When integrating over the suface, you have (curl F dot n) in the centre of the integral. Is that n or normal the normal to the boundary of the surface? Or is it the normal of the whole surface, in which case there will be many different ones so seems wrong because n should be definite. I also like to ask about the normal (I assume it should be constant as well) in Gauss' Divergence theorem (again replacing (F dot ds) by (F dot n dS). How do you determine that? If I have the bottom hemisphere of a ball, z<=0. What would be its normal vector?