I am wondering about the inverted function theorem
$({\bf G}_{\bf F}({\bf x}))^{-1} = {1 \over {\bf G}_{\bf F}({\bf x})}$
If I have an equation
${\bf y} = \left({d {\bf F}({\bf x}) \over d {\bf x}}\right)^{-1}{d^2 ({\bf F}({\bf x}){\bf k}) \over d {\bf x}^2}$
${d {\bf F}({\bf x}) \over d {\bf x}}$ is a Jacobain matrix, ${d^2 {\bf F}({\bf x}) \over d {\bf x}^2}$ is a Hessian. ${\bf k}$ is a vector, independent of and the same size as ${\bf x}$. what can I say about this equation, can I say it is now first order due to the inverted derivative?
Can anyone recommend a book or journal paper?