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About the derivative of the Jacobian in fluid dynamics
ODE for the Jacobian determinant of the flow mapIs Euler's lemma of fluid mechanics a nonlinear version of Liouville's theorem of ODEs?Independency of the frame of reference of the strain rate tensorImplementation of a simulation of an incompressible Newtonian fluid with uniform densityHow can we describe the evolution of a density “injected” into an incompressible Newtonian fluid?find the distance between $P_1$ and $P_2$ as a function of timeODE for the Jacobian determinant of the flow mapfluid dynamics - How to understand material derivative?Finding the Density Change of a FluidProving Euler's identity (fluid mechanics) by differentiating the determinant $J=partial(x,y,z)/partial(X,Y,Z)$Is the inverse of the Jacobian equivalent to the Jacobian of the inverse?
$begingroup$
I was studying a book on the mathematics of fluid dynamics in which there was a lemma on how to find the derivative of the Jacobian. The explanation is as follows (Sorry if it's too long):
There is a region $D$ in Euclidean space where there is a fluid whose velocity at any point $mathbf vec xin D$ at any time $t$ is given by the vector field $mathbfvec u(mathbfvec x, t)$. Let us write $mathbfvec varphi(mathbfvec x, t)$ for the trajectory followed by the particle that is at
point $mathbfvec x$ at time $t = 0$. We will assume $varphi$ is smooth enough so the following
manipulations are legitimate and for fixed $t$, $mathbfvec varphi$ is an invertible mapping.
Let $varphi_t$ denote the map $mathbfvec x rightarrow mathbfvec varphi(mathbfvec x, t)$; that is, with fixed $t$, this map advances
each fluid particle from its position at time $t = 0$ to its position at time $t$.
Here, of course, the subscript does not denote differentiation. We call $mathbfvec varphi$ the
fluid flow map. If $W$ is a region in $D$, then $varphi_t(W) = W_t$ is the volume
$W$ moving with the fluid as shown in figure.
Now let's say that $mathbfvec x=xhat i+yhat j+zhat k$; $mathbfvec varphi=epsilon hat i+etahat j+zeta hat k$ and $mathbfvec u=uhat i+vhat j+what k$. Since $mathbfvec varphi$ is the displacement field and $mathbfvec u$ is the velocity field so we have
$$fracdeltadelta t(mathbfvec varphi(mathbfvec x, t))=mathbfvec u(mathbfvec varphi(mathbfvec x, t), t)$$
So
$$fracdeltaepsilondelta t=u; fracdeltaetadelta t=v; fracdeltazetadelta t=w cdots(i)$$
The Jacobian of $mathbfvec x$ w.r.t $mathbfvec varphi(mathbfvec x, t)$ is given as:
$$J(mathbf vec x,t)=left[beginmatrix
displaystylefracdeltaepsilondelta x&displaystylefracdeltaetadelta x&displaystylefracdeltazetadelta x\
displaystylefracdeltaepsilondelta y&displaystylefracdeltaetadelta y&displaystylefracdeltazetadelta y\
displaystylefracdeltaepsilondelta z&displaystylefracdeltaetadelta z&displaystylefracdeltazetadelta z
endmatrixright]$$
Now the task is to prove that
$$displaystylefracdeltadelta tJ(mathbfvec x, t) = J(mathbfvec x, t)left[textdiv mathbfvec u(mathbfvec varphi(mathbfvec x, t), t)right]$$
What I did:
$$beginalign
displaystylefracdeltadelta tJ&=left[beginmatrix
displaystylefracdeltadelta tfracdeltaepsilondelta x&displaystylefracdeltaetadelta x&displaystylefracdeltazetadelta x\
displaystylefracdeltadelta tfracdeltaepsilondelta y&displaystylefracdeltaetadelta y&displaystylefracdeltazetadelta y\
displaystylefracdeltadelta tfracdeltaepsilondelta z&displaystylefracdeltaetadelta z&displaystylefracdeltazetadelta z
endmatrixright]+left[beginmatrix
displaystylefracdeltaepsilondelta x&displaystylefracdeltadelta tfracdeltaetadelta x&displaystylefracdeltazetadelta x\
displaystylefracdeltaepsilondelta y&displaystylefracdeltadelta tfracdeltaetadelta y&displaystylefracdeltazetadelta y\
displaystylefracdeltaepsilondelta z&displaystylefracdeltadelta tfracdeltaetadelta z&displaystylefracdeltazetadelta z
endmatrixright]+left[beginmatrix
displaystylefracdeltaepsilondelta x&displaystylefracdeltaetadelta x&displaystylefracdeltadelta tfracdeltazetadelta x\
displaystylefracdeltaepsilondelta y&displaystylefracdeltaetadelta y&displaystylefracdeltadelta tfracdeltazetadelta y\
displaystylefracdeltaepsilondelta z&displaystylefracdeltaetadelta z&displaystylefracdeltadelta tfracdeltazetadelta z
endmatrixright]\
&=left[beginmatrix
displaystylefracdelta udelta x&displaystylefracdeltaetadelta x&displaystylefracdeltazetadelta x\
displaystylefracdelta udelta y&displaystylefracdeltaetadelta y&displaystylefracdeltazetadelta y\
displaystylefracdelta udelta z&displaystylefracdeltaetadelta z&displaystylefracdeltazetadelta z
endmatrixright]+left[beginmatrix
displaystylefracdeltaepsilondelta x&displaystylefracdelta vdelta x&displaystylefracdeltazetadelta x\
displaystylefracdeltaepsilondelta y&displaystylefracdelta vdelta y&displaystylefracdeltazetadelta y\
displaystylefracdeltaepsilondelta z&displaystylefracdelta vdelta z&displaystylefracdeltazetadelta z
endmatrixright]+left[beginmatrix
displaystylefracdeltaepsilondelta x&displaystylefracdeltaetadelta x&displaystylefracdelta zdelta x\
displaystylefracdeltaepsilondelta y&displaystylefracdeltaetadelta y&displaystylefracdelta zdelta y\
displaystylefracdeltaepsilondelta z&displaystylefracdeltaetadelta z&displaystylefracdelta zdelta z
endmatrixright] [textBy (i)]
endalign$$
At this point the book says to use the fact that
$$fracdelta udelta x=fracdelta udelta epsilonfracdelta epsilondelta x+fracdelta udelta etafracdelta etadelta x+fracdelta udelta zetafracdelta zetadelta x$$
$$fracdelta udelta y=fracdelta udelta epsilonfracdelta epsilondelta y+fracdelta udelta etafracdelta etadelta y+fracdelta udelta zetafracdelta zetadelta y$$
$$vdots$$
$$fracdelta wdelta z=fracdelta wdelta epsilonfracdelta epsilondelta z+fracdelta wdelta etafracdelta etadelta z+fracdelta wdelta zetafracdelta zetadelta z$$
However I can't quite understand how to use these equations to continue the determinant. Please help me in this problem.
Thanks for the attention.
vector-fields fluid-dynamics jacobian
asked Mar 29 at 9:36
Faiq IrfanFaiq Irfan
8311317
$endgroup$
add a comment |
$begingroup$
I was studying a book on the mathematics of fluid dynamics in which there was a lemma on how to find the derivative of the Jacobian. The explanation is as follows (Sorry if it's too long):
There is a region $D$ in Euclidean space where there is a fluid whose velocity at any point $mathbf vec xin D$ at any time $t$ is given by the vector field $mathbfvec u(mathbfvec x, t)$. Let us write $mathbfvec varphi(mathbfvec x, t)$ for the trajectory followed by the particle that is at
point $mathbfvec x$ at time $t = 0$. We will assume $varphi$ is smooth enough so the following
manipulations are legitimate and for fixed $t$, $mathbfvec varphi$ is an invertible mapping.
Let $varphi_t$ denote the map $mathbfvec x rightarrow mathbfvec varphi(mathbfvec x, t)$; that is, with fixed $t$, this map advances
each fluid particle from its position at time $t = 0$ to its position at time $t$.
Here, of course, the subscript does not denote differentiation. We call $mathbfvec varphi$ the
fluid flow map. If $W$ is a region in $D$, then $varphi_t(W) = W_t$ is the volume
$W$ moving with the fluid as shown in figure.
Now let's say that $mathbfvec x=xhat i+yhat j+zhat k$; $mathbfvec varphi=epsilon hat i+etahat j+zeta hat k$ and $mathbfvec u=uhat i+vhat j+what k$. Since $mathbfvec varphi$ is the displacement field and $mathbfvec u$ is the velocity field so we have
$$fracdeltadelta t(mathbfvec varphi(mathbfvec x, t))=mathbfvec u(mathbfvec varphi(mathbfvec x, t), t)$$
So
$$fracdeltaepsilondelta t=u; fracdeltaetadelta t=v; fracdeltazetadelta t=w cdots(i)$$
The Jacobian of $mathbfvec x$ w.r.t $mathbfvec varphi(mathbfvec x, t)$ is given as:
$$J(mathbf vec x,t)=left[beginmatrix
displaystylefracdeltaepsilondelta x&displaystylefracdeltaetadelta x&displaystylefracdeltazetadelta x\
displaystylefracdeltaepsilondelta y&displaystylefracdeltaetadelta y&displaystylefracdeltazetadelta y\
displaystylefracdeltaepsilondelta z&displaystylefracdeltaetadelta z&displaystylefracdeltazetadelta z
endmatrixright]$$
Now the task is to prove that
$$displaystylefracdeltadelta tJ(mathbfvec x, t) = J(mathbfvec x, t)left[textdiv mathbfvec u(mathbfvec varphi(mathbfvec x, t), t)right]$$
What I did:
$$beginalign
displaystylefracdeltadelta tJ&=left[beginmatrix
displaystylefracdeltadelta tfracdeltaepsilondelta x&displaystylefracdeltaetadelta x&displaystylefracdeltazetadelta x\
displaystylefracdeltadelta tfracdeltaepsilondelta y&displaystylefracdeltaetadelta y&displaystylefracdeltazetadelta y\
displaystylefracdeltadelta tfracdeltaepsilondelta z&displaystylefracdeltaetadelta z&displaystylefracdeltazetadelta z
endmatrixright]+left[beginmatrix
displaystylefracdeltaepsilondelta x&displaystylefracdeltadelta tfracdeltaetadelta x&displaystylefracdeltazetadelta x\
displaystylefracdeltaepsilondelta y&displaystylefracdeltadelta tfracdeltaetadelta y&displaystylefracdeltazetadelta y\
displaystylefracdeltaepsilondelta z&displaystylefracdeltadelta tfracdeltaetadelta z&displaystylefracdeltazetadelta z
endmatrixright]+left[beginmatrix
displaystylefracdeltaepsilondelta x&displaystylefracdeltaetadelta x&displaystylefracdeltadelta tfracdeltazetadelta x\
displaystylefracdeltaepsilondelta y&displaystylefracdeltaetadelta y&displaystylefracdeltadelta tfracdeltazetadelta y\
displaystylefracdeltaepsilondelta z&displaystylefracdeltaetadelta z&displaystylefracdeltadelta tfracdeltazetadelta z
endmatrixright]\
&=left[beginmatrix
displaystylefracdelta udelta x&displaystylefracdeltaetadelta x&displaystylefracdeltazetadelta x\
displaystylefracdelta udelta y&displaystylefracdeltaetadelta y&displaystylefracdeltazetadelta y\
displaystylefracdelta udelta z&displaystylefracdeltaetadelta z&displaystylefracdeltazetadelta z
endmatrixright]+left[beginmatrix
displaystylefracdeltaepsilondelta x&displaystylefracdelta vdelta x&displaystylefracdeltazetadelta x\
displaystylefracdeltaepsilondelta y&displaystylefracdelta vdelta y&displaystylefracdeltazetadelta y\
displaystylefracdeltaepsilondelta z&displaystylefracdelta vdelta z&displaystylefracdeltazetadelta z
endmatrixright]+left[beginmatrix
displaystylefracdeltaepsilondelta x&displaystylefracdeltaetadelta x&displaystylefracdelta zdelta x\
displaystylefracdeltaepsilondelta y&displaystylefracdeltaetadelta y&displaystylefracdelta zdelta y\
displaystylefracdeltaepsilondelta z&displaystylefracdeltaetadelta z&displaystylefracdelta zdelta z
endmatrixright] [textBy (i)]
endalign$$
At this point the book says to use the fact that
$$fracdelta udelta x=fracdelta udelta epsilonfracdelta epsilondelta x+fracdelta udelta etafracdelta etadelta x+fracdelta udelta zetafracdelta zetadelta x$$
$$fracdelta udelta y=fracdelta udelta epsilonfracdelta epsilondelta y+fracdelta udelta etafracdelta etadelta y+fracdelta udelta zetafracdelta zetadelta y$$
$$vdots$$
$$fracdelta wdelta z=fracdelta wdelta epsilonfracdelta epsilondelta z+fracdelta wdelta etafracdelta etadelta z+fracdelta wdelta zetafracdelta zetadelta z$$
However I can't quite understand how to use these equations to continue the determinant. Please help me in this problem.
Thanks for the attention.
vector-fields fluid-dynamics jacobian
asked Mar 29 at 9:36
Faiq IrfanFaiq Irfan
8311317
$endgroup$
$begingroup$
See this answer for a derivation using more compact notation.
$endgroup$
– RRL
Mar 29 at 9:45
add a comment |
$begingroup$
I was studying a book on the mathematics of fluid dynamics in which there was a lemma on how to find the derivative of the Jacobian. The explanation is as follows (Sorry if it's too long):
There is a region $D$ in Euclidean space where there is a fluid whose velocity at any point $mathbf vec xin D$ at any time $t$ is given by the vector field $mathbfvec u(mathbfvec x, t)$. Let us write $mathbfvec varphi(mathbfvec x, t)$ for the trajectory followed by the particle that is at
point $mathbfvec x$ at time $t = 0$. We will assume $varphi$ is smooth enough so the following
manipulations are legitimate and for fixed $t$, $mathbfvec varphi$ is an invertible mapping.
Let $varphi_t$ denote the map $mathbfvec x rightarrow mathbfvec varphi(mathbfvec x, t)$; that is, with fixed $t$, this map advances
each fluid particle from its position at time $t = 0$ to its position at time $t$.
Here, of course, the subscript does not denote differentiation. We call $mathbfvec varphi$ the
fluid flow map. If $W$ is a region in $D$, then $varphi_t(W) = W_t$ is the volume
$W$ moving with the fluid as shown in figure.
Now let's say that $mathbfvec x=xhat i+yhat j+zhat k$; $mathbfvec varphi=epsilon hat i+etahat j+zeta hat k$ and $mathbfvec u=uhat i+vhat j+what k$. Since $mathbfvec varphi$ is the displacement field and $mathbfvec u$ is the velocity field so we have
$$fracdeltadelta t(mathbfvec varphi(mathbfvec x, t))=mathbfvec u(mathbfvec varphi(mathbfvec x, t), t)$$
So
$$fracdeltaepsilondelta t=u; fracdeltaetadelta t=v; fracdeltazetadelta t=w cdots(i)$$
The Jacobian of $mathbfvec x$ w.r.t $mathbfvec varphi(mathbfvec x, t)$ is given as:
$$J(mathbf vec x,t)=left[beginmatrix
displaystylefracdeltaepsilondelta x&displaystylefracdeltaetadelta x&displaystylefracdeltazetadelta x\
displaystylefracdeltaepsilondelta y&displaystylefracdeltaetadelta y&displaystylefracdeltazetadelta y\
displaystylefracdeltaepsilondelta z&displaystylefracdeltaetadelta z&displaystylefracdeltazetadelta z
endmatrixright]$$
Now the task is to prove that
$$displaystylefracdeltadelta tJ(mathbfvec x, t) = J(mathbfvec x, t)left[textdiv mathbfvec u(mathbfvec varphi(mathbfvec x, t), t)right]$$
What I did:
$$beginalign
displaystylefracdeltadelta tJ&=left[beginmatrix
displaystylefracdeltadelta tfracdeltaepsilondelta x&displaystylefracdeltaetadelta x&displaystylefracdeltazetadelta x\
displaystylefracdeltadelta tfracdeltaepsilondelta y&displaystylefracdeltaetadelta y&displaystylefracdeltazetadelta y\
displaystylefracdeltadelta tfracdeltaepsilondelta z&displaystylefracdeltaetadelta z&displaystylefracdeltazetadelta z
endmatrixright]+left[beginmatrix
displaystylefracdeltaepsilondelta x&displaystylefracdeltadelta tfracdeltaetadelta x&displaystylefracdeltazetadelta x\
displaystylefracdeltaepsilondelta y&displaystylefracdeltadelta tfracdeltaetadelta y&displaystylefracdeltazetadelta y\
displaystylefracdeltaepsilondelta z&displaystylefracdeltadelta tfracdeltaetadelta z&displaystylefracdeltazetadelta z
endmatrixright]+left[beginmatrix
displaystylefracdeltaepsilondelta x&displaystylefracdeltaetadelta x&displaystylefracdeltadelta tfracdeltazetadelta x\
displaystylefracdeltaepsilondelta y&displaystylefracdeltaetadelta y&displaystylefracdeltadelta tfracdeltazetadelta y\
displaystylefracdeltaepsilondelta z&displaystylefracdeltaetadelta z&displaystylefracdeltadelta tfracdeltazetadelta z
endmatrixright]\
&=left[beginmatrix
displaystylefracdelta udelta x&displaystylefracdeltaetadelta x&displaystylefracdeltazetadelta x\
displaystylefracdelta udelta y&displaystylefracdeltaetadelta y&displaystylefracdeltazetadelta y\
displaystylefracdelta udelta z&displaystylefracdeltaetadelta z&displaystylefracdeltazetadelta z
endmatrixright]+left[beginmatrix
displaystylefracdeltaepsilondelta x&displaystylefracdelta vdelta x&displaystylefracdeltazetadelta x\
displaystylefracdeltaepsilondelta y&displaystylefracdelta vdelta y&displaystylefracdeltazetadelta y\
displaystylefracdeltaepsilondelta z&displaystylefracdelta vdelta z&displaystylefracdeltazetadelta z
endmatrixright]+left[beginmatrix
displaystylefracdeltaepsilondelta x&displaystylefracdeltaetadelta x&displaystylefracdelta zdelta x\
displaystylefracdeltaepsilondelta y&displaystylefracdeltaetadelta y&displaystylefracdelta zdelta y\
displaystylefracdeltaepsilondelta z&displaystylefracdeltaetadelta z&displaystylefracdelta zdelta z
endmatrixright] [textBy (i)]
endalign$$
At this point the book says to use the fact that
$$fracdelta udelta x=fracdelta udelta epsilonfracdelta epsilondelta x+fracdelta udelta etafracdelta etadelta x+fracdelta udelta zetafracdelta zetadelta x$$
$$fracdelta udelta y=fracdelta udelta epsilonfracdelta epsilondelta y+fracdelta udelta etafracdelta etadelta y+fracdelta udelta zetafracdelta zetadelta y$$
$$vdots$$
$$fracdelta wdelta z=fracdelta wdelta epsilonfracdelta epsilondelta z+fracdelta wdelta etafracdelta etadelta z+fracdelta wdelta zetafracdelta zetadelta z$$
However I can't quite understand how to use these equations to continue the determinant. Please help me in this problem.
Thanks for the attention.
vector-fields fluid-dynamics jacobian
asked Mar 29 at 9:36
Faiq IrfanFaiq Irfan
8311317
$endgroup$
I was studying a book on the mathematics of fluid dynamics in which there was a lemma on how to find the derivative of the Jacobian. The explanation is as follows (Sorry if it's too long):
There is a region $D$ in Euclidean space where there is a fluid whose velocity at any point $mathbf vec xin D$ at any time $t$ is given by the vector field $mathbfvec u(mathbfvec x, t)$. Let us write $mathbfvec varphi(mathbfvec x, t)$ for the trajectory followed by the particle that is at
point $mathbfvec x$ at time $t = 0$. We will assume $varphi$ is smooth enough so the following
manipulations are legitimate and for fixed $t$, $mathbfvec varphi$ is an invertible mapping.
Let $varphi_t$ denote the map $mathbfvec x rightarrow mathbfvec varphi(mathbfvec x, t)$; that is, with fixed $t$, this map advances
each fluid particle from its position at time $t = 0$ to its position at time $t$.
Here, of course, the subscript does not denote differentiation. We call $mathbfvec varphi$ the
fluid flow map. If $W$ is a region in $D$, then $varphi_t(W) = W_t$ is the volume
$W$ moving with the fluid as shown in figure.
Now let's say that $mathbfvec x=xhat i+yhat j+zhat k$; $mathbfvec varphi=epsilon hat i+etahat j+zeta hat k$ and $mathbfvec u=uhat i+vhat j+what k$. Since $mathbfvec varphi$ is the displacement field and $mathbfvec u$ is the velocity field so we have
$$fracdeltadelta t(mathbfvec varphi(mathbfvec x, t))=mathbfvec u(mathbfvec varphi(mathbfvec x, t), t)$$
So
$$fracdeltaepsilondelta t=u; fracdeltaetadelta t=v; fracdeltazetadelta t=w cdots(i)$$
The Jacobian of $mathbfvec x$ w.r.t $mathbfvec varphi(mathbfvec x, t)$ is given as:
$$J(mathbf vec x,t)=left[beginmatrix
displaystylefracdeltaepsilondelta x&displaystylefracdeltaetadelta x&displaystylefracdeltazetadelta x\
displaystylefracdeltaepsilondelta y&displaystylefracdeltaetadelta y&displaystylefracdeltazetadelta y\
displaystylefracdeltaepsilondelta z&displaystylefracdeltaetadelta z&displaystylefracdeltazetadelta z
endmatrixright]$$
Now the task is to prove that
$$displaystylefracdeltadelta tJ(mathbfvec x, t) = J(mathbfvec x, t)left[textdiv mathbfvec u(mathbfvec varphi(mathbfvec x, t), t)right]$$
What I did:
$$beginalign
displaystylefracdeltadelta tJ&=left[beginmatrix
displaystylefracdeltadelta tfracdeltaepsilondelta x&displaystylefracdeltaetadelta x&displaystylefracdeltazetadelta x\
displaystylefracdeltadelta tfracdeltaepsilondelta y&displaystylefracdeltaetadelta y&displaystylefracdeltazetadelta y\
displaystylefracdeltadelta tfracdeltaepsilondelta z&displaystylefracdeltaetadelta z&displaystylefracdeltazetadelta z
endmatrixright]+left[beginmatrix
displaystylefracdeltaepsilondelta x&displaystylefracdeltadelta tfracdeltaetadelta x&displaystylefracdeltazetadelta x\
displaystylefracdeltaepsilondelta y&displaystylefracdeltadelta tfracdeltaetadelta y&displaystylefracdeltazetadelta y\
displaystylefracdeltaepsilondelta z&displaystylefracdeltadelta tfracdeltaetadelta z&displaystylefracdeltazetadelta z
endmatrixright]+left[beginmatrix
displaystylefracdeltaepsilondelta x&displaystylefracdeltaetadelta x&displaystylefracdeltadelta tfracdeltazetadelta x\
displaystylefracdeltaepsilondelta y&displaystylefracdeltaetadelta y&displaystylefracdeltadelta tfracdeltazetadelta y\
displaystylefracdeltaepsilondelta z&displaystylefracdeltaetadelta z&displaystylefracdeltadelta tfracdeltazetadelta z
endmatrixright]\
&=left[beginmatrix
displaystylefracdelta udelta x&displaystylefracdeltaetadelta x&displaystylefracdeltazetadelta x\
displaystylefracdelta udelta y&displaystylefracdeltaetadelta y&displaystylefracdeltazetadelta y\
displaystylefracdelta udelta z&displaystylefracdeltaetadelta z&displaystylefracdeltazetadelta z
endmatrixright]+left[beginmatrix
displaystylefracdeltaepsilondelta x&displaystylefracdelta vdelta x&displaystylefracdeltazetadelta x\
displaystylefracdeltaepsilondelta y&displaystylefracdelta vdelta y&displaystylefracdeltazetadelta y\
displaystylefracdeltaepsilondelta z&displaystylefracdelta vdelta z&displaystylefracdeltazetadelta z
endmatrixright]+left[beginmatrix
displaystylefracdeltaepsilondelta x&displaystylefracdeltaetadelta x&displaystylefracdelta zdelta x\
displaystylefracdeltaepsilondelta y&displaystylefracdeltaetadelta y&displaystylefracdelta zdelta y\
displaystylefracdeltaepsilondelta z&displaystylefracdeltaetadelta z&displaystylefracdelta zdelta z
endmatrixright] [textBy (i)]
endalign$$
At this point the book says to use the fact that
$$fracdelta udelta x=fracdelta udelta epsilonfracdelta epsilondelta x+fracdelta udelta etafracdelta etadelta x+fracdelta udelta zetafracdelta zetadelta x$$
$$fracdelta udelta y=fracdelta udelta epsilonfracdelta epsilondelta y+fracdelta udelta etafracdelta etadelta y+fracdelta udelta zetafracdelta zetadelta y$$
$$vdots$$
$$fracdelta wdelta z=fracdelta wdelta epsilonfracdelta epsilondelta z+fracdelta wdelta etafracdelta etadelta z+fracdelta wdelta zetafracdelta zetadelta z$$
However I can't quite understand how to use these equations to continue the determinant. Please help me in this problem.
Thanks for the attention.
vector-fields fluid-dynamics jacobian
vector-fields fluid-dynamics jacobian
asked Mar 29 at 9:36
Faiq IrfanFaiq Irfan
8311317
asked Mar 29 at 9:36
Faiq IrfanFaiq Irfan
8311317
asked Mar 29 at 9:36
Faiq IrfanFaiq Irfan
8311317
asked Mar 29 at 9:36
Faiq IrfanFaiq Irfan
8311317
asked Mar 29 at 9:36
Faiq IrfanFaiq Irfan
8311317
8311317
$begingroup$
See this answer for a derivation using more compact notation.
$endgroup$
– RRL
Mar 29 at 9:45
add a comment |
$begingroup$
See this answer for a derivation using more compact notation.
$endgroup$
– RRL
Mar 29 at 9:45
$begingroup$
See this answer for a derivation using more compact notation.
$endgroup$
– RRL
Mar 29 at 9:45
$begingroup$
See this answer for a derivation using more compact notation.
$endgroup$
– RRL
Mar 29 at 9:45
add a comment |
0
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$begingroup$
See this answer for a derivation using more compact notation.
$endgroup$
– RRL
Mar 29 at 9:45