Green's theorem

Trades a circulation integral around a closed curve for a double integral of curl over the region it encloses.

Green's Theorem trades a circulation integral $\oint_C\mathbf{F}\cdot d\mathbf{r}$ around a positively oriented (counterclockwise) closed curve $C$ for a double integral $\iint_R (Q_x-P_y)\,dA$ of the curl integrand over the region $R$ it encloses. Total circulation around the boundary equals the sum of every interior point's local spin, weighted by area. This reproves the 2D conservative-field test as a special case: if $P_y=Q_x$ everywhere inside $R$ , the integrand is identically 0, so $\oint_C\mathbf{F}\cdot d\mathbf{r}=0$ for every closed curve, which is exactly path independence.

Examples

For $\mathbf{F}=(-y,x)$ around the unit circle: $Q_x-P_y=2$ everywhere, so $\oint_C \mathbf{F}\cdot d\mathbf{r} = \iint_R 2\,dA = 2\pi$ , using only the constant curl value and the disk's area.