In this question it is worked out that the unit vectors in polar coordinates are a noncoordinate basis. I tried recalculating it, but I get a different result:
$$\hat{r}=cos(\theta)\hat{x}+sin(\theta)\hat{y}$$ $$\hat{\theta}=-sin(\theta)\hat{x}+cos(\theta)\hat{y}$$
I then just calculated the commutator $$[\hat{r},\hat{\theta}]=[cos(\theta)\hat{x}+sin(\theta)\hat{y},-sin(\theta)\hat{x}+cos(\theta)\hat{y}]$$ by using that $$\hat{x}=\partial/\partial x$$ and $$\hat{y}=\partial/\partial y$$
Since $\theta$ is a function of $x$ and $y$. One has to use the product rule when applying the partial derivatives.
Finally I get $$[\hat{r},\hat{\theta}]=-\partial^2/\partial x^2-\partial^2/\partial y^2$$
So they really don't commute, but I get a different result as in the link above? Did I do something wrong? Note that one need not calculate the derivatives of $\theta$ wrt. $x$ and $y$. Many terms cancel when just stating the derivative.
