Es sei

${\displaystyle F\colon G\longrightarrow \mathbb {R} ^{2}}$

ein stetig differenzierbares Vektorfeld auf einer offenen Menge

${\displaystyle {}G\subseteq \mathbb {R} ^{2}\,}$

und es sei

${\displaystyle {}\rho (P):={\frac {\partial F_{2}}{\partial x}}(P)-{\frac {\partial F_{1}}{\partial y}}(P)\,.}$

Zeige

${\displaystyle {}\rho (P)={\frac {1}{\pi }}\cdot \operatorname {lim} _{\epsilon \rightarrow 0}\,{\frac {1}{\epsilon ^{2}}}\int _{\gamma _{\epsilon }}F\,,}$

${\displaystyle {}\gamma _{\epsilon }}$

${\displaystyle {}P}$

${\displaystyle {}\epsilon }$