In the book "Quantum Field Theory Demystified", David McMahon derives the equation of motion for the Lagrangian:
$$
L=\frac{1}{2}(\{\partial{_u\phi})^2-m^2\phi^2\}
$$
where $ \phi $ is the field as a function of $ (x_0, x_1, x_2, x_3) $
and $ \partial{_u} $ is partial differentiation w.r.t the contravariant 4-vector.
During the derivation, he says that $ \partial{_u}\phi $ can be treated as an independent variable so that
$$
\frac{\partial}{\partial[\partial_u\phi]}(m^2\phi^2) =0
$$
I don't understand this assumption since, just taking $ x_0 $ (which is t), $ \phi $ depends on t so that $ \frac{\partial^2}{\partial^2 t} (m^2\phi^2) $ may have a non-zero value?
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