Time evolution of a Jacobian determinant

In summary, the problem involves finding the derivative of the Jacobian J with respect to time, where the functions f1, f2, and f3 have a time-dependent functional form but their explicit time dependence is unknown. The first step is to correctly formulate the Jacobian as the determinant of a 3x3 matrix. Then, using the given operator for the derivative, the expression for the derivative of J can be found. Finally, the remaining task is to find the divergence of the velocity vector in order to fully solve the problem.
  • #1
Apashanka
429
15
In this paper ##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_2(X_2),f_1(X_1),f_3(X_3)## evolves with time.
IMG_20190330_222629.jpg

Now using this ##\dot J=\frac{d}{dt}(\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})## and using ##\frac{d}{dt}=\frac{\partial}{\partial t}+(v•\nabla)## and ##\frac{\partial X_i}{\partial t}=0## this comes as ##3(v•\nabla)J## but it is given as ##J\theta## where ##\theta=\nabla • v##
Will anyone please help me in sort out this...
 
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  • #2
Or another one will be to take derivative of ##d^3x=Jd^3X## where ##dx,dX## are the volume element ,but then how to proceed??
 
  • #3
Apashanka said:
Or another one will be to take derivative of ##d^3x=Jd^3X## where ##dx,dX## are the volume element ,but then how to proceed??
As noted in the other thread, taking the derivative of the volume element isn't meaningful.

The snippet you've posted in this problem omits a lot of context, and the notation used is fairly dense, but I believe that you don't have a correct formulation for the Jacobian and its determinant. First, let's unpack some of the notation: ##\vec x = \vec f (\vec X)##. As noted in the image, ##\vec X## represents the position of an arbitrary particle in space (##\mathbb R^3##) at time ##t_0##, and ##\vec x## represents the position of the same particle at a later time ##t##. The function ##f## is the mapping between the two positions.

The Jacobian J should be the determinant of a 3x3 matrix, not a 1x3 matrix as you have written several times. I believe it should look like this:
##J = \begin{vmatrix} \frac{\partial f(X_1)}{\partial X_1} &\frac{\partial f(X_1)}{\partial X_2} & \frac{\partial f(X_1)}{\partial X_3}\\ \frac{\partial f(X_2)}{\partial X_1} &\frac{\partial f(X_2)}{\partial X_2} & \frac{\partial f(X_2)}{\partial X_3} \\ \frac{\partial f(X_3)}{\partial X_1} &\frac{\partial f(X_3)}{\partial X_2} & \frac{\partial f(X_3)}{\partial X_3} \end{vmatrix}##

Also, since the coordinates ##X_1, X_2,## and ##X_3## are independent (I believe), all of the entries off the main diagonal are zero, so the Jacobian determinant reduces to this:
##J = \begin{vmatrix} \frac{\partial f(X_1)}{\partial X_1} & 0 & 0\\ 0 &\frac{\partial f(X_2)}{\partial X_2} & 0 \\ 0 & 0 & \frac{\partial f(X_3)}{\partial X_3} \end{vmatrix}##

So J is just the product of the entries on the main diagonal; i.e., ##J =
\frac{\partial f(X_1)}{\partial X_1} \cdot \frac{\partial f(X_2)}{\partial X_2} \cdot \frac{\partial f(X_3)}{\partial X_3}##, hence ##\dot J = \frac d{dt}\left( \frac{\partial f(X_1)}{\partial X_1} \cdot \frac{\partial f(X_2)}{\partial X_2} \cdot \frac{\partial f(X_3)}{\partial X_3} \right)##, which you show in your previous post.

You also show
and using ##\frac{d}{dt}=\frac{\partial}{\partial t}+(v•\nabla)##
I don't know how you came up with what you have for the operator ##\frac d{dt}## -- it seems incorrect to me.

Now that we have an expression for ##\dot J##, what remains is to find ##\nabla \cdot \vec v##. Wouldn't ##\vec v## be the vector ##<\frac {dx_1}{dt}, \frac {dx_2}{dt}, \frac {dx_3}{dt}>##?
 
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  • #4
Mark44 said:
As noted in the other thread, taking the derivative of the volume element isn't meaningful.

The snippet you've posted in this problem omits a lot of context, and the notation used is fairly dense, but I believe that you don't have a correct formulation for the Jacobian and its determinant. First, let's unpack some of the notation: ##\vec x = \vec f (\vec X)##. As noted in the image, ##\vec X## represents the position of an arbitrary particle in space (##\mathbb R^3##) at time ##t_0##, and ##\vec x## represents the position of the same particle at a later time ##t##. The function ##f## is the mapping between the two positions.

The Jacobian J should be the determinant of a 3x3 matrix, not a 1x3 matrix as you have written several times. I believe it should look like this:
##J = \begin{vmatrix} \frac{\partial f(X_1)}{\partial X_1} &\frac{\partial f(X_1)}{\partial X_2} & \frac{\partial f(X_1)}{\partial X_3}\\ \frac{\partial f(X_2)}{\partial X_1} &\frac{\partial f(X_2)}{\partial X_2} & \frac{\partial f(X_2)}{\partial X_3} \\ \frac{\partial f(X_3)}{\partial X_1} &\frac{\partial f(X_3)}{\partial X_2} & \frac{\partial f(X_3)}{\partial X_3} \end{vmatrix}##

Also, since the coordinates ##X_1, X_2,## and ##X_3## are independent (I believe), all of the entries off the main diagonal are zero, so the Jacobian determinant reduces to this:
##J = \begin{vmatrix} \frac{\partial f(X_1)}{\partial X_1} & 0 & 0\\ 0 &\frac{\partial f(X_2)}{\partial X_2} & 0 \\ 0 & 0 & \frac{\partial f(X_3)}{\partial X_3} \end{vmatrix}##

So J is just the product of the entries on the main diagonal; i.e., ##J =
\frac{\partial f(X_1)}{\partial X_1} \cdot \frac{\partial f(X_2)}{\partial X_2} \cdot \frac{\partial f(X_3)}{\partial X_3}##, hence ##\dot J = \frac d{dt}\left( \frac{\partial f(X_1)}{\partial X_1} \cdot \frac{\partial f(X_2)}{\partial X_2} \cdot \frac{\partial f(X_3)}{\partial X_3} \right)##, which you show in your previous post.

You also show I don't know how you came up with what you have for the operator ##\frac d{dt}## -- it seems incorrect to me.

Now that we have an expression for ##\dot J##, what remains is to find ##\nabla \cdot \vec v##. Wouldn't ##\vec v## be the vector ##<\frac {dx_1}{dt}, \frac {dx_2}{dt}, \frac {dx_3}{dt}>##?
##\dot J=\frac{d}{dt}[\frac{\partial f_1}{\partial X_1}\frac{\partial f_2}{\partial X_2}\frac{\partial f_3}{\partial X_3}]## where the functional form of ##f_1(X_1),f_2(X_2),f_3(X_3)## changes with time but their explicit time dependence is not known...then how to proceed this??
 
  • #5
I do not see how you managed to get a factor ##3## in here. Where does it come from? As I understand it do we have
\begin{equation*}
D_v\Phi = \frac{d_v \Phi}{d t}=(\frac{\partial}{\partial t}+ v \cdot \nabla)(\Phi) = \dot V + (v\cdot \nabla)(\Phi)
\end{equation*}
and the local or comoving behaviour affects only the first term, not the convection. The change in the scalar volume is then as given in the text.

Maybe @Orodruin or someone can explain it better from a physicist's point of view. The framework here is less a differentiation as it is a model for fluid dynamics.
 
  • #6
fresh_42 said:
I do not see how you managed to get a factor 3 in here.
Sorry I didn't get you..
 
  • #7
Apashanka said:
Sorry I didn't get you..
Apashanka said:
... and ##\frac{\partial X_i}{\partial t}=0## this comes as ##\mathbf{3}(v•\nabla)J## but it is given as ##J\theta## where ##\theta=\nabla • v##
Will anyone please help me in sort out this...

But to be honest, I'm not really an expert here. I just tried to adapt the example in here into a context, where we have time dependent spatial coordinates (and there is no additional factor in sight):
https://www.physicsforums.com/insights/pantheon-derivatives-part-iv/
 
  • #8
fresh_42 said:
But to be honest, I'm not really an expert here. I just tried to adapt the example in here into a context, where we have time dependent spatial coordinates (and there is no additional factor in sight):
https://www.physicsforums.com/insights/pantheon-derivatives-part-iv/
Actually I need some help or hints in proving this...
Apashanka said:
##\dot J=\frac{d}{dt}[\frac{\partial f_1}{\partial X_1}\frac{\partial f_2}{\partial X_2}\frac{\partial f_3}{\partial X_3}]## where the functional form of ##f_1(X_1),f_2(X_2),f_3(X_3)## changes with time but their explicit time dependence is not known...then how to proceed this??
 
  • #9
Apashanka said:
Actually I need some help or hints in proving this...
Have you looked at the example of the river flow on the page I quoted? Now assume that the spatial components are also time dependent. This does not affect the the spatial differentials, only the time differential, i.e. only the comoving part. As we have a change in volume which we examine here, we get the time differential of said volume as the term for local behaviour. This is at least how I see it.
 
  • #10
fresh_42 said:
Have you looked at the example of the river flow on the page I quoted? Now assume that the spatial components are also time dependent. This does not affect the the spatial differentials, only the time differential, i.e. only the comoving part. As we have a change in volume which we examine here, we get the time differential of said volume as the term for local behaviour. This is at least how I see it.
##d^3x=Jd^3X##, since the Jacobian changes with time as the functional form ##f_1,f_2,f_3## changes therefore the volume element ##d^3x## also changes with time,so what's the harm in calculating it's rate of change e.g ##\frac{d}{dt}d^3x## which gives ##dv_xdydz+dv_ydxdz+dv_zdxdy=\dot Jd^3X##,now is it possible to prove from this ##\dot J=J\nabla•\vec v##
 
  • #11
Do you have a reasonable definition for ##\theta##? Also what should ##J\nabla v## be? The gradient times a direction is a scalar, what do you mean by the Jacobian of a scalar?
 
  • #12
fresh_42 said:
Do you have a reasonable definition for ##\theta##? Also what should ##J\nabla v## be? The gradient times a direction is a scalar, what do you mean by the Jacobian of a scalar?
##\theta=\vec\nabla•\vec v## and ##\dot J=J(\vec \nabla•\vec v)##
 
  • #13
Apashanka said:
##\theta=\vec\nabla•\vec v## and ##\dot J=J(\vec \nabla•\vec v)##
That makes no sense. If ##\theta## is the slope, what is the Jacobian of it? A simple time derivative? It is a bit of a disadvantage that your notation neither mentions the vector field nor the point of evaluation or the variable. ##J## as standalone doesn't say very much, and the ##J(scalar)## even less.
 
  • #14
fresh_42 said:
That makes no sense. If ##\theta## is the slope, what is the Jacobian of it? A simple time derivative? It is a bit of a disadvantage that your notation neither mentions the vector field nor the point of evaluation or the variable. ##J## as standalone doesn't say very much, and the ##J(scalar)## even less.
Here ##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}## and ##J\theta=\dot J## ...so what's the problem??
IMG_20190504_221933.jpg

The snap is from the paper by @thomas buchert and @Jurgen ehlers on averaging Newtonian cosmology,here is the link
Web results
Averaging inhomogeneous Newtonian cosmologies
https://arxiv.org/abs/astro-ph/9510056
 
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  • #15
Apashanka said:
...so what's the problem??
The problem is, that we have a ##3\times 3## matrix and no vector. So at least it should be ##\operatorname{diag}\left( \frac{\partial f_i}{\partial x_i} \right)##, even better ##J(f)##. And is it a function of location, because the differentials are not evaluated, or is it the linear differential of ##f##, and if, at which point? Next you defined the result of an inner product, called it ##\theta## and wrote a ##J## in front of it. But what is the Jacobian of a scalar? Zero?

The problem is, that it is mathematically so sloppy, that only people can read it, who are supposed to know what the authors mean. I tried to figure out if you knew what is meant, but obviously my guesswork isn't sufficient.
 
  • #16
Mark44 said:
You also show I don't know how you came up with what you have for the operator ##\frac d{dt}## -- it seems incorrect to me.
I think it's just a consequence of the chain rule. Written out more explicitly:
$${d \over dt} f(\vec{x}) = {\partial f \over \partial t} + {d\vec{x} \over dt} \cdot \vec{\nabla} f(\vec{x})$$

fresh_42 said:
I do not see how you managed to get a factor ##3## in here. Where does it come from?
It looks like it stemmed from an assumption that all elements of the dot product were identical, thus adding them together results in three times the value of a single element of the dot product. This often occurs for homogeneous fluids, but you have to be careful that you're applying the assumption correctly. I'm not sure I have enough information right now to see whether it was applied correctly here.
 
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  • #17
The text is somewhat confusing (or at least the parts you have posted are without their surrounding context).

There are two ways of describing fluid flows:
  • The more widely used Eulerian specification describes the fluid by the instantaneous velocity [itex]\mathbf{v}[/itex] of the fluid element which at time [itex]t[/itex] is at position [itex]\mathbf{x}[/itex]. As time changes, different fluid elements will occupy this position. The independent variables are [itex]\mathbf{x}[/itex] and [itex]t[/itex].
  • The Lagrangian specification, used in the post in question, describes the fluid by the position [itex]\mathbf{x}[/itex] at time [itex]t[/itex] of the fluid element originally at position [itex]\mathbf{X}[/itex]. The independent variables are [itex]\mathbf{X}[/itex] and [itex]t[/itex].

The two specifications are related by [tex]
\mathbf{v}(\mathbf{x}(\mathbf{X},t),t) = \left.\frac{\partial \mathbf{x}}{\partial t}\right|_{(\mathbf{X},t)}.
[/tex]

The material derivative [tex]
\frac{D}{Dt} = \frac{\partial}{\partial t}+ \mathbf{v} \cdot \nabla[/tex] is relevant to the Eulerian specification, in which partial differentiation with respect to time is at a fixed position occupied by successive different fluid elements. Finding the rate of change with respect to time experienced by a particular fluid element requires the addition of the advective [itex]\mathbf{v} \cdot \nabla[/itex] term. (To be clear, the spatial derivative is with respect to [itex]\mathbf{x}[/itex].) But in the Lagrangian specification, which the text is using here, a particular fluid element is labelled by its initial position [itex]\mathbf{X}[/itex], and partial differentiation with respect to time holding [itex]\mathbf{X}[/itex] constant does "follow the fluid" without the need for an advective term.

The complication here is that when the author writes [itex]\theta = \nabla \cdot \mathbf{v}[/itex] they apparently intend differentiation not with respect to the initial position [itex]\mathbf{X}[/itex] but with respect to the current position [itex]\mathbf{x}[/itex]. Allowing for this and using the normal partial time derivative yields the result (at least in the 2D case, which is the only case I could be bothered to check).

Starting with @Mark44's expression for [itex]J[/itex] you can take the partial derivative with respect to time and then simplify some terms using [tex]
\frac{\partial^2 x_i}{\partial X_j\,\partial t} = \frac{\partial v_i}{\partial X_j}[/tex] since in the Lagrangian specificattion the velocity field is just the partial derivative of the current position with respect to time. You can then use the chain rule [tex]
\frac{\partial v_i}{\partial X_j} = \sum_k \frac{\partial v_i}{\partial x_k} \frac{\partial x_k}{\partial X_j}[/tex] and you will find some terms will cancel, giving a result equal to [tex]
J \theta = J \sum_i \frac{\partial v_i}{\partial x_i}.[/tex]
 
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  • #18
pasmith said:
The text is somewhat confusing (or at least the parts you have posted are without their surrounding context).

There are two ways of describing fluid flows:
  • The more widely used Eulerian specification describes the fluid by the instantaneous velocity [itex]\mathbf{v}[/itex] of the fluid element which at time [itex]t[/itex] is at position [itex]\mathbf{x}[/itex]. As time changes, different fluid elements will occupy this position. The independent variables are [itex]\mathbf{x}[/itex] and [itex]t[/itex].
  • The Lagrangian specification, used in the post in question, describes the fluid by the position [itex]\mathbf{x}[/itex] at time [itex]t[/itex] of the fluid element originally at position [itex]\mathbf{X}[/itex]. The independent variables are [itex]\mathbf{X}[/itex] and [itex]t[/itex].

The two specifications are related by [tex]
\mathbf{v}(\mathbf{x}(\mathbf{X},t),t) = \left.\frac{\partial \mathbf{x}}{\partial t}\right|_{(\mathbf{X},t)}.
[/tex]

The material derivative [tex]
\frac{D}{Dt} = \frac{\partial}{\partial t}+ \mathbf{v} \cdot \nabla[/tex] is relevant to the Eulerian specification, in which partial differentiation with respect to time is at a fixed position occupied by successive different fluid elements. Finding the rate of change with respect to time experienced by a particular fluid element requires the addition of the advective [itex]\mathbf{v} \cdot \nabla[/itex] term. (To be clear, the spatial derivative is with respect to [itex]\mathbf{x}[/itex].) But in the Lagrangian specification, which the text is using here, a particular fluid element is labelled by its initial position [itex]\mathbf{X}[/itex], and partial differentiation with respect to time holding [itex]\mathbf{X}[/itex] constant does "follow the fluid" without the need for an advective term.

The complication here is that when the author writes [itex]\theta = \nabla \cdot \mathbf{v}[/itex] they apparently intend differentiation not with respect to the initial position [itex]\mathbf{X}[/itex] but with respect to the current position [itex]\mathbf{x}[/itex]. Allowing for this and using the normal partial time derivative yields the result (at least in the 2D case, which is the only case I could be bothered to check).

Starting with @Mark44's expression for [itex]J[/itex] you can take the partial derivative with respect to time and then simplify some terms using [tex]
\frac{\partial^2 x_i}{\partial X_j\,\partial t} = \frac{\partial v_i}{\partial X_j}[/tex] since in the Lagrangian specificattion the velocity field is just the partial derivative of the current position with respect to time. You can then use the chain rule [tex]
\frac{\partial v_i}{\partial X_j} = \sum_k \frac{\partial v_i}{\partial x_k} \frac{\partial x_k}{\partial X_j}[/tex] and you will find some terms will cancel, giving a result equal to [tex]
J \theta = J \sum_i \frac{\partial v_i}{\partial x_i}.[/tex]
Okk but where is then ##\dot J##
 
  • #19
Apashanka said:
Okk but where is then ##\dot J##

[itex]\dot J[/itex] is [itex]\frac{\partial J}{\partial t}[/itex] where, as @Mark44 has already said,
[tex]J = \begin{vmatrix} \frac{\partial x_1}{\partial X_1} &\frac{\partial x_1}{\partial X_2} & \frac{\partial x_1}{\partial X_3}\\ \frac{\partial x_2}{\partial X_1} &\frac{\partial x_2}{\partial X_2} & \frac{\partial x_2}{\partial X_3} \\ \frac{\partial x_3}{\partial X_1} &\frac{\partial x_3}{\partial X_2} & \frac{\partial x_3}{\partial X_3} \end{vmatrix}.[/tex]

The algebra is not difficult, just tedious. You can however see the idea by looking at the 2D case.
 
  • #20
pasmith said:
[itex]\dot J[/itex] is [itex]\frac{\partial J}{\partial t}[/itex] where, as @Mark44 has already said,
[tex]J = \begin{vmatrix} \frac{\partial x_1}{\partial X_1} &\frac{\partial x_1}{\partial X_2} & \frac{\partial x_1}{\partial X_3}\\ \frac{\partial x_2}{\partial X_1} &\frac{\partial x_2}{\partial X_2} & \frac{\partial x_2}{\partial X_3} \\ \frac{\partial x_3}{\partial X_1} &\frac{\partial x_3}{\partial X_2} & \frac{\partial x_3}{\partial X_3} \end{vmatrix}.[/tex]

The algebra is not difficult, just tedious. You can however see the idea by looking at the 2D case.
Here ##\dot J=\frac{dJ}{dt}##,##J## is here the Jacobian determinant which is ##\frac{\partial x_1}{\partial X_1}\frac{\partial x_2}{\partial X_2}\frac{\partial x_3}{\partial X_3}##
 
  • #21
It is, I think, a sufficiently common abuse of notation for even partial derivatives with respect to time to be indicated by a dot. However the context here absolutely requires a partial derivative: [itex]J[/itex] is a function of time and a vector which indicates an initial position. It cannot sensibly be made a function of time so as to enable a total time derivative to be computed.

In general [itex]\frac{\partial x_i}{\partial X_j}[/itex] will not be diagonal, and its determinant [itex]J[/itex] will not be the product of the diagonal entries. If [itex]\frac{\partial x_i}{\partial X_j}[/itex] is diagonal, or there is a reason why the non-diagonal entries don't contribute to the determinant, then the authors must have made further assumptions which you have not communicated to us.
 
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  • #22
pasmith said:
It is, I think, a sufficiently common abuse of notation for even partial derivatives with respect to time to be indicated by a dot. However the context here absolutely requires a partial derivative: [itex]J[/itex] is a function of time and a vector which indicates an initial position. It cannot sensibly be made a function of time so as to enable a total time derivative to be computed.

In general [itex]J[/itex] will not be diagonal, and its determinant will not be the product of the diagonal entries. If [itex]J[/itex] is diagonal, or there is a reason why the non-diagonal entries don't contribute to the determinant, then the authors must have made further assumptions which you have not communicated to us.
##x_1(t)=f_1(X_1),x_2(t)=f_2(X_2),x_3=f_3(X_3)## or,##dx_1dx_2dx_3=\frac{\partial x_1}{\partial X_1}\frac{\partial x_2}{\partial X_2}\frac{\partial x_3}{\partial X_3}dX_1dX_2dX_3##,or ##d^3x=Jd^3X## as defined by author.
The functional form of ##f_1(X_1),f(X_2),f(X_3)## changes with time.
Therefore the Jacobian matrix is of course diagonal where ##J## is the determinant of it.
And if ##J## only depends upon time then ##\dot J=\frac{\partial J}{\partial t}##
 
  • #23
pasmith said:
The text is somewhat confusing (or at least the parts you have posted are without their surrounding context).

There are two ways of describing fluid flows:
  • The more widely used Eulerian specification describes the fluid by the instantaneous velocity [itex]\mathbf{v}[/itex] of the fluid element which at time [itex]t[/itex] is at position [itex]\mathbf{x}[/itex]. As time changes, different fluid elements will occupy this position. The independent variables are [itex]\mathbf{x}[/itex] and [itex]t[/itex].
  • The Lagrangian specification, used in the post in question, describes the fluid by the position [itex]\mathbf{x}[/itex] at time [itex]t[/itex] of the fluid element originally at position [itex]\mathbf{X}[/itex]. The independent variables are [itex]\mathbf{X}[/itex] and [itex]t[/itex].

The two specifications are related by [tex]
\mathbf{v}(\mathbf{x}(\mathbf{X},t),t) = \left.\frac{\partial \mathbf{x}}{\partial t}\right|_{(\mathbf{X},t)}.
[/tex]

The material derivative [tex]
\frac{D}{Dt} = \frac{\partial}{\partial t}+ \mathbf{v} \cdot \nabla[/tex] is relevant to the Eulerian specification, in which partial differentiation with respect to time is at a fixed position occupied by successive different fluid elements. Finding the rate of change with respect to time experienced by a particular fluid element requires the addition of the advective [itex]\mathbf{v} \cdot \nabla[/itex] term. (To be clear, the spatial derivative is with respect to [itex]\mathbf{x}[/itex].) But in the Lagrangian specification, which the text is using here, a particular fluid element is labelled by its initial position [itex]\mathbf{X}[/itex], and partial differentiation with respect to time holding [itex]\mathbf{X}[/itex] constant does "follow the fluid" without the need for an advective term.

The complication here is that when the author writes [itex]\theta = \nabla \cdot \mathbf{v}[/itex] they apparently intend differentiation not with respect to the initial position [itex]\mathbf{X}[/itex] but with respect to the current position [itex]\mathbf{x}[/itex]. Allowing for this and using the normal partial time derivative yields the result (at least in the 2D case, which is the only case I could be bothered to check).

Starting with @Mark44's expression for [itex]J[/itex] you can take the partial derivative with respect to time and then simplify some terms using [tex]
\frac{\partial^2 x_i}{\partial X_j\,\partial t} = \frac{\partial v_i}{\partial X_j}[/tex] since in the Lagrangian specificattion the velocity field is just the partial derivative of the current position with respect to time. You can then use the chain rule [tex]
\frac{\partial v_i}{\partial X_j} = \sum_k \frac{\partial v_i}{\partial x_k} \frac{\partial x_k}{\partial X_j}[/tex] and you will find some terms will cancel, giving a result equal to [tex]
J \theta = J \sum_i \frac{\partial v_i}{\partial x_i}.[/tex]
Thanks @pasmith ,the result is coming true
 

1. What is a Jacobian determinant?

A Jacobian determinant is a mathematical concept used in calculus and multivariate analysis. It represents the change in volume when a transformation is applied to a set of variables. In simpler terms, it measures how much a function is stretched or compressed at a specific point.

2. How is the Jacobian determinant used in time evolution?

In time evolution, the Jacobian determinant is used to describe how a system changes over time. It is often used in the study of dynamical systems and differential equations to understand how a system's state changes over time.

3. Why is the Jacobian determinant important in science?

The Jacobian determinant is important in science because it allows us to understand and analyze complex systems and their behavior over time. It is used in a variety of fields, including physics, engineering, and economics, to model and predict the evolution of systems.

4. How is the time evolution of a Jacobian determinant calculated?

The time evolution of a Jacobian determinant is calculated by taking the partial derivatives of the system's equations with respect to each variable and then multiplying them together. This results in a single value that represents the change in the system over time.

5. What are some real-world applications of the time evolution of a Jacobian determinant?

The time evolution of a Jacobian determinant has many applications, including weather forecasting, population dynamics, and financial modeling. It is also used in the study of biological systems, such as the growth and development of organisms. Additionally, it is used in control theory to design systems that can adapt and respond to changes over time.

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