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The Gauss-Green theorem. (English) Zbl 0732.26013
Since the early eighties, there is a lot of interest in defining integrals for which the Gauss-Green theorem holds for (not necessarily continuously) differentiable vector fields. The challenge is on the properties of the integrals, and on the generality of the manifold of integration. In this interesting paper, a variational integral is constructed which allows to prove the Gauss-Green theorem for any bounded set of bounded variation in $${\mathbb{R}}^ m$$ and any bounded vector field continuous outside a set of (m-1)-dimensional Hausdorff measure and almost differentiable outside a set of $$\sigma$$-finite (m-1)-dimensional Hausdorff measure. Perron and Riemann-type definitions of the integral are also given.

##### MSC:
 26B20 Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.)
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