How do I find the equation for the binding surface of a vector field?

Hi,

this is Vector calculus. I am working on finding stokes' theorem for different three dimensional surfaces. As we have to perform the surface integral along the binding surface, we must first find what that binding surface is.

For a unit half sphere, a cylinder and a cone, the bounding surface is the same: a unit circle in the x-y plane.

Why is this?

Why is *that* the binding surface and not the rest?

Also, why is it that when converting to spherical polar coordinates, the limits of integration of the cita variable are 0 to pi/2 and not pi?

I don't know if i expressed myself clearly enough,

thanks a million!