What is Stokes theorem?
Where Green's theorem is a two-dimensional theorem that relates a line integral to the region it surrounds, Stokes theorem is a three-dimensional version relating a line integral to the surface it surrounds. For that reason, Green's theorem is actually a special case of Stokes Theorem.
Again, Stokes theorem is a relationship between a line integral and a surface integral. Before you use Stokes theorem, you need to make sure that you're dealing with a surface S that's an oriented smooth surface, and you need to make sure that the curve C that bounds S is a simple, closed smooth boundary curve with positive orientation.
Since Stokes theorem can be evaluated both ways, we'll look at two examples. In one example, we'll be given information about the line integral and we'll need to evaluate the surface integral. In the other example, we'll be given information about the surface integral and we'll need to evaluate the line integral.
0:00 // About Stokes theorem
2:41 // Example 1
10:46 // Example 2