How to find the curl of a vector field which points in the theta direction?

[Delta2]

Well we all are humans, we can do mistakes. No one is perfect.

@benorin The fact that you admitted your mistake I believe it says a lot about your ethos.

 

[benorin]

@Delta2 thanks. Bonus points for quoting he who was my hero during my romance with philosophy in college: Socrates.

 

[etotheipi]

I think a Cartesian coordinate system is actually defined as an affine coordinate system with origin $O$ and an orthonormal basis {$\vec{e}_x, \vec{e}_y, \vec{e}_z$} where the coordinates of $P$ are a tuple containing the components of the position vector $\overrightarrow{OP} = x\vec{e}_x + y\vec{e}_y + z\vec{e}_z$.

Part of the confusion might be because it is not obvious how a polar/spherical coordinate system is defined, because the basis vectors depend on the position. I.e. we can only say $\overrightarrow{OP} = r\hat{r}$, and the components of this vector evidently don't give us the coordinates.

Perhaps someone (@Orodruin/@Delta2) might be able to elaborate on how the mapping from $A$ to $\mathbb{R}_n$ is achieved for polar coordinates?

 

[Orodruin]

Part of the confusion might be because it is not obvious how a polar/spherical coordinate system is defined, because the basis vectors depend on the position. I.e. we can only say $\overrightarrow{OP} = r\hat{r}$, and the components of this vector evidently don't give us the coordinates.

This is the key issue. In a Cartesian coordinate system, the coordinates also happen to be the components of the position vector. This is generally not true. A coordinate transformation is defined as a 1-to-1 map between two different sets of coordinates. It has no a priori relationship to the components of the position vector, but is just a function from an n-tuple of numbers to another n-tuple of numbers. For example, relating polar coordinates on $\mathbb R^2$ to Cartesian coordinates,
$$
x = \rho \cos\phi, \quad y = \rho \sin\phi.
$$