The Patchwork Divergence Theorem

TL;DR

The paper introduces a generalized divergence theorem applicable to piecewise smooth vector fields, including those with changing metric signatures, by deriving a metric-independent 'patchwork divergence theorem' that extends classical results.

Contribution

It provides an elegant, metric-independent derivation of the patchwork divergence theorem for piecewise smooth vector fields with potential signature changes.

Findings

Valid for vector fields with boundary discontinuities

Applicable to regions with different metric signatures

Extends classical divergence theorem to more general settings

Abstract

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. (PACS numbers 4.20.Cv, 04.20.Me, 11.30.-j, 02.40.Hw)