Green's Theorem

$$\oint_C (P\,dx + Q\,dy) = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)dA$$

where $C$ is a positively oriented, simple, closed curve bounding $D$.

When to use

• Line integral is hard, curl expression is simple → double integral
• Double integral is hard, boundary is simple → line integral

Area formulas

• $A = \oint_C x\,dy$
• $A = -\oint_C y\,dx$
• $A = \frac{1}{2}\oint_C (x\,dy - y\,dx)$

Connection to curl — The integrand is the $z$-component of $\nabla\times\vec{F}$ for $\vec{F} = (P, Q, 0)$.

Key insight: Green's theorem is Stokes' theorem in $\mathbb{R}^{2}$.

Common pitfall: Green’s theorem requires a closed curve, positive (counterclockwise) orientation, and a field smooth throughout the enclosed region. A single interior singularity — a vortex at the origin — breaks the equality outright.