$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Useful formulas for computing the Jacobian
$$
\begin{align*}
\frac{\partial u}{\partial c_j} &= \frac{\partial}{\partial c_j}
\sum_kc_k\baspsi_k = \baspsi_j\\
\frac{\partial u^{\prime}}{\partial c_j} &= \frac{\partial}{\partial c_j}
\sum_kc_k\baspsi_k^{\prime} = \baspsi_j^{\prime}
\end{align*}
$$
