Let say we have two sets of variables $\{x,y\}$ and $\{u,v\}$ and we have an expression for $u$ and $v$ in terms of $x$ and $y$. If we want to find the partial derivatives $\partial_u x$, $\partial_u y$, $\partial_v x$ and $\partial_v y$ we can find them from $\partial_x u$, $\partial_x v$, $\partial_y u$ and $\partial_y v$ by inverting the jacobian, see: Partial derivatives inverse question.
Can we also find the second partial derivatives in a similar way, without ever explicitly writing $x$ and $y$ as a function of $u$ and $v$? How?
