Proposition: Covariance of the Euler operators on the plateau

For $F\in {\mathcal{G}}_U$ and all smooth $(q,\varphi )$,

as a contravariant symmetric tensor density and a scalar density.

Proof. Apply Lemma lem:invariance to $(q+s\delta q,\varphi +s\delta \varphi )$ with compactly supported $(\delta q,\delta \varphi )$ and differentiate at $s=0$:

Change variables by $F$ on the left; the pairing of a tensor density with a tensor is invariant. Since $\delta q,\delta \varphi$ are arbitrary, the densities agree.