Trying to calculate the time derivative of a position differential

[Apashanka]

Homework Statement

Trying to calculate $\frac{d}{dt}dx$

Relevant Equations

Trying to calculate $\frac{d}{dt}dx$

here I am trying to find $\frac{d}{dt}dx$ where $x(t)$ is the position vector
Now $\frac{d}{dt}(v_x(x,y,z,t)dt)=\frac{dv_x}{dt}dt=\frac{\partial v_x}{\partial t}dt+\frac{\partial v_x}{\partial x}dx+\frac{\partial v_x}{\partial y}dy+\frac{\partial v_x}{\partial z}dz$
Now dividing by $dx$
$\frac{\partial v_x}{\partial t}\frac{dt}{dx}+\frac{\partial v_x}{\partial x}$
Other terms goes to zero.
It therefore becomes $\frac{\partial v_x}{\partial t}\frac{\partial t}{\partial x}+\frac{\partial v_x}{\partial x}=\frac{\partial v_x}{\partial x}+\frac{\partial v_x}{\partial x}=2\frac{\partial v_x}{\partial x}$
Am I right in doing so??

 

[Mark44]

Problem Statement: Trying to calculate $\frac{d}{dt}dx$
Relevant Equations: Trying to calculate $\frac{d}{dt}dx$

here I am trying to find $\frac{d}{dt}dx$ where $x(t)$ is the position vector

As stated, your problem doesn't make sense to me.
One thing that is confusing is that if $x(t)$ is a position vector, are its components x, y, and z values?
If so, the usual way to describe it is as $\vec r(t)$ where $\vec r(t) = <x(t), y(t), z(t)>$ if we're talking about a vector in $\mathbb R^3$.
In this case, $\frac{d}{dt}\left(\vec r(t)\right) = <x'(t), y'(t), z'(t)>$, with the primes indicating the derivative with respect to t.

Now $\frac{d}{dt}(v_x(x,y,z,t)dt)=\frac{dv_x}{dt}dt=\frac{\partial v_x}{\partial t}dt+\frac{\partial v_x}{\partial x}dx+\frac{\partial v_x}{\partial y}dy+\frac{\partial v_x}{\partial z}dz$
Now dividing by $dx$
$\frac{\partial v_x}{\partial t}\frac{dt}{dx}+\frac{\partial v_x}{\partial x}$
Other terms goes to zero.
It therefore becomes $\frac{\partial v_x}{\partial t}\frac{\partial t}{\partial x}+\frac{\partial v_x}{\partial x}=\frac{\partial v_x}{\partial x}+\frac{\partial v_x}{\partial x}=2\frac{\partial v_x}{\partial x}$
Am I right in doing so??

 

[BvU]

In physics (maybe also in math ) $\mathrm d$ has a special meaning: take a small chunk of what follows and let that chunk shrink until it is infinitesimally small. When we speak of ${\mathrm d} x$ we use the term 'infinitesimal'.

If $x$ is a function of $t$, then ${\mathrm d} x$ is also a function of $t$ and you can form a mental idea by thinking of ${\mathrm d} x(t)$ as $\ \displaystyle \lim_{h_\downarrow 0} \Bigl ( x(t+h) - x(t) \Bigr )$. But only as a reminder (basically it is zero).

Because that way, for ${\mathrm d} t$ you would get $\ \displaystyle\lim_{h_\downarrow 0} t+h \ - t= 0\$. So we write ${\mathrm d} t$ as a reminder, like: something we will divide by later on and then take a limit letting it go to zero.

Things only becomes non-infinitesimal in the division -- where we write a fraction and the (single) limit of the ratio is taken and not the ratio of two limits:$${{\mathrm d} x\over dt }\equiv \lim_{h_\downarrow 0} {x(t+h) - x(t) \over h} $$Continuing in that jargon, $\ {{\mathrm d} x\over {\mathrm d} t}\ = {{\mathrm d} \over {\mathrm d} t }(x) \ = v(t)$ is a derivative.
One can ask for a second derivative, but there is no point in asking for a derivative of an infinitesimal. Can you provide some more context of what you try to do ?

 

[Apashanka]

As stated, your problem doesn't make sense to me.
One thing that is confusing is that if $x(t)$ is a position vector, are its components x, y, and z values?
If so, the usual way to describe it is as $\vec r(t)$ where $\vec r(t) = <x(t), y(t), z(t)>$ if we're talking about a vector in $\mathbb R^3$.
In this case, $\frac{d}{dt}\left(\vec r(t)\right) = <x'(t), y'(t), z'(t)>$, with the primes indicating the derivative with respect to t.

Ok the thing is it ,I came across an equation which is $dx(t)dy(t)dz(t)=JdXdYdZ...(1)$ where $x,y,z$ are cartesian components of the position vector and $X,Y,Z$ are constt of time.
and it is written that $\dot J=J\theta$ where $\theta=\vec \nabla•\vec v$ and $\vec v$ is the velocity vector.
In order to prove it I took time derivative of both sides of (1) for which Rhs become $\dot J dXdYdZ$ and for one part of LHS out of three parts it came as $\frac{\partial v_x}{\partial x}+\frac{\partial v_x}{\partial t}\frac{\partial t}{\partial x}$ and combining three terms it came as $\theta$+extra terms
Then how to prove it??

 

[Mark44]

In physics (maybe also in math ) $\mathrm d$ has a special meaning

Yeah, it has the same meaning in mathland -- the differential of something.

One can ask for a second derivative, but there is no point in asking for a derivative of an infinitesimal.

I agree.

 

[Mark44]

Ok the thing is it ,I came across an equation which is $dx(t)dy(t)dz(t)=JdXdYdZ...(1)$ where $x,y,z$ are cartesian components of the position vector and $X,Y,Z$ are constt of time.

constt (sic) of time?
What this looks like to me is a change of coordinates using the Jacobian. On the left side is a volume element $dx(t)dy(t)dz(t)$. The corresponding volume element in a different coordinate system would be $JdX(t)dY(t)dZ(t)$.
If X, Y, and Z are constants, as you seem to be saying, then dX = dY = dZ = 0.

and it is written that $\dot J=J\theta$ where $\theta=\vec \nabla•\vec v$ and $\vec v$ is the velocity vector.

 

[Mark44]

BTW, it seems to me that you're asking the same question (or similar) in this thread - https://www.physicsforums.com/threads/jacobian-determinant.969073/

 

[Apashanka]

constt (sic) of time?
What this looks like to me is a change of coordinates using the Jacobian. On the left side is a volume element $dx(t)dy(t)dz(t)$. The corresponding volume element in a different coordinate system would be $JdX(t)dY(t)dZ(t)$.

Here is the snap where I tried to prove equation 3

 

[Apashanka]

In physics (maybe also in math ) $\mathrm d$ has a special meaning: take a small chunk of what follows and let that chunk shrink until it is infinitesimally small. When we speak of ${\mathrm d} x$ we use the term 'infinitesimal'.

If $x$ is a function of $t$, then ${\mathrm d} x$ is also a function of $t$ and you can form a mental idea by thinking of ${\mathrm d} x(t)$ as $\ \displaystyle \lim_{h_\downarrow 0} \Bigl ( x(t+h) - x(t) \Bigr )$. But only as a reminder (basically it is zero).

Because that way, for ${\mathrm d} t$ you would get $\ \displaystyle\lim_{h_\downarrow 0} t+h \ - t= 0\$. So we write ${\mathrm d} t$ as a reminder, like: something we will divide by later on and then take a limit letting it go to zero.

Things only becomes non-infinitesimal in the division -- where we write a fraction and the (single) limit of the ratio is taken and not the ratio of two limits:$${{\mathrm d} x\over dt }\equiv \lim_{h_\downarrow 0} {x(t+h) - x(t) \over h} $$Continuing in that jargon, $\ {{\mathrm d} x\over {\mathrm d} t}\ = {{\mathrm d} \over {\mathrm d} t }(x) \ = v(t)$ is a derivative.
One can ask for a second derivative, but there is no point in asking for a derivative of an infinitesimal. Can you provide some more context of what you try to do ?

Trying to prove eq.3 ...
Any hints to arrive at the result??

 

[Apashanka]

What I have tried is taking time derivative of $d^3x=Jd^3X$ which gives RHS $\dot J d^3X$,for LHS simplying one term out of three e.g $\frac{d(dx(t))}{dt}dy(t)dz(t)=dv_xdydz=(\frac{\partial v_x}{\partial x}dx+\frac{\partial v_y}{\partial y}dy+\frac{\partial v_z}{\partial z}dz+\frac{\partial v_x}{\partial t}dt)dydz$
Multiplying throughout by $d^3x=Jd^3X$
Rhs becomes $\frac{\dot J}{J}$ and LHS becomes $\frac{(\frac{\partial v_x}{\partial x}dx+\frac{\partial v_y}{\partial y}dy+\frac{\partial v_z}{\partial z}dz+\frac{\partial v_x}{\partial t}dt)}{dx}=\frac{\partial v_x}{\partial x}+\frac{\partial v_x}{\partial t}\frac{\partial t}{\partial x}$
Combining all three terms we get
$\frac{\partial v_x}{\partial x}+\frac{\partial v_x}{\partial t}\frac{\partial t}{\partial x}+\frac{\partial v_y}{\partial y}+\frac{\partial v_y}{\partial t}\frac{\partial t}{\partial y}+\frac{\partial v_z}{\partial z}+\frac{\partial v_z}{\partial t}\frac{\partial t}{\partial z}$
Hence $\dot J=J(\nabla•\vec v+$extra terms).
Can anyone please help me out to get rid of these extra terms??

 

[Mark44]

From post #1:

here I am trying to find $\frac{d}{dt}dx$ where $x(t)$ is the position vector
Now $\frac{d}{dt}(v_x(x,y,z,t)dt)=\frac{dv_x}{dt}dt=\frac{\partial v_x}{\partial t}dt+\frac{\partial v_x}{\partial x}dx+\frac{\partial v_x}{\partial y}dy+\frac{\partial v_x}{\partial z}dz$
Now dividing by $dx$
$\frac{\partial v_x}{\partial t}\frac{dt}{dx}+\frac{\partial v_x}{\partial x}$
Other terms goes to zero.
It therefore becomes $\frac{\partial v_x}{\partial t}\frac{\partial t}{\partial x}+\frac{\partial v_x}{\partial x}=\frac{\partial v_x}{\partial x}+\frac{\partial v_x}{\partial x}=2\frac{\partial v_x}{\partial x}$
Am I right in doing so??

I don't think you're on the right track here.
You are trying to prove that $\dot J = J \nabla \cdot \vec v$, right?
What are $J$, $\dot J$, and $\nabla \cdot \vec v$ in this case?

The reason I believe you're on the wrong track is that you are workiing with the volume elements $dx~dy~dz$ and $JdX~dY~dZ$, and trying to find the derivatives or differentials of them, when you should be working directly with the Jacobian J and showing that its derivative $\dot J$ is equal to J times the divergence of $\vec v$; i.e. $J \nabla \cdot \vec v$. See https://en.wikipedia.org/wiki/Del.

 

[Apashanka]

From post #1:I don't think you're on the right track here.
You are trying to prove that $\dot J = J \nabla \cdot \vec v$, right?
What are $J$, $\dot J$, and $\nabla \cdot \vec v$ in this case?

The reason I believe you're on the wrong track is that you are workiing with the volume elements $dx~dy~dz$ and $JdX~dY~dZ$, and trying to find the derivatives or differentials of them, when you should be working directly with the Jacobian J and showing that its derivative $\dot J$ is equal to J times the divergence of $\vec v$; i.e. $J \nabla \cdot \vec v$. See https://en.wikipedia.org/wiki/Del.

$J=\frac{\partial f_1(X_1)}{\partial X_1}\frac{\partial f_2(X_2)}{\partial X_2}\frac{\partial f_3(X_3)}{\partial X_3}$ where $f_1,f_2$ and $f_3$ changes with time, but they doesn't have any explicit time dependence.
And $X_1,X_2$ and $X_3$ doesn't have time dependence,they are initial coordinates at time $t_0$ fixed.
Then how to prove it??

 

[Mark44]

Thread closed as being a duplicate of this thread:
https://www.physicsforums.com/threads/time-evolution-of-jacobian-determinant.969136/@Apashanka , please don't start new threads with the same question.