Chapter 11 Two Interpretations of Green's Theorem
There are several possibilities to interpret Green's theorem. We can interpret the line integral as the flux of a field across the boundary of a domain. This leads to what is usually called the Divergence Theorem (or Gauss' Theorem). The other possibility is to look at the circulation of the vector field along the “rim” of the surface and to see how it relates to the integral over the interior of the domain. This will lead to a special case of Stokes' Theorem.
The fact that both the Divergence Theorem and Stokes' Theorem can be obtained from the same formula shows that both are closely related. In fact, using differential forms, one can indeed unify the two and write them in the form
\begin{equation*}
\int_Sd\omega=\int_{\partial S}\omega\text{.}
\end{equation*}
(note the similarity to the fundamental theorem of calculus.) However this is beyond the scope of this course. A good reference for this theory is the book by Spivak [5].
