浸在$\Bbb R^m$。证明了该积分是勒贝格积分的推广，并满足推广的Stokes定理.
英文摘要：
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 [{\it Geometric integration theory}, Princeton Univ. Press, Princeton, NJ, 1957] and by geometric measure theorists because we extend the class of integrable {\it domains}. Let $ω$ be an $n$-form defined on $\Bbb R^m$. We show that if $ω$ is sufficiently smooth, it may be integrated over sufficiently controlled, but nonsmooth, domains $γ$. The smoother is $ ω$, the rougher may be $γ$. Allowable domains include a large class of nonsmooth chains and topological $n$-manifolds immersed in $\Bbb R^m$. We show that our integral extends the Lebesgue integral and satisfies a generalized Stokes' theorem.
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