Polynomring/Formales partielles Ableiten/Einführung/Textabschnitt

Es ist

${}\partial ^{\nu }(X_{1}\partial ^{\lambda })=X_{1}\partial ^{\nu +\lambda }+\nu _{1}\partial ^{\nu +\lambda -e_{1}}\,.$

Induktion über ${}\nu _{1}$. Bei ${}\nu _{1}=0$ kann man ${}X_{1}$ vorziehen. Der Induktionsschritt ergibt sich aus

${}{\begin{aligned}\partial ^{\nu }(X_{1}\partial ^{\lambda })&=\partial _{1}\partial ^{\nu -e_{1}}(X_{1}\partial ^{\lambda })\\&=\partial _{1}{\left(X_{1}\partial ^{\nu -e_{1}+\lambda }+(\nu _{1}-1)\partial ^{\nu +\lambda -2e_{1}}\right)}\\&=\partial _{1}{\left(X_{1}\partial ^{\nu -e_{1}+\lambda }\right)}+(\nu _{1}-1)\partial ^{\nu +\lambda -e_{1}}\\&=X_{1}\partial ^{\nu +\lambda }+\partial ^{\nu -e_{1}+\lambda }+(\nu _{1}-1)\partial ^{\nu +\lambda -e_{1}}\\&=X_{1}\partial ^{\nu +\lambda }+\nu _{1}\partial ^{\nu +\lambda -e_{1}}.\end{aligned}}$