What does Jacobian mean in this compactification context?

Question

Referring to Pg. 2 and 4 of this paper: the advection equation $$ \partial_tu + \partial_xu = 0 $$ under the compactifying coordinates $$ \rho(x) = \dfrac{x}{1+x} $$ becomes $$ \partial_tu + (1-\rho)^2\partial_\rho u = 0. $$ Then a time coordinate transformation is introduced, $$ \tau = t - \bigg(x + \dfrac{C}{1 + x}\bigg). $$ So I'm confused by the following sentence: "With the compactification we get the Jacobian $$ \begin{split} \partial_\tau &= \partial_t\\ \partial_x &= (-1+C\Omega^2)\partial_\tau + \Omega^2\partial_\rho, \end{split} $$ where $\Omega:=(1-\rho)$." What does the Jacobian here refer to? It doesn't make sense to me that it is the Jacobian matrix and determinant (Wikipedia). Thanks.

Answer 1

I suppose that the therm Jacobian is used here as synonymous of total derivative of the function $u$ that, with the given substitutions becomes: $$ u(t,x)=u\left(\tau+\frac{\rho}{1-\rho}+C(1-\rho),\frac{\rho}{1-\rho}\right) $$

So, for the total derivative (that is a $2\times 1$ matrix) we have: $$ (\partial_\tau,\partial_\rho)=\left(\partial_t\frac{\partial t}{\partial\tau}+\partial_x\frac{\partial x}{\partial\tau},\partial_t\frac{\partial t}{\partial\rho}+\partial_x\frac{\partial x}{\partial\rho} \right) $$

that, with a bit of algebra, gives:

$ \partial_\tau=\partial t $

$ \partial_\rho=\partial_x \frac{1}{(1-\rho)^2}+\partial_t \left(\frac{1}{(1-\rho)^2} -C \right) $

From which we have your final result.