Green’s Theorem on Regions with Holes

Green’s Theorem can be used on regions with holes when the boundary is oriented correctly. The theorem links the circulation of a vector field around a closed curve to the behavior of the field across the region inside that curve. It also replaces a line integral around the boundary with a double integral over the interior. ...

Green's Theorem connects a line integral around a closed curve to a double integral over the region it encloses.

While typically applied to simple, solid shapes, the theorem extends to unions of simple regions and regions with holes.

A useful example is evaluating an integral over the region of an island with a lake at its center. Th...