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Curl vs exterior derivative in spherical coords

by pellman
Tags: coords, curl, derivative, exterior, spherical
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pellman
#1
Dec15-11, 02:25 PM
P: 582
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 [tex]dx \wedge dy[/tex] with the basis vector [tex]\mathbf{\hat{e}}_z[/tex], the justification of which in general is not clear to me.

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

[tex]\mathbf{A}=A'^r\mathbf{\hat{r}} +A'^{\theta}\mathbf{\hat{\theta}} +A'^{\phi}\mathbf{\hat{\phi}} [/tex]

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

[tex]\mathbf{A}=A^r\mathbf{\hat{e}}_r +A^{\theta}\mathbf{\hat{e}}_\theta +A^{\phi}\mathbf{\hat{e}}_\phi [/tex]

where [tex]\mathbf{\hat{e}}_r = \mathbf{\hat{r}}[/tex] [tex]\mathbf{\hat{e}}_\theta=r\mathbf{\hat{\theta}}[/tex] [tex]\mathbf{\hat{e}}_\phi=r\sin\theta\mathbf{\hat{\phi}}[/tex]

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

[tex]\nabla\times\mathbf{A}=\frac{1}{r\sin\theta}\left(\frac{\partial}{\part ial\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}}[/tex]

We can form a basis [tex]dr,d\theta,d\phi[/tex] dual to the coordinate basis and consider the 1-form [tex]A=A_r dr + A_\theta d\theta + A_\phi d\phi[/tex] with exterior derivative [tex]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[/tex] 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 [tex]A'^r = A_r[/tex][tex]A'^\theta = \frac{1}{r} A_\theta[/tex][tex]A'^\phi = \frac{1}{r\sin\theta} A_\phi[/tex]

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:

[tex]\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 [/tex]
[tex]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[/tex]

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

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