Let $\varphi (u,v,w)$ be a orthogonal coordinate transformation from the $u,v,w$ coordinate system to the $x,y,$z coordinate system.
Let $\mathbf{F}(x,y,z)$ be a vector field defined in $x,y,z$ space, and $f(x,y,z)$ be a scalar field defined in $x,y,z$ space.
Let $\xi$ be a differential operator acting on a function or vector field defined in x,y,z space.
(i.e, The $\xi$ is a differential operator that can act on $\mathbf{F}(x,y,z)$ or $f$ above)
My question.
Am I right if I understand that a coordinate transformation by $\varphi$ of a differential operator $\xi$ is to find $\xi '$ such that;
- $\xi'$ is a differential operator acting on a function or vector field defined in $u,v,w$ space.
- $\xi'$ satisfies
$\xi\left[f\right]\left(\varphi\left(u,v,w\right)\right)=\left(\xi^\prime\left[\varphi^{*}f\right]\right)\left(u,v,w\right) $ (for scalar field $f(x,y,z)$)
$\xi\left[\mathbf{F}\right]\left(\varphi\left(u,v,w\right)\right)=\left(\xi^\prime\left[\varphi^{*}\mathbf{F}\right]\right)\left(u,v,w\right) $ (for vector field $\mathbf{F}(x,y,z)$)
Here,
- $\varphi^{*}f := f\circ \varphi$,
- $\varphi^{*}\mathbf{F}= F_u \mathbf{t}_u + F_v \mathbf{t}_v + F_w \mathbf{t}_w$ ,
$\mathbf{t}_u(u,v,w):= \frac{1}{h_u(u,v,w)}\frac{\partial \varphi}{\partial u}(u,v,w) ,\cdots ,$
$h_u(u,v,w):=\left\|\frac{\partial \varphi}{\partial u}(u,v,w)\right\| , \cdots ,$
$ F_u(u,v,w):= <\mathbf{F}\circ\varphi (u,v,w)|\mathbf{t}_u(u,v,w)> ,\cdots ,$
Where
- $< | >$ represents inner product ,
- $\left\| \right\|$ represents norm ,
- $\xi\left[f\right]\left(\varphi\left(u,v,w\right)\right)$ is the value of at $\varphi\left(u,v,w\right)$ of $\xi\left[f\right]$, $\xi\left[f\right]$ is what $\xi$ acts on $f$. (The same manner for vector fields.)
- $\cdots$