$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Transformation to a reference cell in 2D/3D (3)
Can derive
$$
\begin{align*}
\nabla_{\X}\refphi_r &= J\cdot\nabla_{\x}\basphi_i\\
\nabla_{\x}\basphi_i &= \nabla_{\x}\refphi_r(\X)
= J^{-1}\cdot\nabla_{\X}\refphi_r(\X)
\end{align*}
$$
Integral transformation from physical to reference coordinates:
$$
\begin{equation*}
\int_{\Omega^{(e)}} a(\x)\nabla_{\x}\basphi_i\cdot\nabla_{\x}\basphi_j\dx =
\int_{\tilde\Omega^r} a(\x(\X))(J^{-1}\cdot\nabla_{\X}\refphi_r)\cdot
(J^{-1}\cdot\nabla\refphi_s)\det J\dX
\end{equation*}
$$
