Converting a vector field from spherical to cylindrical coordinates

I have a problem where I need to verify Stoke's Theorem. Doing that isn't a big deal but the problem is the vector field has been given in spherical coordinates and we are forced to use another type of coordinate system (since the solution to the problem in spherical coordinates is already available).

Given the region of integration (shown below) I thought that cylindrical coordinates would be the best bet.

I unfortunately don't know how to convert the unit vectors, as I have pointed out below. And say with the conversion from theta to cylindrical coordinates, you get an arctan and having an arctan inside a sin or cos, which is what I'll get given the vector field **f** in the question.

Is there some easier way around this? Am I missing something? I understand that in the region of integration some things will be made easier by having various coordinates zero or constant or something but the whole thing still does look like it's getting rather ugly.

Thank you in advance for help.