If I have a second-order, multivariable Taylor series expansion
$\mathbf{f}(\mathbf{u}_2)=\mathbf{0}=\mathbf{f}(\mathbf{u}_1)+\frac{\partial\mathbf{f}}{\partial \mathbf{u}}\mathbf{e}+\mathcal{O}(||\mathbf{e}||^2)$,
where $\mathbf{e}\equiv\mathbf{u}_2-\mathbf{u}_1$, I can express $\mathbf{e}$ as
$\mathbf{e}=-\left(\frac{\partial\mathbf{f}}{\partial \mathbf{u}}\right)^{-1}\mathbf{f}(\mathbf{u}_1)+\mathcal{O}(||\mathbf{e}||^2)$.
If I want to approximate $\mathbf{e}^T\mathbf{e}$, then I can obtain from the previous equation
$\mathbf{e}^T\mathbf{e}=\mathbf{f}^T(\mathbf{u}_1)\left(\frac{\partial\mathbf{f}}{\partial \mathbf{u}}\right)^{-T}\left(\frac{\partial\mathbf{f}}{\partial \mathbf{u}}\right)^{-1}\mathbf{f}(\mathbf{u}_1)+\mathcal{O}(||\mathbf{e}||^2)$.
Is this a valid approximation, or is there an issue with $\mathbf{e}^T\mathbf{e}$ being the same order as $\mathcal{O}(||\mathbf{e}||^2)$? Are there any ways to better justify such an approximation?