Examples of Green's Theorem

Example 1. This example shows an alternative approach to calculating the area of a region R that is enclosed by a curve C. Look at the circulation of the vector field $\vec{F} = x\hat{j}$. By Green's Theorem,

$$\oint_C \vec{F} \cdot d\vec{r} = \oint_C xdy = \int \int_R dx dy = A$$ (9)

where A is the area of R. Similarly, the circulation of the vector field $\vec{G} = -y\hat{i}$ aound C is

$$\oint_C \vec{G} \cdot d\vec{r} = \oint_C (-y)dx = \int \int_R dx dy = A.$$ (10)

Adding these expressions together and solving for A gives us

$$A = \oint_C (xdy - ydx).$$ (11)

Example 2. Let C be the circle x² + y² = 1 in the xy plane and compute the circulation of $\vec{F} = (y^2 - 7y)\hat{i} + (2xy + 2x)\hat{j}$ around C.

$$\oint_C \vec{F} \cdot d\vec{r} = \int \int_R [(2y + 2) - (2y - 7)]dx dy = \int \int_R 9 dx dy$$

$$= 9 \int \int_R dx dy = 9(\mbox{area of R}) = 9\pi.$$