suppose to have a function $F(x,y,z) = [ f_1(x,y,z),f_2(x,y,z),f_3(x,y,z)]$ and that $f_1$ depend only by x, $f_2$ depends only by y and $f_3$ depends only by z. Now if I apply newton method I can write $[d_x,d_y,d_z] = -J^{-1}(x,y,z)*F(x,y,z)$.
The question is if I exchange the rows of $F$ for example in this way $F_{new}(x,y,z) = [ f_2(x,y,z),f_1(x,y,z),f_3(x,y,z)]$ it seems I can write in the same way this $[d_x,d_y,d_z] = -J^{-1}(x,y,z)*F_{new}(x,y,z)$ , but intuitively I expect something like this $[d_y,d_x,d_z] = -J^{-1}(x,y,z)*F_{new}(x,y,z)$.
Can you please explain me why the inversion of rows doesn't affect the steps retrieved?
This is because the permutation that swaps these rows is also applied to the Jacobian. Let this permutation be $P$, then we have $$ F_{new} = PF\implies J_{new} = PJ. $$ The Newton step $s_{new}$ is then $$ s_{new} = -J_{new}^{-1}F_{new}(x_k) = -J^{-1}P^{-1}PF = -J^{-1}F(x_k) = s $$