No, I calculated the jacobian directly from the transforms (54)(55). Had you done the same thing instead of swallowing Shubert's false claims of "decomposition" you would have learned about your error.
*sigh*. Fine. Step by step. In units where
k = 1, Eugene's non-linear transformation, copied almost directly out of page 9 of his essay, is:
$$
\begin{align}
t^{\prime} \,&=\, \gamma \bigl[ t \,-\, vx \,-\, S(x) \bigr]
\,+\, S \Bigl( \gamma \bigl[ x \,-\, vt \,+\, v S(x) \bigr] \Bigr)
\\ \\ \\
x^{\prime} \,&=\, \gamma \bigl[ x \,-\, vt \,+\, v S(x) \bigr] \;.
\end{align}
$$
Now, let's calculate the partial derivatives:
$$
\begin{align}
\frac{\partial t^{\prime}}{\partial t} \,&=\,
\gamma \,+\, \frac{\partial}{\partial t}
S \Bigl( \gamma \bigl[ x \,-\, vt \,+\, v S(x) \bigr] \Bigr)
\\ \\ \\
\,&=\, \gamma \,-\, \gamma v S^{\prime}(y) \;,
\end{align}
$$
where I've set $$y \,\equiv\, \gamma \bigl[ x \,-\, vt \,+\, v S(x) \bigr]$$. Moving on,
$$
\begin{align}
\frac{\partial t^{\prime}}{\partial x} \,&=\, - \, \gamma v \,-\, \gamma S^{\prime}(x)
\,+\, \frac{\partial}{\partial x}
S \Bigl( \gamma \bigl[ x \,-\, vt \,+\, v S(x) \bigr] \Bigr)
\\ \\ \\
\,&=\, -\, \gamma v \,-\, \gamma S^{\prime} (x)
\,+\, \gamma \bigl( 1 \,+\, v S^{\prime}(x) \bigr) S^{\prime}(y) \;.
\end{align}
$$
The partial derivatives of $$x^{\prime}$$ are easier to evaluate:
$$
\begin{align}
\frac{\partial x^{\prime}}{\partial t} \,&=\, -\, \gamma v
\\ \\ \\
\frac{\partial x^{\prime}}{\partial x} \,&=\, \gamma
\,+\, \gamma v S^{\prime}(x) \;.
\end{align}
$$
Easy, right? There should be no arguments here, since apart from setting
c = 1 these turn out to be the same partial derivatives you found in post #319. Congratulations, it turns out you
can apply the chain rule correctly after all! You had us all worried there, with your talk about differentiating
S with respect to itself and all that.
Now onto the part you
didn't bother to calculate: the Jacobian determinant. It's given by
$$
J_{C} \,=\, \frac{\partial t^{\prime}}{\partial t} \,
\frac{\partial x^{\prime}}{\partial x} \,-\,
\frac{\partial t^{\prime}}{\partial x} \,
\frac{\partial x^{\prime}}{\partial t} \;.
$$
The first part is:
$$
\begin{align}
\frac{\partial t^{\prime}}{\partial t} \,
\frac{\partial x^{\prime}}{\partial x}
\,&=\, \gamma^{2} \bigl( 1 \,-\, v S^{\prime}(y) \bigr)
\bigl( 1 \,+\, v S^{\prime}(x) \bigr)
\\ \\ \\
\,&=\, \gamma^{2} \Bigl[ 1 \,+\, v S^{\prime}(x) \,-\, v S^{\prime}(y)
\,-\, v^{2} S^{\prime}(x) S^{\prime}(y) \Bigr] \;.
\end{align}
$$
The second part is:
$$
\begin{align}
\frac{\partial x^{\prime}}{\partial t} \,
\frac{\partial t^{\prime}}{\partial x}
\,&=\, \gamma^{2} v \Bigl[ v \,+\, S^{\prime}(x) \,-\, S^{\prime}(y)
\,-\, v S^{\prime}(x) S^{\prime}(y) \Bigr]
\\ \\ \\
\,&=\, \gamma^{2} \Bigl[ v^{2} \,+\, v S^{\prime}(x) \,-\, v S^{\prime}(y)
\,-\, v^{2} S^{\prime}(x) S^{\prime}(y) \Bigr]
\end{align}
$$
The Jacobian determinant -- **drumroll** -- is just the difference of these two terms:
$$
\begin{align}
J_{C} \,&=\, \gamma^{2} ( 1 \,-\, v^{2} )
\\ \\ \\
\,&=\, 1 \;.
\end{align}
$$
Surprise!
You bought Shubert's claim on the decomposition. The claim is FALSE.
Er, wrong again:
$$
\theta^{-1}(t,\,x) \,=\,
\begin{pmatrix} t \,-\, S(x) \\ \\ \\ x \end{pmatrix} \;.
$$
Composing with the Lorentz transformation
$$
\Lambda(t,\,x) \,=\,
\begin{pmatrix}
\gamma ( t \,-\, vx ) \\ \\ \\ \gamma ( x \,-\, vt )
\end{pmatrix}
$$
yields
$$
\Lambda \,\circ\, \theta^{-1}(t,\, x) \,=\,
\begin{pmatrix}
\gamma \bigl[ t \,-\, S(x) \,-\, vx \bigr]
\\ \\ \\
\gamma \bigl[ x \,-\, vt \,+\, v S(x) \bigr]
\end{pmatrix} \;.
$$
Finally, $$\theta$$ is given by
$$
\theta^{-1}(t,\,x) \,=\,
\begin{pmatrix} t \,+\, S(x) \\ \\ \\ x \end{pmatrix} \;,
$$
so the composition $$\theta \,\circ\, \Lambda \,\circ\, \theta^{-1}$$ is given by:
$$
\theta \,\circ\, \Lambda \,\circ\, \theta^{-1} (t,\,x) \,=\,
\begin{pmatrix}
\gamma \bigl[ t \,-\, S(x) \,-\, vx \bigr]
\,+\, S \Bigl( \gamma \bigl[ x \,-\, vt \,+\, v S(x) \bigr] \Bigr)
\\ \\ \\
\gamma \bigl[ x \,-\, vt \,+\, v S(x) \bigr]
\end{pmatrix} \;,
$$
which is just Eugene's transformation. Surprise again!
Now, you were calling someone
lazy...?