- #1
TroyElliott
- 59
- 3
In Lagrangians we often take derivatives (##\frac{\partial}{\partial (\partial_{\mu}\phi)}##) of terms like ##(\partial_{\nu}\phi \partial^{\nu}\phi)##. We lower the ##\partial^{\nu}## term with the metric and do the usual product rule. My question is why do we do this? Isn't ##\partial^{\nu}\phi## an element of a different vector space? I would think that ##\frac{\partial}{\partial (\partial_{\mu}\phi)}## is really just shorthand for a tensor product of operators, with one being an identity operator in the ##\partial^{\nu}## space. Can anyone explain why this isn't the correct way of looking at doing such a derivative? Thanks!