# Cross product with surface in polar coordinates

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 assume 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