Der Weg
ist durch
-
gegeben. Somit ist
![{\displaystyle {}{\begin{aligned}{\frac {1}{\epsilon ^{2}}}\int _{\gamma _{\epsilon }}F&={\frac {1}{\epsilon ^{2}}}\int _{0}^{2\pi }\left\langle F{\left(P+\epsilon {\begin{pmatrix}\cos t\\\sin t\end{pmatrix}}\right)},\epsilon {\begin{pmatrix}-\sin t\\\cos t\end{pmatrix}}\right\rangle dt\\&={\frac {1}{\epsilon }}\int _{0}^{2\pi }\left\langle F(P)+\epsilon DF_{P}{\begin{pmatrix}\cos t\\\sin t\end{pmatrix}}+\Vert {\epsilon {\begin{pmatrix}\cos t\\\sin t\end{pmatrix}}}\Vert r{\left(\epsilon {\begin{pmatrix}\cos t\\\sin t\end{pmatrix}}\right)},{\begin{pmatrix}-\sin t\\\cos t\end{pmatrix}}\right\rangle dt\\&={\frac {1}{\epsilon }}\int _{0}^{2\pi }\left\langle F(P),{\begin{pmatrix}-\sin t\\\cos t\end{pmatrix}}\right\rangle dt+{\frac {1}{\epsilon }}\int _{0}^{2\pi }\left\langle \epsilon DF_{P}{\begin{pmatrix}\cos t\\\sin t\end{pmatrix}},{\begin{pmatrix}-\sin t\\\cos t\end{pmatrix}}\right\rangle dt+{\frac {1}{\epsilon }}\int _{0}^{2\pi }\left\langle \Vert {\epsilon {\begin{pmatrix}\cos t\\\sin t\end{pmatrix}}}\Vert r{\left(\epsilon {\begin{pmatrix}\cos t\\\sin t\end{pmatrix}}\right)},{\begin{pmatrix}-\sin t\\\cos t\end{pmatrix}}\right\rangle dt\\&={\frac {1}{\epsilon }}\int _{0}^{2\pi }\left\langle F(P),{\begin{pmatrix}-\sin t\\\cos t\end{pmatrix}}\right\rangle dt+\int _{0}^{2\pi }\left\langle DF_{P}{\begin{pmatrix}\cos t\\\sin t\end{pmatrix}},{\begin{pmatrix}-\sin t\\\cos t\end{pmatrix}}\right\rangle dt+\int _{0}^{2\pi }\left\langle r{\left(\epsilon {\begin{pmatrix}\cos t\\\sin t\end{pmatrix}}\right)},{\begin{pmatrix}-\sin t\\\cos t\end{pmatrix}}\right\rangle dt.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dc025fe740671624934a8584a352e8f33f14989)
Wir analysieren die einzelnen Summanden getrennt. Ganz rechts wird der Integrand für
aufgrund der Eigenschaften von
und der Stetigkeit des Skalarproduktes beliebig klein, was sich auf das Integral überträgt. Dieser Term spielt also im Limes keine Rolle. Das linke Integral ist
-
![{\displaystyle {}\int _{0}^{2\pi }-F_{1}(P)\sin t+F_{2}(P)\cos tdt=0\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b73a2c3b1207ee8ce2ab5b382f0db9a6b733da1f)
so dass alles vom mittleren Summanden abhängt. Der Integrand ist
![{\displaystyle {}{\begin{aligned}\left\langle DF_{P}{\begin{pmatrix}\cos t\\\sin t\end{pmatrix}},{\begin{pmatrix}-\sin t\\\cos t\end{pmatrix}}\right\rangle &=\left\langle {\begin{pmatrix}{\frac {\partial F_{1}}{\partial x}}(P)\cos t+{\frac {\partial F_{1}}{\partial y}}(P)\sin t\\{\frac {\partial F_{2}}{\partial x}}(P)\cos t+{\frac {\partial F_{2}}{\partial y}}(P)\sin t\end{pmatrix}},{\begin{pmatrix}-\sin t\\\cos t\end{pmatrix}}\right\rangle \\&=-{\frac {\partial F_{1}}{\partial x}}(P)\cos t\sin t-{\frac {\partial F_{1}}{\partial y}}(P)\sin t\sin t+{\frac {\partial F_{2}}{\partial x}}(P)\cos t\cos t+{\frac {\partial F_{2}}{\partial y}}(P)\sin t\cos t.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eaa81fef216e8b0851dbd0770f990fdccc2200f5)
Wegen
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![{\displaystyle {}\int _{0}^{2\pi }\cos t\sin tdt={\frac {1}{2}}{\left(\sin ^{2}t\right)}_{0}^{2\pi }=0\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7aca803b29e65dff70e5ff05bd979eff6dcd037)
fallen diese Terme weg. Übrig bleiben
-
![{\displaystyle {}-{\frac {\partial F_{1}}{\partial y}}(P)\int _{0}^{2\pi }\sin t\sin tdt=-{\frac {\partial F_{1}}{\partial y}}(P){\frac {1}{2}}{\left(x-\sin t\cos t\right)}_{0}^{2\pi }=-\pi {\frac {\partial F_{1}}{\partial y}}(P)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6ad62a47a758e40b9ee4350ab1fa991d6d221fa)
und
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![{\displaystyle {}{\frac {\partial F_{2}}{\partial x}}(P)\int _{0}^{2\pi }\cos t\cos tdt=\pi {\frac {\partial F_{2}}{\partial x}}(P)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29a4b80d239d24a504eae9cf7a391efee4daf770)
Alles zusammen ergibt
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![{\displaystyle {}\operatorname {lim} _{\epsilon \rightarrow 0}\,{\frac {1}{\epsilon ^{2}}}\int _{\gamma _{\epsilon }}F=\pi {\left({\frac {\partial F_{2}}{\partial x}}(P)-{\frac {\partial F_{1}}{\partial y}}(P)\right)}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/648743f29557fe715deb884a17577014f3f65532)