Subsection 1.6.1 The Gradient in Curvilinear Coordinates
Let \(\phi\) be the scalar function of orthogonal curvilinear coordinates \(u,v,w\text{,}\) then,
\begin{equation}
\,grad \phi=\vec{\nabla}\phi = f_{1}\hat{e}_{1}+f_{2}\hat{e}_{2}+f_{3}\hat{e}_{3}\tag{1.6.7}
\end{equation}
where \(f_{1},f_{2},f_{3}\) are to be determined. We know that
\begin{equation*}
\vec{\,dr}= h_{1}\hat{e}_{1}\,du+h_{2}\hat{e}_{2}\,dv+h_{3}\hat{e}_{3}\,dw
\end{equation*}
Now,
\begin{equation}
\,d\phi=\vec{\nabla}\phi\cdot\vec{\,dr}=h_{1}f_{1}\,du+h_{2}f_{2}\,dv+h_{3}f_{3}\,dw\tag{1.6.8}
\end{equation}
As, \(\phi=\phi(u,v,w)\text{.}\)
\begin{equation}
\,d\phi= \frac{\partial\phi}{\partial u}\,du+\frac{\partial\phi}{\partial v}\,dv+\frac{\partial\phi}{\partial w}\,dw\tag{1.6.9}
\end{equation}
\begin{equation*}
h_{1}f_{1}=\frac{\partial\phi}{\partial u}\Rightarrow f_{1} = \frac{1}{h_{1}}\frac{\partial\phi}{\partial u}
\end{equation*}
Similarly,
\begin{equation*}
f_{2} = \frac{1}{h_{2}}\frac{\partial\phi}{\partial v},
\end{equation*}
and
\begin{equation*}
f_{3} = \frac{1}{h_{3}}\frac{\partial\phi}{\partial w}
\end{equation*}
\begin{equation*}
\therefore \quad \vec{\nabla}\phi =\frac{1}{h_{2}}\frac{\partial\phi}{\partial u}\hat{e_{1}}+ \frac{1}{h_{2}}\frac{\partial\phi}{\partial v}\hat{e_{2}} + \frac{1}{h_{3}}\frac{\partial\phi}{\partial w}\hat{e_{3}}
\end{equation*}
This indicates the del operator as
\begin{equation*}
\left[\vec{\nabla} = \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]
\end{equation*}
