Green's Theorem, Curl, and Divergence

The Fundamental Theorem for Line Integrals handled conservative fields by reducing a line integral to endpoint values. Green's Theorem handles every field but requires a closed curve: it equates the circulation of $\vec{F}$ around a closed curve $C$ with a double integral of a derivative quantity over the region $C$ encloses. That single equation explains the component test for conservative fields, computes areas from boundary data, and, once its two derivative quantities are named curl and divergence, generalizes into Stokes' Theorem and the Divergence Theorem.

Green's Theorem

Fix a region $D$ in the plane bounded by a simple closed curve $C$. The boundary gets an orientation.

With that convention the theorem reads as follows.

The left side integrates the field around the boundary; the right side integrates a derivative of the field over the interior. It is the two-dimensional counterpart of the Fundamental Theorem of Calculus ${\int }_a^b{F}^{\prime }(x)\mathbf{d}x=F(b)-F(a)$: an integral of a derivative over a region equals an integral of the field over the oriented boundary of that region.

Green's Theorem and the component test. For a conservative field $Q_x=P_y$, so the integrand on the right vanishes and every closed-loop integral is zero — exactly the closed-loop characterization of conservative fields. Green's Theorem is the reason the cross-partial test detects conservativeness on a simply connected domain.

Parametrizing the circle and integrating directly would involve four unpleasant trigonometric integrals; Green's Theorem replaces them with the area of the disk.

Area from the boundary

Run the theorem backward. Choosing $P,Q$ so that $Q_x-P_y=1$ turns the double integral into the area of $D$, computable from the boundary alone. Three standard choices:

The symmetric third form is what a planimeter mechanizes: trace the boundary and read off the enclosed area. For the ellipse $\vec{r}(t)=⟨a\cos t,b\sin t⟩$, $0\le t\le 2\pi$,

The proof idea

Green's Theorem holds for a simple region (both type I and type II) by two direct computations, one matching $\underset{C}{\oint }P\mathbf{d}x$ to $-\underset{D}{\iint }{P}_y\mathbf{d}A$ and the other matching $\underset{C}{\oint }Q\mathbf{d}y$ to $\underset{D}{\iint }{Q}_x\mathbf{d}A$. A general region is cut into simple pieces; along each internal cut the boundary is traversed twice in opposite directions, so those line integrals cancel and only the outer boundary remains.

Curl

The quantity $Q_x-P_y$ that appears in Green's Theorem measures how much the field circulates near a point. In three dimensions the corresponding object is a vector, the curl, whose $\hat{k}$-component is exactly $Q_x-P_y$.

Introduce the del operator $\nabla =\hat{ı}{\partial }_x+\hat{ȷ}{\partial }_y+\hat{k}{\partial }_z$. Then the curl is the formal cross product

a mnemonic that reproduces all three components. Physically, if $\vec{F}$ is a fluid velocity field, $\mathrm{curl}\vec{F}$ at a point is twice the local angular velocity: drop a tiny paddle wheel into the flow and it spins about the axis $\mathrm{curl}\vec{F}$ at a rate proportional to its magnitude. Where the curl is zero, the flow is irrotational — the paddle wheel translates but does not turn.

The curl connects to conservative fields. Taking the curl of a gradient always gives zero, because the mixed partials cancel in pairs by Clairaut's Theorem.

This is the three-dimensional component test. In the plane it reduces to $Q_x-P_y=0$, the planar condition on $P_y=Q_x$.

Divergence

Curl measures rotation; divergence measures expansion. It is the formal dot product of del with the field, a scalar.

If $\vec{F}$ is a fluid velocity field, $\mathrm{div}\vec{F}$ at a point is the net rate at which fluid flows out of a tiny box around the point, per unit volume. Positive divergence marks a source (fluid produced, box emptying); negative divergence marks a sink; zero divergence means incompressible flow, whatever enters a region also leaves it.

Divergence pairs with curl through a second identity: the divergence of any curl is zero.

The six terms cancel in pairs by Clairaut's Theorem. A field that is itself a curl therefore has no sources or sinks; this is why magnetic fields, which are curls of a vector potential, are divergence-free.

The two vector forms of Green's Theorem

Green's Theorem, written with curl and divergence, is the two-dimensional case of Stokes' Theorem and the Divergence Theorem. Regard the planar field $\vec{F}=P\hat{ı}+Q\hat{ȷ}$ as living in space with zero $\hat{k}$-component.

• Circulation form. Since $(\mathrm{curl}\vec{F})\cdot \hat{k}=Q_x-P_y$, Green's Theorem is$$\underset{C}{\oint }\vec{F}\cdot \mathbf{d}\vec{r}=\underset{D}{\iint }(\mathrm{curl}\vec{F})\cdot \hat{k}\mathbf{d}A.$$
  The circulation around the boundary equals the integrated curl over the region. Lifting $C$ and $D$ off the plane into space gives Stokes' Theorem.
• Flux form. With the outward normal $\hat{n}$ on $C$, a parallel computation gives$$\underset{C}{\oint }\vec{F}\cdot \hat{n}\mathbf{d}s=\underset{D}{\iint }\mathrm{div}\vec{F}\mathbf{d}A.$$
  The outward flux across the boundary equals the integrated divergence over the region. Replacing the plane region by a solid and its boundary curve by a boundary surface gives the Divergence Theorem.

The two identities $\mathrm{curl}(\nabla f)=\vec{0}$ and $\mathrm{div}(\mathrm{curl}\vec{F})=0$ organize these three operators into a chain: gradient, then curl, then divergence, with each composition vanishing.¹