The stokes theorem allows you to find the circulation of the field along a closed loop. It says that the circulation of the field along the loop equals the integral flux of the curl taken over *any* surface that is bound by that loop. That is, you can choose whatever surface you want to integrate the curl over, the result won't change, as long as the surface is delimited by the loop around which you want to find the circulation.

Originally Posted by **maple_tree** Hi,
For a unit half sphere, a cylinder and a cone, the bounding surface is the same: a unit circle |

I think you mistakenly inverted the two things in the sentence. The circle would be the loop, and the half sphere, etc. would be the surface.

If you want to calculate the circulation of the field around the circle, the stokes theorem tells you that it equals the flux of the curl of the field over any surface that is bound by that circle. That means that you can calculate the flux of the rotor of E over the surface of the half sphere lying on that circle, of a cone, of a ctylinder, etc. as long as they have their basis on the circle you are interested in. The number you get will always be the same.

As for the coordinates question, I will leave it to someone else because I'm having coordinates-related problems myself and I guess I would end up confusing you too.