Wegintegral/Vektorfeld/Archimedische Spirale/Senkrechtes Feld/Aufgabe/Pseudolösung

Es ist

${}{\begin{aligned}\int _{0}^{2\pi }\left\langle {\begin{pmatrix}t\sin t\\t\cos t\end{pmatrix}},{\begin{pmatrix}t\sin t-\cos t\\t\cos t+\sin t\end{pmatrix}}\right\rangle dt&=\int _{0}^{2\pi }t^{2}\sin tCheckmeplease+t\sin t\cos t-t^{2}\cos tCheckmeplease-t\sin t\cos tdt\\&=\int _{0}^{2\pi }-t^{2}dt\\&=-{\frac {1}{3}}t^{3}|_{0}^{2\pi }\\&={\frac {-8\pi ^{3}}{3}}.\,\end{aligned}}$