Subsection 1.6.4 Laplacian in Curvilinear Coordinate Systems
we know that \(\nabla^{2}\psi=\vec{\nabla}\cdot\vec{\nabla \psi}\text{,}\) put \(\vec{\nabla \psi}=\vec{F}\text{,}\) then
\begin{equation*}
\vec{\nabla}\cdot\vec{F}
\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]\tag{1.6.10}
\end{equation}
also
\begin{equation*}
\vec{F} = \vec{\nabla \psi} = \left(\frac{\hat{e_{1}}}{h_{1}}\frac{\partial \psi}{\partial u}
+ \frac{\hat{e_{2}}}{h_{2}}\frac{\partial \psi}{\partial v}
+\frac{\hat{e_{3}}}{h_{3}}\frac{\partial \psi}{\partial w}\right)
\end{equation*}
Hence,
\begin{equation*}
F_{1}= \frac{1}{h_{1}}\frac{\partial \psi}{\partial u}, \quad F_{2}
= \frac{1}{h_{2}}\frac{\partial \psi}{\partial v},\quad F_{3}
= \frac{1}{h_{3}}\frac{\partial \psi}{\partial w}
\end{equation*}
and from eqn. (1.6.10),
\begin{equation*}
\vec{\nabla}\cdot\vec{F} = \vec{\nabla}\cdot\vec{\nabla \psi} = \nabla^{2}\psi
\end{equation*}
\begin{equation*}
= \frac{1}{h_{1}h_{2}h_{3}}\left[\frac{\partial}{\partial u}\left(\frac{h_{2}h_{3}}{h_{1}}\frac{\partial \psi}{\partial u}\right)
+\frac{\partial}{\partial v}\left(\frac{h_{3}h_{1}}{h_{2}}\frac{\partial \psi}{\partial v}\right)
+\frac{\partial}{\partial w}\left(\frac{h_{1}h_{2}}{h_{3}}\frac{\partial \psi}{\partial w}\right)\right]
\end{equation*}
which is the required expression.
Note: If \(h_{1}=h_{2}=h_{3}=1\) and \(\hat{e}_{1}, \hat{e}_{2}, \hat{e}_{3}\text{,}\) are replaced by \(\hat{i}, \hat{j}, \hat{k}\) then all the above expressions reduce into usual expressions in rectangular coordinates where \((u,v,w)\) is replaced by \((x,y,z)\text{.}\)
