Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It only takes a minute to sign up.
Sign up to join this community
for the Lagrangian density $$\mathscr{L}=\frac{1}{2}(\partial_{\mu}A^{\mu})^2$$
how can I get this $$\frac{\partial{\mathscr{L}}}{\partial(\partial_{\mu}A_\nu)}=(\partial_\rho A^\rho)\eta^{\mu\nu}$$
or I can only get $\frac{\partial{\mathscr{L}}}{\partial(\partial_{\mu}A_\nu)}=\partial_\rho A^\rho$ but where the $\eta^{\mu\nu}$ comes from.
Writing $$\mathscr{L}=\frac{1}{2}(\partial_{\mu}A^{\mu})^2=\frac{1}{2}(\partial_{\mu}A_{\nu}\eta^{\mu\nu})^2$$ we can write $$\frac{\partial{\mathscr{L}}}{\partial(\partial_{\mu}A_\nu)}=\frac{1}{2}2(\partial_{\rho}A_{\sigma}\eta^{\rho\sigma})\eta^{\mu\nu}=(\partial_{\rho}A^{\rho})\eta^{\mu\nu} .$$
