Green’s theorem: the hole method

This video demonstrates the 'hole method' for applying Green's Theorem when a vector field has a singularity within the integration region. It begins by showing that direct application fails because the origin is undefined for the given field. By introducing a small circle around the singularity, an annular region free of singularities is created. Green's Theorem is then applied to this new region to equate the line integral over the outer boundary to that over the inner circular boundary. Finally, parameterizing the small circle yields the exact value of the integral as $2\pi$.

Before you watch

• Multivariable Calculus Basics
• Line Integrals and Double Integrals
• Standard Form of Green's Theorem