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
<br /> \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,<br />
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.