$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Newton's method for \( F(u)=0 \)
Linearization of \( F(u)=0 \) equation via multi-dimensional Taylor series:
$$ F(u) = F(u^{-}) + J(u^{-}) \cdot (u - u^{-}) + \mathcal{O}(||u - u^{-}||^2) $$
where \( J \) is the Jacobian of \( F \), sometimes denoted \( \nabla_uF \), defined by
$$ J_{i,j} = \frac{\partial F_i}{\partial u_j}$$
Approximate the original nonlinear system \( F(u)=0 \) by
$$ \hat F(u) = F(u^{-}) + J(u^{-}) \cdot \delta u =0,\quad
\delta u = u - u^{-}$$
which is linear vector equation in \( u \)
