Green's Theorem in Summary

This leads us to Green's Theorem in the Plane:

If the assumptions that

1.

C is a closed curve that doesn't cross itself,

2.

C encloses a region R in the xy plane,

3.

C is oriented so that R is on the left as you travel in the forward direction around C, and

4.

the vector field $\vec{F} = F_1 \hat{i} + F_2 \hat{j}$ has continuous partial derivatives as every point of R,

hold, then we have the result that

$$\oint_C \vec{F} \cdot d\vec{r} = \int \int_R \left( -\frac{\... ...1}{\partial y} - \frac{\partial F_2}{\partial x} \right) dx dy.$$

This allows us to convert between line integrals in the plane and area integrals over the region enclosed by the curve. Note that, for a conservative vector field this will again result in zero since the $\hat{k}$ component of the curl of a conservative vector field is $\left( -\frac{\partial F_1}{\partial y} - \frac{\partial F_2}{\partial x} \right)$ which is zero since the curl is identically the zero vector.