On the divergence theorem on manifolds

Stokes' Theorem on Manifolds: A Kurzweil-Henstock Approach

“…Furthermore, if F is continuous on M , then ϕ is KH-integrable on M . We remark that in [3] Corollary 1, the claim that f is HK-integrable with respect to chart α if and only if f is HK-integrable with respect to any other chart α is not correct.…”

“…The value of the integral does not depend on α, more precisely, if a k-form ϕ is KH-integrable on M with respect to a chart α and another chart β, then the values of these two integrals are equal, see [3]. Hence the integral value is uniquely determined.…”

“…The Kurzweil-Henstock integration on Manifolds has been studied in [2,3]. For easy reference, we give a brief introduction here.…”

“…Suppose dϕ is KH-integrable on M and ϕ is KH-integrable on ∂M . Using the Henstock Lemma; see [8, p. 81] or following the proof of Lemma 3 in [3], we can prove that for each > 0, there exists δ(x) > 0 such that whenever…”

“…where I is an interval containing the point p with surface ∂I and n is the exterior normal to ∂I. Recently, this physical definition of the divergence has been used by Acker, Macdonald, Hubbard and Boonpogkrong; see [1,3,5,9]. In this paper, we shall use the physical definition of an exterior derivative and k-forms to prove Stokes' theorem by the Kurzweil-Henstock approach.…”