- #1
spaghetti3451
- 1,344
- 34
I know that ##\frac{\partial}{\partial (\partial_{\mu}\phi)} \big( \partial_{\mu} \phi\ \partial^{\mu} \phi \big) = \partial_{\mu} \phi##.
Now, I need to prove this to myself.
So, here goes nothing.
##\frac{\partial}{\partial (\partial_{\mu}\phi)} \big( \partial_{\mu} \phi\ \partial^{\mu} \phi \big)##
## = \frac{\partial}{\partial (\partial_{\mu}\phi)} \big( \eta^{\mu\nu}\partial_{\mu} \phi\ \partial_{\nu} \phi \big)##
##= \eta^{\mu\nu}\ \partial_{\nu} \phi + \eta_{\mu\nu} \eta^{\mu\nu}\ \partial_{\mu} \phi##,
where I first differentiated the factor ##\partial_{\mu}\phi## with respect to ##\partial_{\mu}\phi## and then I differentiated the factor ##\partial_{\nu}\phi## with respect to ##\partial_{\mu}\phi##.
Am I correct so far?
Now, I need to prove this to myself.
So, here goes nothing.
##\frac{\partial}{\partial (\partial_{\mu}\phi)} \big( \partial_{\mu} \phi\ \partial^{\mu} \phi \big)##
## = \frac{\partial}{\partial (\partial_{\mu}\phi)} \big( \eta^{\mu\nu}\partial_{\mu} \phi\ \partial_{\nu} \phi \big)##
##= \eta^{\mu\nu}\ \partial_{\nu} \phi + \eta_{\mu\nu} \eta^{\mu\nu}\ \partial_{\mu} \phi##,
where I first differentiated the factor ##\partial_{\mu}\phi## with respect to ##\partial_{\mu}\phi## and then I differentiated the factor ##\partial_{\nu}\phi## with respect to ##\partial_{\mu}\phi##.
Am I correct so far?