Subsection 1.6.3 The Curl in Curvilinear Coordinates
\begin{equation*}
\,curl\vec{F} =\vec{\nabla}\times\vec{F} = \vec{\nabla}\times F_{1}\hat{e}_{1}+F_{2}\hat{e}_{2}+F_{3}\hat{e}_{3}
\end{equation*}
\begin{equation*}
=\left(\frac{\hat{e}_{1}}{h_{1}}\frac{\partial}{\partial u}+ \frac{\hat{e}_{2}}{h_{2}}\frac{\partial}{\partial v}+\frac{\hat{e}_{3}}{h_{3}}\frac{\partial}{\partial w}\right)\times(F_{1}\hat{e}_{1}+F_{2}\hat{e}_{2}+F_{3}\hat{e}_{3})
\end{equation*}
\begin{equation*}
= {\begin{vmatrix}
\hat{e}_{1} & \hat{e}_{2} & \hat{e}_{3} \\
\frac{1}{h_{1}}\frac{\partial}{\partial u} & \frac{1}{h_{1}}\frac{\partial}{\partial v} & \frac{1}{h_{1}}\frac{\partial}{\partial w} \\
F_{1} & F_{2} & F_{3}
\end{vmatrix}}
\end{equation*}
\begin{equation*}
=\hat{e}_{1}\left[\frac{1}{h_{2}}\frac{\partial F_{3}}{\partial v}
-\frac{1}{h_{3}}\frac{\partial F_{2}}{\partial w}\right] -
\hat{e}_{2}\left[\frac{1}{h_{1}}\frac{\partial F_{3}}{\partial u}
-\frac{1}{h_{3}}\frac{\partial F_{1}}{\partial w}\right]
\end{equation*}
\begin{equation*}
+ \hat{e}_{3}\left[\frac{1}{h_{1}}\frac{\partial F_{2}}{\partial u}
-\frac{1}{h_{2}}\frac{\partial F_{1}}{\partial v}\right]
\end{equation*}
\begin{equation*}
=\frac{\hat{e}_{1}}{h_{2}h_{3}}\left[\frac{\partial}{\partial v}(F_{3}h_{3})
-\frac{\partial}{\partial w}(F_{2}h_{2})\right]
\end{equation*}
\begin{equation*}
+ \frac{\hat{e}_{2}}{h_{3}h_{1}}\left[\frac{\partial}{\partial w}(F_{1}h_{1})
-\frac{\partial}{\partial u}(F_{3}h_{3})\right]
\end{equation*}
\begin{equation*}
+\frac{\hat{e}_{3}}{h_{1}h_{2}}\left[\frac{\partial}{\partial u}(F_{2}h_{2})
-\frac{\partial}{\partial v}(F_{1}h_{1})\right]
\end{equation*}
\begin{equation*}
\therefore \vec{\nabla}\times\vec{F}= {\begin{vmatrix}
h_{1}\hat{e}_{1} & h_{2}\hat{e}_{2} & h_{3}\hat{e}_{3} \\
\frac{\partial}{\partial u} & \frac{\partial}{\partial v} & \frac{\partial}{\partial w} \\
F_{1}h_{1} & F_{2}h_{2} & F_{3}h_{3}
\end{vmatrix}}
\end{equation*}