Corollary: Classical plateau identity

For $F\in {\mathcal{G}}_p$,

off shell. Varying $\varphi$ with a compact test recovers the density-valued Euler covariance (eq:scalarcov).

Proof. Every term in (eq:flatvariation) is supported where the gauge transformation acts; there $1-g=0$. The current is supported there as well, and $\partial g=0$. Thus (eq:varied) vanishes.

For a compact variation $\chi$, vary the transformed scalar by $F^{\ast }\chi$. The Euler variation and change of variables identify the two density-valued Euler operators against every test, proving their equality.