Aplicando las ecuaciones de Euler-Lagrange a la teoría de Maxwell

Tenemos$\frac12 (\partial_{\mu}A^{\mu})^2 = \frac12 (\partial_{\alpha} A^{\alpha})(\partial_{\beta}A^{\beta})= \frac12 (\partial_{\alpha} A_{\sigma}) \eta^{\sigma\alpha}(\partial_{\beta}A_{\rho}) \eta^{\rho\beta}$por lo que la derivada wrt$\partial_{\mu} A_{\nu}$es

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

donde he etiquetado y reetiquetado libremente índices ficticios.