Anwendung der Euler-Lagrange-Gleichungen auf Maxwells Theorie

Wir haben$\frac12 (\partial_{\mu}A^{\mu})^2 = \frac12 (\partial_{\alpha} A^{\alpha})(\partial_{\beta}A^{\beta})= \frac12 (\partial_{\alpha} A_{\sigma}) \eta^{\sigma\alpha}(\partial_{\beta}A_{\rho}) \eta^{\rho\beta}$also die Ableitung bzgl$\partial_{\mu} A_{\nu}$Ist

$$\frac12\delta_{\alpha}^{\mu} \delta_{\sigma}^{\nu} \eta^{\sigma\alpha}(\partial_{\beta}A_{\rho}) \eta^{\rho\beta}+\frac12(\partial_{\alpha} A_{\sigma}) \eta^{\sigma\alpha}\delta_{\beta}^{\mu} \delta_{\rho}^{\nu} \eta^{\rho\beta}= \frac12 \eta^{\mu\nu} (\partial_{\beta} A^{\beta})+\frac12 (\partial_{\alpha}A^{\alpha}) \eta^{\mu\nu} = (\partial_{\rho}A^{\rho})\eta^{\mu\nu}$$

wo ich Dummy-Indizes frei beschriftet und umbenannt habe.