Seeing this formula for the Hessian matrix under a coordinate transformation, I am confused as to why there is no product rule involved to give extra terms. As an example (2 dimensions)
If
$f:\mathbb{R^2}\to \mathbb{R}$
$(x,y) \to (u(x,y),v(x,y))$ is a coordinate transformation. Consider
$f(u(x,y),v(x,y))$ and its Hessian w.r.t. $(x,y)$ and $(u,v)$. Then
$H_{u,v} = J^t H_{x,y} J$ where $J$ is the Jacobian of the transformation. But assuming $x_i, x_j \in \{x,y\}$ shouldn't
$f_{x_i} = f_u u_{x_i} + f_v v_{x_i}$ (total derivative),
$f_{x_i x_j} = f_{uu}u_{x_i}u_{x_j} + f_{uv}u_{x_i}v_{x_j} + f_u u_{x_i x_j} +
f_{vu}v_{x_i}u_{x_j} + f_{vv}v_{x_i}v_{x_j} + f_v v_{x_i x_j}$ (product rule)?
Where do the $f_u u_{x_i x_j}$ and $f_v v_{x_i x_j}$ terms go?
Best Answer
Just realised the transformation formula only necessarily holds at critical points, where
$f_u = 0 = f_v$.