Are partitial differential operators commutative? I am trying to convert $\Delta=\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}$ to polar coordinates. If anyone has any references on how to do that I would appreciate it.
In my evaluation, I am messing with a lot of partials. Alghough, I am not sure as to if partials are commutative, and google has not been helping me. Enough of a prompt; is this true?
$$
\frac{\partial u}{\partial x}\frac{\partial y}{\partial r}=\frac{\partial u}{\partial r}\frac{\partial y}{\partial x}
$$
In the previous equation I just switched the denominators.

What you're looking for is the "Polar Laplacian".  Googling "Polar Laplacian Derivation" leads to many papers such as this one that go through step-by-step with how to derive it.
If after reading this you're still interested in a challenge, try deriving the Laplacian for Spherical or Cylindrical coordinates.
