Condition for delta operator and total time differential to commute

In summary, the conversation discusses the application of the continuity equation in fluid mechanics and the order of taking total time derivative and applying delta operator. The question is raised about the condition for the equation ##\left[d/dt, \delta\right]= d\delta/dt- \delta d/dt=0## to hold, and whether the assumption that ##\dot x_i## is independent of ##x_i## is correct. The derivation of the condition is attempted, considering the fixed grid points ##x_{i+h}## and ##x_i##.
  • #1
Abhishek11235
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TL;DR Summary
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?
 
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  • #2
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##:
$$\frac d {dt} \delta f(x_i) = \frac d {dt} (f(x_{i+h})-f(x_i))=\frac d {dt} f(x_{i+h})-\frac d {dt}f(x_i)=\delta \frac d {dt} f(x_i)$$
 
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Related to Condition for delta operator and total time differential to commute

1. What is a delta operator?

A delta operator is a mathematical operator used in the field of calculus to represent a change in a variable over a small interval of time or space. It is often denoted by the symbol Δ and is used to calculate derivatives and integrals.

2. What is the condition for the delta operator and total time differential to commute?

The condition for the delta operator and total time differential to commute is that the function being operated on must be continuous and differentiable at the point of interest.

3. What is the significance of the delta operator and total time differential commuting?

When the delta operator and total time differential commute, it means that the order in which they are applied does not affect the final result. This can be useful in simplifying calculations and solving certain types of problems.

4. Can the delta operator and total time differential commute for all functions?

No, the delta operator and total time differential can only commute for functions that satisfy the condition for commutativity, which is that they are continuous and differentiable at the point of interest. For functions that do not meet this condition, the order of operations can affect the final result.

5. How is the delta operator related to the concept of infinitesimals?

The delta operator is closely related to the concept of infinitesimals, which are quantities that are infinitely small but not equal to zero. The delta operator can be thought of as representing an infinitesimal change in a variable, allowing for the calculation of derivatives and integrals.

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