${}$
Green's theorem is the planar realization of the laws of balance expressed by the Divergence and Stokes' theorems. There are two different expressions of Green's theorem, one that expresses the balance law of the Divergence theorem, and one that expresses the balance law of Stokes' theorem. The two forms of Green's theorem are listed in Table 9.10.1.
Divergence form

$\int {\int}_{R}\left({f}_{x}\+{g}_{y}\right)\mathrm{dA}$ = ${\u2233}_{C}f\mathrm{dy}g\mathrm{dx}$

Stokes' form

$\int {\int}_{R}\left({g}_{x}{f}_{y}\right)\mathrm{dA}$ = ${\u2233}_{C}f\mathrm{dx}plus;g\mathrm{dy}$

Table 9.10.1 Two forms of Green's theorem



${}$
In the "divergence" form of Green's theorem, the double integral on the left is the integral of the divergence of $\mathbf{F}\=f\left(x\,y\right)\mathbf{i}plus;g\left(xcomma;y\right)\mathbf{j}$ over the plane region $R$ bounded by the simple, closed, rectifiable, orientable curve $C$. The components of F are assumed to have continuous first partial derivatives. The line integral on the right is the flux of F through $C$.
In the "Stokes" form of Green's theorem, the double integral on the left is the integral of the normal component of the curl of F. To obtain the curl of F, augment it with a third, but zero component. The normal to $R$ is the unit vector k, so $\left(\nabla \times \mathbf{F}\right)\xb7k\={g}_{x}{f}_{y}$. The line integral on the right is the integral of the tangential component of F, sometimes called the circulation, and sometimes the work done by the field on a unit particle.
${}$
The two "forms" of Green's theorem stated in Table 9.10.1 are related, since one can be converted to the other by simply changing the names of the components of F. For example, call $f$ by the new name $G$ and call $g$ by the new name $F$. Then the divergenceform in Table 9.10.1 would read
$\int {\int}_{R}\left({G}_{x}{F}_{y}\right)\mathrm{dA}$ = ${\u2233}_{C}G\mathrm{dx}plus;F\mathrm{dy}$
which is the Stokes' form of the theorem.
${}$
By judicious choices for F, Green's theorem can be used to show that the area inside $C$ is given by any one of the line integrals
${\u2233}_{C}x\mathrm{dy}$, ${\u2233}_{C}y\mathrm{dx}$, $\frac{1}{2}{\u2233}_{C}x\mathrm{dy}y\mathrm{dx}$
${}$
The third integral suggests taking $\mathbf{F}\=x\mathbf{i}plus;y\mathbf{j}$ so that
$\int {\int}_{R}\left({f}_{x}\+{g}_{y}\right)\mathrm{dA}$ = $\int {\int}_{R}2\mathrm{dA}$
which is twice the area of $R$. The divergenceform of Green's theorem then gives the area inside $R$, that is, $\int {\int}_{R}\mathrm{dA}$ , as $\frac{1}{2}{\u2233}_{C}x\mathrm{dy}y\mathrm{dx}$. The other two expressions for the area inside $R$ are developed in the Examples.