Deriving Notations for Differentials in Time-Evolving Systems

[Apashanka]

if $d^3x=Jd^3X...(1)$ where $x's$ evolves with time and $X's$ are constt. and $x_i=f(X_i)$(for $i^{th}$ coordinate) where the functional form of $f(X_i)$ changes with the time evolution of $x_i$.
Now taking time derivative of (1) and dividing throughout by (1) it is coming $\dot J=J(\nabla•v)$($x$ and $X$ are coordinates) which is consistent.
But another thing is approximated while doing this $\frac{d}{dt}(dx_i)=d(\frac{dx_i}{dt})=dv_i$ actually I try to prove it by hand but can't...
Will anyone provide me any hints...

 

[WWGD]

D , as usually defined acts on differential forms which includes functions, but the expression d/dx is neither unless you have a special definition. How do you define this operation?

 

[Apashanka]

D , as usually defined acts on differential forms which includes functions, but the expression d/dx is neither unless you have a special definition. How do you define this operation?

$\frac{d}{dt}$ is defined as $\frac{\partial }{\partial t}+(v•\nabla)$

 

[WWGD]

This is defined for time series ( for context) or a general definition?

 

[Apashanka]

This is defined for time series ( for context) or a general definition?

Here is the case where $i^{th}$ component of velocity depends on $v_i(x,y,z,t)$ ,now $\frac{d v_i}{dt}=\frac{\partial v_i}{\partial t}+(v•\nabla)v_i$ for which $\frac{d}{dt}=\frac{\partial }{\partial t}+v•\nabla$ using this I tried to find $d(\frac{dx}{dt})=dv_x$ and trying to equate it to $\frac{d}{dt}(dx)$...but can't...

 

[Apashanka]

Okk I have done it like this ,is it the correct procedure??
$\frac{d}{dt}(dx)=\frac{d}{dt}(v_x(x,y,z,t)dt)=\frac{dv_x(x,y,z,t)}{dt}dt=dt(\frac{\partial v_x}{\partial t}+(v•\nabla)v_x)=dt[\frac{\partial v_x}{dt}+(v_x\frac{\partial }{dx}+v_y\frac{\partial }{dy}+v_z\frac{\partial }{dz})v_x]$

And taking $d(\frac{dx}{dt})=dv_x(x,y,z,t)=\frac{\partial v_x}{\partial x}dx+\frac{\partial v_x}{\partial y}dy+\frac{\partial v_x}{\partial z}dz+\frac{\partial v_x}{\partial t}dt$ which are same...