Graduate Hessian as "Square" of Jacobian?

[WWGD]

Hi,
Is there a way of representing the Laplacian ( Say for 2 variables, to start simple) $\partial^2(f):= f_{xx}+f_{yy}$ as a "square of Jacobians" ( More precisely, as $JJ^T ; J^T$ is the transpose of J, for dimension reasons)? I am ultimately trying to use this to show that the Laplacian is rotationally-invariant, using a rotation matrix and manipulating the product.

 

[BvU]

Don't understand. $\Delta \equiv \nabla^2$ but even for 2 variables the Hessian is not as you write it !? it is the matrix product of $\nabla$ and $\nabla$: $H_{ij} = \nabla_i\nabla_j$

Ah, my bad: Hessian = det$H$