Green's Theorem

Green's theorem relates a double integral over a planar region to a line integral around its boundary, providing the two-dimensional version of the fundamental theorem of calculus for vector fields.

The Theorem

Theorem10.2Green's Theorem

Let $D$ be a bounded region in $\mathbb{R}^2$ with piecewise smooth, positively oriented boundary $\partial D = C$ . If $P$ and $Q$ have continuous first partial derivatives on an open set containing $D$ , then $\oint_C P\,dx + Q\,dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA$

The left side is the circulation of the vector field $\mathbf{F} = (P, Q)$ around the boundary $C$ , while the right side integrates the "curl" $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$ over the enclosed region.

Applications

ExampleArea via line integrals

Setting $P = 0$ , $Q = x$ gives $\oint_C x\,dy = \iint_D dA = \text{Area}(D)$ . More symmetrically: $\text{Area}(D) = \frac{1}{2}\oint_C (x\,dy - y\,dx)$ For an ellipse $x = a\cos t$ , $y = b\sin t$ : $\text{Area} = \frac{1}{2}\int_0^{2\pi} (ab\cos^2 t + ab\sin^2 t)\,dt = \pi ab$

ExampleComputing a line integral via Green's theorem

Evaluate $\oint_C (y^2\,dx + 3xy\,dy)$ where $C$ is the boundary of the triangle with vertices $(0,0)$ , $(1,0)$ , $(1,2)$ (counterclockwise). $\frac{\partial(3xy)}{\partial x} - \frac{\partial(y^2)}{\partial y} = 3y - 2y = y$ $\oint_C = \iint_D y\,dA = \int_0^1 \int_0^{2x} y\,dy\,dx = \int_0^1 2x^2\,dx = \frac{2}{3}$

Divergence Form

Theorem10.3Green's Theorem (Flux Form)

With the same hypotheses, $\oint_C \mathbf{F} \cdot \hat{\mathbf{n}}\,ds = \iint_D \nabla \cdot \mathbf{F}\,dA$ where $\hat{\mathbf{n}}$ is the outward unit normal to $C$ . This equates the outward flux through the boundary to the total divergence inside.

RemarkGreen's theorem as a special case

Green's theorem is the two-dimensional special case of both Stokes' theorem (circulation form) and the divergence theorem (flux form). It is the essential link between the one-dimensional fundamental theorem of calculus and the higher-dimensional integral theorems of vector analysis.