What is the cross product in spherical coordinates?

The cross product in spherical coordinates is given by the rule,

$$ \hat{\phi} \times \hat{r} = \hat{\theta},$$

$$ \hat{\theta} \times \hat{\phi} = \hat{r},$$

$$ \hat{r} \times \hat{\theta} = \hat{\phi},$$

this would result in the determinant,

$$ \vec{A} \times \vec{B} = \left| \begin{array}{ccc} \ \hat{r} & \hat{\theta} & \hat{\phi} \\ A_r & A_\theta & A_\phi \\ B_r & B_\theta & B_\phi \\ \end{array}\right|$$

This rule can be verified by writing these unit vectors in Cartesian coordinates.

The scale factors are only present in the determinant for the curl. This has to do with the definition of the curl and its use of length and area.