$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Newton's method for \( A(u)u=b(u) \)
For
$$ F_i = \sum_k A_{i,k}(u)u_k - b_i(u)$$
one gets
$$
J_{i,j} = \frac{\partial F_i}{\partial u_j}
= \sum_k \frac{\partial A_{i,k}}{\partial u_j}u_k
+ A_{i,j} -
\frac{\partial b_i}{\partial u_j}
$$
Matrix form:
$$ (A + A^{\prime}u + b^{\prime})\delta u = -Au + b$$
$$ (A(u^{-}) + A^{\prime}(u^{-})u^{-} + b^{\prime}(u^{-}))\delta u
= -A(u^{-})u^{-} + b(u^{-})$$
