Help with quite simple change of variables?

Question

Suppose I have $t=pt'+qx'$ and $x=rt'+sx'$ with $\phi(x,t)\rightarrow \phi(x',t')$

Would I be right in saying that $$\frac{\partial \phi}{\partial x}=\frac{\partial \phi}{\partial x'}\frac{1}{s}$$

Thanks

Answer 1

You need to be careful about the notation here.

I assume you are trying to find

$$\frac{\partial \phi}{\partial x }=\frac{\partial}{\partial x} \phi(x'(x,t),t'(x,t)),$$

where $\phi$ is some function from $R^2$ into $R$. Let $\phi_1$ and $\phi_2$ denote the partial derivatives of that function (with respect to first and second arguments). Then using the chain-rule,

$$\frac{\partial \phi}{\partial x }=\phi_1(x'(x,t),t'(x,t))\frac{\partial x'}{\partial x} + \phi_2(x'(x,t),t'(x,t))\frac{\partial t'}{\partial x}.$$

The partials $\frac{\partial x'}{\partial x}$ and $\frac{\partial t'}{\partial x}$ are found by inverting the transformation. This can be done at points where the Jacobian $ps-qr$ does not vanish.

In this case, we find

$$\frac{\partial \phi}{\partial x }=\frac{\phi_1(x'(x,t),t'(x,t))(-q) + \phi_2(x'(x,t),t'(x,t))p}{ps-qr}.$$