I'm struggling with a problem concerning cross products. I have a perturbation to a background magnetic field given by

$\delta E \times B$

where $B(r(\phi),\phi)= \frac{B_0R_E^3}{r(\phi)^3} + \Delta B(r)\cos\phi$, so $r$ is a polar curve in terms of $\phi$. I was wondering how to go about the cross product? I know that

$\delta E = \delta E_r \hat{r} + \delta E_\phi \hat{\phi}$

but how can I decompose the functional form of B into its vector form to complete the cross product?

EDIT Am I correct in this approach?:

$\vec{B} = (r(\phi),\phi,B(r,\phi))$

\begin{vmatrix} \hat{r} & \hat{\phi} &\hat{z} \\ \delta E_r & \delta E_\phi & 0 \\ r & \phi & B \end{vmatrix}

where the zero is present because I am considering B which has no height component

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