Extended Versions of Green’s Theorem

Green’s Theorem connects the circulation of a vector field around a closed curve with the behavior of the field across the region enclosed by that curve. It provides a way to replace a line integral around a boundary with a double integral over the interior region, making it especially useful in plane geometry, fluid flow, and vector calculus.

Although Green’s Theorem is often introduced using simple regions without gaps, it can also be applied to regions made from several simple parts. This includes regions with holes, such as an island surrounding a lake. In this case, the land region has two boundaries: the outer coastline and the inner shoreline of the lake.

For Green’s Theorem to apply correctly, the total boundary must be positively oriented. This means that as one moves along each boundary curve, the region must remain on the left. The outer boundary is therefore traced counterclockwise, while the inner boundary around the hole is traced clockwise.

A region with a hole can be divided into simpler regions by adding imaginary cuts between the outer and inner boundaries. When Green’s Theorem is applied to each simpler region, the integrals along these shared artificial cuts occur in opposite directions. Since these contributions cancel, only the original boundary curves remain. This shows that Green’s Theorem extends naturally to regions with holes when all boundaries are given the correct orientation.