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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*}