Wegintegral/x^3dx-yzdy+xz^2dz/(-t^2,t^3-1,t+2)/-1 bis 0/Aufgabe/Lösung

Es ist

${}{\begin{aligned}\gamma ^{*}\omega &=(-t^{2})^{3}d(-t^{2})-(t^{3}-1)(t+2)d(t^{3}-1)-t^{2}(t+2)^{2}d(t+2)\\&=2t^{7}dt-3t^{2}(t^{4}+2t^{3}-t-2)dt-t^{2}(t^{2}+4t+4)dt\\&=(2t^{7}-3t^{6}-6t^{5}+3t^{3}+6t^{2}-t^{4}-4t^{3}-4t^{2})dt\\&=(2t^{7}-3t^{6}-6t^{5}-t^{4}-t^{3}+2t^{2})dt.\end{aligned}}$

Daher ist

${}{\begin{aligned}\int _{\gamma }\omega &=\int _{-1}^{0}(2t^{7}-3t^{6}-6t^{5}-t^{4}-t^{3}+2t^{2})dt\\&=({\frac {1}{4}}t^{8}-{\frac {3}{7}}t^{7}-t^{6}-{\frac {1}{5}}t^{5}-{\frac {1}{4}}t^{4}+{\frac {2}{3}}t^{3})|_{-1}^{0}\\&=-({\frac {1}{4}}+{\frac {3}{7}}-1+{\frac {1}{5}}-{\frac {1}{4}}-{\frac {2}{3}})\\&=-{\frac {3}{7}}+1-{\frac {1}{5}}+{\frac {2}{3}}\\&={\frac {-45+105-21+70}{105}}\\&={\frac {109}{105}}.\end{aligned}}$

Zur gelösten Aufgabe