Green's Theorem - Relationship To The Divergence Theorem

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.