2 Vector derivatives in orthogonal coordinates

Given an orthogonal coordinate system $u,v,w$ with unit vectors $\underset{̲}{û}$ , $\underset{̲}{\stackrel{̂}{v}}$ and $\underset{̲}{ŵ}$ and scale factors, $h_u$ , $h_v$ and $h_w$ , it is possible to find the derivatives $\underset{̲}{\nabla }f$ , $\underset{̲}{\nabla }\cdot \underset{̲}{F}$ and $\underset{̲}{\nabla }\times \underset{̲}{F}$ .

It is found that

grad $f=\underset{̲}{\nabla }f=\frac{1}{h_u}\frac{\partial f}{\partial u}\underset{̲}{û}+\frac{1}{h_v}\frac{\partial f}{\partial v}\underset{̲}{\stackrel{̂}{v}}+\frac{1}{h_w}\frac{\partial f}{\partial w}\underset{̲}{ŵ}$

If $\underset{̲}{F}=F_u\underset{̲}{û}+F_v\underset{̲}{\stackrel{̂}{v}}+F_w\underset{̲}{ŵ}$ then

div $\underset{̲}{F}=\underset{̲}{\nabla }\cdot \underset{̲}{F}=\frac{1}{h_u{h}_v{h}_w}\left[\frac{\partial }{\partial u}\left(F_u{h}_v{h}_w\right)+\frac{\partial }{\partial v}\left(F_v{h}_u{h}_w\right)+\frac{\partial }{\partial w}\left(F_w{h}_u{h}_v\right)\right]$

Also if $\underset{̲}{F}=F_u\underset{̲}{û}+F_v\underset{̲}{\stackrel{̂}{v}}+F_w\underset{̲}{ŵ}$ then

curl $\underset{̲}{F}=\underset{̲}{\nabla }\times \underset{̲}{F}=\frac{1}{h_u{h}_v{h}_w}\left|\begin{array}{ccc}\hfill h_u\underset{̲}{û}\hfill & \hfill h_v\underset{̲}{\stackrel{̂}{v}}\hfill & \hfill h_w\underset{̲}{ŵ}\hfill \\ \hfill \hfill \\ \hfill \frac{\partial }{\partial u}\hfill & \hfill \frac{\partial }{\partial v}\hfill & \hfill \frac{\partial }{\partial w}\hfill \\ \hfill \hfill \\ \hfill h_u{F}_u\hfill & \hfill h_v{F}_v\hfill & \hfill h_w{F}_w\hfill \end{array}\right|$