Let $\phi:\Omega\to\Omega^'$ be an invertible mapping between two bounded domains in $\mathbb{R}^{n}$ (typically with $n=2$ or $n=3$), and let $F$ be its derivative (i.e., the Jacobian matrix). Let $M:\Omega\to\mathbb{R}^{n\times n}$ be a matrix field. I am wondering if the following is true: $$ F^{-1}\Delta' (FMF^T)F^{-T} = \Delta M, $$ where $\Delta$ acts elementwise, and $\Delta'$ is the same on $\Omega'$. If it is convenient, you can assume $F$ is orthogonal and $M$ is symmetric.
More generally, if the above is not true, is it possible to correct $\Delta$ to $D$, by adding lower order terms, so that
$$ F^{-1}D' (FMF^T)F^{-T} = D M, $$ is true?
Update: Robert Bryant commented that it is not true in general, but is true if $F$ is orthogonal, because then $F$ would have to be constant. I would like to relax the orthogonality condition as it is too restrictive for my purposes. Now I am wondering what happens if we change the metric on $\Omega'$ by the induced metric coming from $\phi$, and replace $\Delta'$ by some appropriate geometric Laplacian. Or in other words, now supposing that $\Omega$ and $\Omega'$ are Riemannian manifolds, and $\Delta$ and $\Delta'$ are suitable Laplacians defined on them, what would be a condition on $\phi$ to guarantee the above invariance?
