The Motivation

The previous sections outline a method for converting between line and area integrals in the case where $\vec{F}$ is two dimensional and C lies in the xy plane. With some work (proof at the end of this chapter) we can generalize this result to work for any vector field and any curve.

To get a start on this, notice that the integrand of the area integral in Green's Theorem is the $\hat{k}$ component of the curl of $\vec{F}$. Also notice that the $\hat{k}$ vector is perpendicular to the plane in which C lies. What if C were in the yz plane? You might suppose that we would get the $\hat{i}$ component of the curl. Carrying this further, it is natural to assume that the generalization of Green's theorem will involve all components of the curl. The question is, what does the region R in Green's Theorem become in this case?