Proposition: Complete Hessian ledger

For smooth metric directions $h,h_1,h_2$ and scalar directions $\psi ,{\psi }_1,{\psi }_2$,

and

The mixed blocks are symmetric under exchanging the derivative slots.

Proof. The stress tensor (eq:Tpolar) is symmetric bilinear. Differentiating $-\frac{1}{2}\int g\gamma T(\varphi ,\varphi )$ once in each indicated direction gives (eq:VHessian).

Each of $\mathcal{C}$ and $\mathcal{D}$ is symmetric bilinear in its metric arguments. Each occurrence of ${\mathcal{A}}_1$ is linear in both displayed arguments. The first two terms of (eq:polarO) are linear in $\varphi$, while its last term is cubic in $\varphi$ and independent of $\gamma$. Applying the product rule gives all three formulas in (eq:OHessian), with the stated factors of two.