Cross product expressed in the cylindrical basis

[fluidistic]

Homework Statement

It's not a homework question but a doubt I have.
Say I want to write $\vec A \times \vec B$ in the basis of the cylindrical coordinates.
I already know that the cross product is a determinant involving $\hat i$, $\hat j$ and $\hat k$.
And that it's worth in my case $(A_yB_z-B_yA_z) \hat i +(A_xB_z-B_xA_z)\hat j+(A_xB_y-B_xA_y)\hat k$.
In order to reach what I'm looking for, can I "translate" $A_x$, $B_y$, etc. into cylindrical coordinates and then replace $\hat i$, $\hat j$ and $\hat z$ by what they are worth when translated in cylindrical coordinates and in function of the cylindrical unit vectors $\hat r$, $\hat \theta$ and $\hat z$?

 

[lanedance]

i think this is probably ok for the cross product of 2 static vectors as the cylindrical basis is orthonormal 9try it and check)

however as soon as you look at vectors changing with time or derivatives, you will have a problem as the basis vector change with position, (eg. div, curl, etc.). The reason the cartesian coords are so useful is the basis does not change with position