Relationship To The Divergence Theorem
Considering only two-dimensional vector fields, Green's theorem is equivalent to the following two-dimensional version of the divergence theorem:
where is the outward-pointing unit normal vector on the boundary.
To see this, consider the unit normal in the right side of the equation. Since in Green's theorem is a vector pointing tangential along the curve, and the curve C is the positively-oriented (i.e. counterclockwise) curve along the boundary, an outward normal would be a vector which points 90° to the right, which would be . The length of this vector is . So
Now let the components of . Then the right hand side becomes
which by Green's theorem becomes
The converse can also easily shown to be true.
Read more about this topic: Green's Theorem
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