Math Help - Converting a vector field from spherical to cylindrical coordinates

1. 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.

2. Whoops, stupid me forgot to link the image containing the question data. Here it is:

Cylindrical Coordinates -- from Wolfram MathWorld

Also its a common google search, you might find useful information.

Finally, at worse. use the link mathworld link to convert to cartesian and then to cylindrical

4. Del in cylindrical and spherical coordinates - Wikipedia, the free encyclopedia

I tried there but, conveniently, the conversion from spherical to cylindrical has a a typo in it (it has a double ~ where there should be one of the coordinates). That's the last thing I need :-(

Also I have tried a fair few Google searches. This forum wasn't my first resort.

5. Okay, so here is my working so far in a fair bit of detail. Given that my workings in spherical and cylindrical coordinates don't match, I must have made an error at some point. Some help would be much appreciated. Remember that I am forced to use cylindrical coordinates. I am only using spherical coordinates as a check.

6. Anything guys? I'm still rather stuck on this :-(

7. Anyone able to shed some light on this? Can anyone see an error in my thinking or logic?