Green's Theorem

Green's Theorem bridges a 1D boundary trip and a 2D interior sweep:

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

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

Think of it as 2D Stokes' theorem. It converts hard line integrals into easy double integrals, and vice versa.

Area and Pitfalls

Use line integrals to sweep out 2D area ($A$) using these formulas:

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

Common Pitfall: Green's theorem strictly requires a closed curve, positive counterclockwise orientation, and a vector field with continuous partial derivatives throughout the entire enclosed region $D$.

If your field has a single interior singularity—like a vortex or point charge at the origin where the denominator hits zero—the equality breaks down completely. Always check your interior domain for holes or undefined points before applying the theorem!