Proposition: The symmetric correction changes the prescription

The quadratic functional

has Hessian

It is not in general the source operator $N_w$. Switching a chosen local density before variation likewise produces derivatives of the onset, with the result depending on that density representative.

Proof. Vary the two copies of $r$ and transpose the operator in the first variation. The two second-variation contributions are $N_w$ and ${N_w}^{\mathrm{T}}$. For a density depending on derivatives of $r$, the Euler integration by parts differentiates its switch as well as its original coefficients.