Do partial derivatives of different coordinate systems commute? Consider an arbitrary set of coordinates $x^\mu$ and another set of coordinates $y^{\mu}$, which is a (lorentzian) transformation from $x^\mu$ given by $y^\mu = f(x^\mu)$.
So I want to know whether $\frac{\partial}{\partial x^\alpha}\frac{\partial}{\partial y^\beta} = \frac{\partial}{\partial y^\beta}\frac{\partial}{\partial x^\alpha}$ holds true or false?
 A: It depends on the transformation at hand, but in general the answer is no. It boils down to whether $x^\alpha$ can change while $y^\beta$ is kept constant.
Denote by $J_{\nu}^\mu(x) = \frac{\partial y^\mu}{\partial x^\nu} $ the Jacobian of the change of coordinates.
By the chain rule
$$ \frac{\partial}{\partial x^\alpha} = J_{\alpha}^\nu(y)\frac{\partial}{\partial y^\nu} $$
It is not difficult to see that it makes a difference whether this expression is acted upon by $\partial_{y^\beta}$ from the left or the right. The former generates an additional term
$$ \frac{\partial J_{\alpha}^\nu(y)}{\partial y^\beta} \frac{\partial}{\partial y^\nu} $$ which needn't vanish.
A: No they don't commute in general. At least not with the usual understanding that
$$
\frac{\partial}{\partial x^2}
$$
is the partial derivative with $x^1$, $x^3$ etc held fixed.
Here is a counterexample:
Let $x= r \cos \theta$,  $y= r \sin \theta$ be cartesian and polar coordinates. Then
$$
\frac{\partial x}{\partial y}=0\Rightarrow \frac{ \partial}{\partial r}  \frac{\partial x}{\partial y}=0.
$$
but
$$
\frac{\partial x}{\partial r}\equiv \left(\frac{\partial x}{\partial r}\right)_\theta= \cos \theta
$$
is the derivative with $\theta$ being held fixed. Now
$$
\frac {\partial} {\partial y} \frac{\partial x}{\partial r}= \frac {\partial} {\partial y}\cos \theta =\frac {\partial} {\partial y}\frac{x}{\sqrt{x^2+y^2}}\ne 0.
$$
so
$$
\left(\frac {\partial} {\partial y}\left( \frac{\partial x}{\partial r}\right)_\theta\right)_x \ne 
\left(\frac {\partial} {\partial r}\left( \frac{\partial x}{\partial y}\right)_x\right)_\theta,
$$
where I have made it explicit what is being held fixed for each derivative.
It may be that they commute if the transformation is linear, but even then I have doubts. Why don't you try and see if it it's OK in this restricted case?
