Curl vs exterior derivative in spherical coords

[pellman]

I am trying to get a good grasp of the relation between the curl of a vector field and the exterior derivative of a 1-form field. In cartesian coordinates for flat R^3 the relationship is misleadingly simple. However, it still requires us to make an identification of the 2-form basis $$dx \wedge dy$$ with the basis vector $$\mathbf{\hat{e}}_z$$, the justification of which in general is not clear to me.

Consider instead spherical coordinates on R^3. Given a vector field

$$\mathbf{A}=A'^r\mathbf{\hat{r}} +A'^{\theta}\mathbf{\hat{\theta}} +A'^{\phi}\mathbf{\hat{\phi}}$$

The basis vectors here are unit vectors. We have put a prime on the components to distinguish them from the components with respect to the coordinate basis of the tangent space. In the coordinate basis we have

$$\mathbf{A}=A^r\mathbf{\hat{e}}_r +A^{\theta}\mathbf{\hat{e}}_\theta +A^{\phi}\mathbf{\hat{e}}_\phi$$

where $$\mathbf{\hat{e}}_r = \mathbf{\hat{r}}$$ $$\mathbf{\hat{e}}_\theta=r\mathbf{\hat{\theta}}$$ $$\mathbf{\hat{e}}_\phi=r\sin\theta\mathbf{\hat{\phi}}$$

The $$\mathbf{\hat{e}}_j$$ basis vectors are essentially identical to $$\partial_j$$. (The relationship between $$\mathbf{\hat{e}}_r,\mathbf{\hat{e}}_\theta,\mathbf{\hat{e}}_\phi$$ and $$\mathbf{\hat{e}}_x=\mathbf{\hat{x}},\mathbf{\hat{e}}_y=\mathbf{\hat{y}}$$$$\mathbf{\hat{e}}_z=\mathbf{\hat{z}}$$ is identical to the relationship between $$\partial_r,\partial_\theta,\partial_\phi$$ and $$\partial_x,\partial_y,\partial_z$$) Now the curl of A is usually given with respect to the unit-vector (orthonormal) basis:

$$\nabla\times\mathbf{A}=\frac{1}{r\sin\theta}\left(\frac{\partial}{\partial\theta}(A'^\phi\sin\theta)-\frac{\partial A'^\theta}{\partial\phi}\right)\mathbf{\hat{r}} + \frac{1}{r} \left( \frac{1}{\sin\theta} \frac{\partial A'^r}{\partial \phi}- \frac{\partial}{\partial r}(rA'^{\phi}) \right)\mathbf{\hat{\theta}} + \frac{1}{r} \left( \frac{\partial}{\partial r}(rA'^{\theta}) - \frac{\partial A'^r}{\partial \theta} \right)\mathbf{\hat{\phi}}$$

We can form a basis $$dr,d\theta,d\phi$$ dual to the coordinate basis and consider the 1-form $$A=A_r dr + A_\theta d\theta + A_\phi d\phi$$ with exterior derivative $$dA = \left( \frac{\partial A_\phi}{\partial\theta} - \frac{\partial A_\theta}{\partial\phi} \right)d\theta \wedge d\phi + \left( \frac{\partial A_r}{\partial\phi} - \frac{\partial A_\phi}{\partial r} \right)d\phi \wedge dr + \left( \frac{\partial A_\theta}{\partial r} - \frac{\partial A_r}{\partial\theta} \right)dr \wedge d\theta$$ the components of which are not clearly related to those of the curl.

It can be shown that the 1-form components (coordinate basis) are related to the vector components (normalized basis) by $$A'^r = A_r$$$$A'^\theta = \frac{1}{r} A_\theta$$$$A'^\phi = \frac{1}{r\sin\theta} A_\phi$$

If we write the expression for the curl in terms of the 1-form components and the coordinate basis vectors we get fairly close to the exterior derivative expression:

$$\nabla\times\mathbf{A}= \left( \frac{\partial A_\phi}{\partial\theta} - \frac{\partial A_\theta}{\partial\phi} \right) \frac{1}{r^2 \sin\theta} \mathbf{\hat{e}}_r + \left( \frac{\partial A_r}{\partial\phi} - \frac{\partial A_\phi}{\partial r} \right)\frac{1}{r^2 \sin\theta} \mathbf{\hat{e}}_\theta +\left( \frac{\partial A_\theta}{\partial r} - \frac{\partial A_r}{\partial\theta} \right) \frac{1}{r^2 \sin\theta} \mathbf{\hat{e}}_\phi$$
$$dA = \left( \frac{\partial A_\phi}{\partial\theta} - \frac{\partial A_\theta}{\partial\phi} \right)d\theta \wedge d\phi + \left( \frac{\partial A_r}{\partial\phi} - \frac{\partial A_\phi}{\partial r} \right)d\phi \wedge dr + \left( \frac{\partial A_\theta}{\partial r} - \frac{\partial A_r}{\partial\theta} \right)dr \wedge d\theta$$

So what is the relationship between these two expressions. How do I understand that they are equivalent?

$$\frac{1}{r^2 \sin\theta}$$ is equal to the Jacobian determinant $$\left| \frac{\partial (r,\theta,\phi)}{\partial (x,y,z)} \right|$$ but I am not sure if that is coincidental or relevant here.

 

[Almsco]

The relationship between the curl of a vector field and the exterior derivative of a 1-form field in different coordinate systems can be tricky to understand. You are correct in noting that the relationship between the two expressions for the curl of \mathbf{A} in spherical coordinates is not immediately clear.

To better understand the relationship between the two expressions, it is helpful to recognize the relationship between the unit-vector basis and the coordinate basis. The unit-vector basis is essentially identical to the coordinate basis as you noted. This means that the components of the vector \mathbf{A} with respect to the unit-vector basis are related to the components with respect to the coordinate basis. Additionally, the Jacobian determinant is equal to the normalizing factor when converting from the unit-vector basis to the coordinate basis.

Once these relationships are recognized, it becomes easier to see the equivalence between the curl of \mathbf{A} and the exterior derivative of A. In particular, the expression for the curl of \mathbf{A} in terms of the 1-form components and the coordinate basis vectors is equal to the exterior derivative of A multiplied by the Jacobian determinant. Thus, we can conclude that the two expressions are equivalent.