The perturbed part of magnetic field is $\mathbf{\delta B}$ where $\mathbf{\delta B} = \delta B_x(x,y), \delta B_y(x,y)$ and $\nabla \cdot \mathbf{\delta B} = 0$.
To prove $\mathbf{\delta B} = \delta B(-sin\phi, cos\phi)$ in spherical coordinates?
I was trying : $\nabla \cdot \mathbf{\delta B} = 0$,
$\mathbf{\delta B}$ in terms of spherical coordinates $(r, \theta, \phi)$.
in Cartesian coordinates: $ \mathbf{\delta B} = \delta B_x \mathbf{\hat{i}} + \delta B_y \mathbf{\hat{j}} $
In spherical coordinates, the Cartesian unit vectors are:
$ \mathbf{\hat{i}} = \sin\phi\cos\theta \mathbf{\hat{r}} + \cos\phi\cos\theta \mathbf{\hat{\theta}} - \sin\theta \mathbf{\hat{\phi}} $
$ \mathbf{\hat{j}} = \sin\phi\sin\theta \mathbf{\hat{r}} + \cos\phi\sin\theta \mathbf{\hat{\theta}} + \cos\theta \mathbf{\hat{\phi}} $
but the actual expression $\mathbf{\delta B} = \delta B(-sin\phi, cos\phi)$ is not coming.
Thank you for your help!