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