The patchwork divergence theorem

The divergence theorem in its usual form applies only to suitably smooth vector fields. For vector fields which are merely piecewise smooth, as is natural at a boundary between regions with different physical properties, one must patch together the divergence theorem applied separately in each region. We give an elegant derivation of the resulting patchwork divergence theorem which is independent of the metric signature in either region, and which is thus valid if the signature changes.