Undergrad Condition for delta operator and total time differential to commute

[Abhishek11235]

TL;DR

Under what condition do total time derivative operator and delta operator commute?

While deriving continuity equation in Fluid mechanics, our professor switched the order of taking total time derivative and then applying delta operator to the function without stating any condition to do so(Of course I know it is Physics which alows you to do so) . So,I began to think mathematical reason for this.The defination of both operators are:

$\delta(f(x_i)) := f(x_{i+h})- f(x_i)$
$d f(x_i)/dt := \partial f(x_i)/\partial x_i \ \dot x_i + \partial f(x_i)/\partial t$

Here $x_i$ are coordinates of position and where the last one is due to the chain rule. Now the question is when(All are operators here and the bracket denotes Lie bracket) :

$\left[d/dt, \delta\right]= d\delta/dt- \delta d/dt=0 ?$

I tried to derive the condition of above but I made assumption that $\dot x_i$ is independent of $x_i$. Is this right and are there additional conditions for this?

 

[tnich]

Aren't $x_{i+h}$ and $x_i$ fixed grid points? If that is the case, their time derivatives are zero and you can just apply the definition of $\delta$:
$$\frac d {dt} \delta f(x_i) = \frac d {dt} (f(x_{i+h})-f(x_i))=\frac d {dt} f(x_{i+h})-\frac d {dt}f(x_i)=\delta \frac d {dt} f(x_i)$$