Extended Versions of Green’s Theorem

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. This region is the land between two boundaries: the outer coastline, C_1, and the inner shoreline, C_2.

For Green’s Theorem to apply, the boundary must be positively oriented, meaning the region stays on the left along the curve. As a result, the outer coastline is traversed counterclockwise, while the lake's inner shoreline is traversed clockwise.

By introducing imaginary cuts, labeled C_3 and C_4, that connect the two shores, the island is temporarily divided into simpler subregions.

When Green’s Theorem is applied to each subregion, the shared boundary appears twice in opposite directions. So, the two line integrals along that boundary are negatives of each other and cancel out.

The remaining integrals come from the visible boundaries. This shows how Green’s Theorem extends to a region with a hole.