Proposition: Weighting changes an integration-by-parts quotient

A compact total divergence need not remain zero after multiplication by a nonconstant switch.

Proof. For a compact vector density $v$,

If ${\partial }_{1}g\ne 0$ on a neighborhood, choose $v^1=\chi {\partial }_{1}g$, other components zero, with $\chi \ge 0$ a nonzero compact bump there. The weighted integral is $-\int \chi ({\partial }_{1}g{)}^2<0$.