Changing Variables in PDEs: Understanding the Chain Rule

[AxiomOfChoice]

Suppose you start with a function $f(x,y,t)$ which satisfies some partial differential equation in the variables $x,y,t$. Suppose you make a change of variables $x,y,t \to \xi,z,\tau$, where $\tau = g_\tau(x,y,t)$ and similarly for $\xi$ and $z$. If you want to know what the differential operators $\partial_t, \partial_x$, and $\partial_y$ look like in these variables, don't you need to do something like
$$\frac{\partial}{\partial t} = \frac{\partial}{\partial \tau}\frac{\partial \tau}{\partial t} + \frac{\partial}{\partial \xi}\frac{\partial \xi}{\partial t} + \frac{\partial}{\partial z}\frac{\partial z}{\partial t} = \frac{\partial}{\partial \tau}\frac{\partial}{\partial t}g_\tau + \frac{\partial}{\partial \xi}\frac{\partial}{\partial t} g_\xi+ \frac{\partial}{\partial z}\frac{\partial}{\partial t}g_z,$$
and similarly for the other variables?

 

[HallsofIvy]

Yes, that's right- you change variables in a differential equation (ordinary or partial) by using the chain rule just as you did.