Why stokes theorem is applied to only open surfaces?

Hello student,
Please find the answer to your question below
Stokes theorem is always applied to open surfaces becausea closed surface is a surface that has no boundary.
In other words, a closed surface S has no “edge” floating around. Another way to say this isthat its boundary is empty: ∂S = ∅. In general, the boundary of a surface will be a curve, orpossibly several curves.
A consequence of Stokes’ theorem is that integrating a vector field which is a curl along aclosed surface S automatically yields zero.
In case the idea of integrating over an empty set feels uncomfortable – though it
shouldn’t – here is another way of thinking about the statement. If S is a closed surface, cut itinto two parts S1 and S2 along some curve C. For example, we can cut the sphere S into theupper hemisphere S1 and lower hemisphere S2 along the equator C. Applying Stokes’s theoremto each part yields= 0
where the opposite signs come from the orientation convention.
In fact, property characterizes curls: A vector field is the curl of some vector field if andonly if its integral along any closed surface is zero.
So stokes theorem is applied only to open surfaces.