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Divergence measure fields are integrable vector fields whose distributional divergence is a measure. Some versions are derived of the divergence theorem for divergence measure fields on sets of finite perimeter. Using these results, it is shown that Cauchy fluxes from the theory of Cauchy´s stress theorem can be extended to a class of surfaces that includes singular surfaces of continuum mechanics (shock waves and phase boundaries). On the singular surfaces, the divergence of the stress has a surface delta type singularity, which tractions on a surface and its opposite different from each other.