Affects of a coordinate transformation

Question

I am attempting to solve a PDE of $f(r,t)$, where $r\in[0,g(t)]$ is a spacial coordinate and $t$ is time. The PDE is coupled to to an ODE for $g(t)$.

I wish to simplify the problem by defining a new independent variable

$$ y \equiv \frac{r}{g(t)}. $$

Is it true that, since $y$ is an independent variable,

$$ \frac{dy}{dt} = 0 = -\frac{r}{\left(g(t)\right)^2}\frac{dg}{dt},$$

therefore $dg/dt = 0$ in this new coordinate system / reference frame? Since the PDE for $f$ involves $dg/dt$, this seems to make the problem too easy.

Additionally, is there any recommended reading on this subject?

Answer 1

No, it is not true that $dy/dt = 0$. The independent variable must be restated in terms of the new variable, so the complete change of variables is described by

$$\begin{align} q(y,t) &= f(r,t) \\ y &= r/R \end{align}$$

where by the chain rule

$$\frac{\partial f}{\partial t} = \frac{\partial q}{\partial t} + \frac{\partial y}{\partial t}\frac{\partial q}{\partial y}.$$

Recommended reading on a formal treatment of change of variables is still unbeknownst to me.