On the Gauss-Green theorem

Abstract

In a previous paper [1], Green's theorem for line integrals in the plane was proved, for Riemann integration, assuming the integrability of Qx−Py, where P(x, y) and Q(x, y) are the functions involved, but not the integrability of the individual partial derivatives Qx and Py. In the present paper, this result is extended to a proof of the Gauss-Green theorem for p-space (p ≥ 2), for Lebesgue integration, under analogous hypotheses. The theorem is proved in the form where Ω is a bounded open set in Rp (p-space), with boundary Ω; g(x) =(g(x1)…, g(xp)) is a p-vector valued function of x = (x1,…,xp), continuous in the closure of Ω; μv,(x) is p-dimensional Lebesgue measure; v(x) = (v1(x),…, vp(x)) and Φ(x) are suitably defined unit exterior normal and surface area on the ‘surface’ ∂Ω and g(x) · v(x) denotes inner product of p-vectors.