Subsection 1.6.2 The Divergence in Curvilinear Coordinates
Let \(\vec{F}=F_{1}\hat{e}_{1}+F_{2}\hat{e}_{2}+F_{3}\hat{e}_{3}\) be the vector function of orthoganal curvilinear coordinates \(u,v,w\text{,}\) then
\begin{equation*}
div\vec{F} =\vec{\nabla}\cdot\vec{F}
\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)\cdot(F_{1}\hat{e_{1}}+F_{2}\hat{e_{2}}+F_{3}\hat{e_{3}})
\end{equation*}
\begin{equation*}
=\left(\frac{1}{h_{1}}\frac{\partial F_{1}}{\partial u}+ \frac{1}{h_{2}}\frac{\partial F_{2}}{\partial v}+\frac{1}{h_{3}}\frac{\partial F_{3}}{\partial w}\right)
\end{equation*}
\begin{equation*}
= \frac{1}{h_{1}h_{2}h_{3}}\left[\frac{\partial}{\partial u}(F_{1}h_{2}h_{3})+\frac{\partial}{\partial v}(F_{2}h_{3}h_{1})+\frac{\partial}{\partial w}(F_{3}h_{1}h_{2})\right]
\end{equation*}
