Second derivatives under a two-variable coordinate change ($f_{ss}=f_{uu}{u}_s^2+2f_{uv}{u}_s{v}_s+f_{vv}{v}_s^2+f_u{u}_{ss}+f_v{v}_{ss}$)

For twice continuously differentiable coordinate functions $u,v$ and an outer scalar function $f$, apply the multivariable chain rule twice along an original coordinate $s$. The first three terms differentiate the outer function, while the last two differentiate the coordinate coefficients. They cannot be omitted for a nonlinear coordinate map. In matrix notation this is the Hessian matrix formula $H_{f\circ g}=J_g^T{H}_f{J}_g+f_u{H}_u+f_v{H}_v$ for $g=(u,v)$. The extra terms vanish for an affine map.