Editing Green's theorem (section)

== Multiply-connected regions ==
'''Theorem.''' Let <math>\Gamma_0,\Gamma_1,\ldots,\Gamma_n</math> be positively oriented rectifiable Jordan curves in <math>\R^{2}</math> satisfying
<math display="block">\begin{aligned}
\Gamma_i \subset R_0,&&\text{if } 1\le i\le n\\
\Gamma_i \subset \R^2 \setminus \overline{R}_j,&&\text{if }1\le i,j \le n\text{ and }i\ne j,
\end{aligned}</math>
where <math>R_i</math> is the inner region of <math>\Gamma_i</math>. Let
<math display="block">D = R_0 \setminus (\overline{R}_1 \cup \overline{R}_2 \cup \cdots \cup \overline{R}_n).</math>

Suppose <math>p: \overline{D} \to \R</math> and <math>q: \overline{D} \to \R</math> are continuous functions whose restriction to <math>D</math> is Fréchet-differentiable. If the function
<math display="block">(x,y)\longmapsto\frac{\partial q}{\partial e_1}(x,y)-\frac{\partial p}{\partial e_2}(x,y)</math>
is Riemann-integrable over <math>D</math>, then
<math display="block">\begin{align}
& \int_{\Gamma_0} p(x,y)\,dx+q(x,y)\,dy-\sum_{i=1}^n \int_{\Gamma_i} p(x,y)\,dx + q(x,y)\,dy \\[5pt]
= {} & \int_D\left\{\frac{\partial q}{\partial e_1}(x,y)-\frac{\partial p}{\partial e_2}(x,y)\right\} \, d(x,y).
\end{align}</math>