Chapter 10 Green's Theorem
If \(f\) is a continuously differentiable function of one variable, defined on a closed bounded interval \([a,b]\text{,}\) then the \textit{fundamental theorem of calculus} asserts that
\begin{equation*}
\int_a^bf'(x)\,dx=f(b)-f(a)\text{.}
\end{equation*}
The above formula expresses the integral of the derivative \(f'\) over an interval by means of the values of \(f\) at the boundary points \(a\) and \(b\) of the interval \([a,b]\text{.}\)
Green's formula provides a generalisation of this fact to functions of two variables. For a function of two variables, the domain is a region, \(D\text{,}\) in the plane \(\mathbb R^2\text{,}\) and the boundary, \(\partial D\text{,}\) of \(D\) is a curve. The formula will apply to vector fields \(\vect f\) and have the following form.
\begin{equation*}
\iint_D\text{}
=\int_{\partial D}\text{}
\end{equation*}
The interval is replaced by the domain \(D\text{,}\) and the boundary points \(a\) and \(b\) by the rim (or boundary) \(\partial D\) of \(D\text{.}\) In two dimensions the degree of freedom is much higher than in one dimension, and therefore we need more “structure” for a formula like the above to be true. Not surprisingly, the key ingredient in the proof of Green's Theorem will be the fundamental theorem of calculus.
