Stokes' Theorem is the equivalent of Green's Theorem applied in three dimensions. It involves finding the total curl, circulation around the boundary surface enclosing a region by adding up alll little bits of curl at every point in the region.

In essence, Stokes' Theorem turns a tricky closed integral into a simpler surface integral or vice-versa, whichever is easier to solve.

The surface itself doesn't matter. The boundary does. Many surfaces that share the same boundary will have the same result. The boundary curve orientation should follow the right hand rule, meaning we consider counterclockwise to be positively-oriented.

In the case that there is no boundary, meaning the surface is closed, all these bits of circulation will cance leachother out, meaning the line integral over the boundary curve is zero, therefore we have
And we recall that the unit normal vector for a 3D surfae is defined as

With the surface integral again also being defined