Relationship To The Stokes Theorem
Green's theorem is a special case of the Kelvin–Stokes theorem, when applied to a region in the xy-plane:
We can augment the two-dimensional field into a three-dimensional field with a z component that is always 0. Write F for the vector-valued function . Start with the left side of Green's theorem:
Kelvin–Stokes Theorem:
The surface is just the region in the plane, with the unit normals pointing up (in the positive z direction) to match the "positive orientation" definitions for both theorems.
The expression inside the integral becomes
Thus we get the right side of Green's theorem
Read more about this topic: Green's Theorem
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