I was reading Schwartz's QFT, I came across a lagrangian density,
$$ \mathcal{L} = -\frac12 h \Box h + \frac13 \lambda h^3 + Jh ,\tag{3.69} $$
Calculating the Euler-Lagrange equation, $$ \partial_{\mu} \frac{\partial \mathcal{L}}{\partial(\partial_{\mu}h)} - \frac{\partial \mathcal{L}}{\partial h} = 0 \tag{1}, $$
the equation of motion is given as $$ \Box h -\lambda h^2 - J = 0. \tag{3.70} $$
I tried calculation. After I integrated by parts, I got
$$ \mathcal L'=\frac{1}{2}\partial_\mu h\partial^\mu h+\frac{1}{3}\lambda h^3+Jh $$
Now I have a problem to calculate $$ \frac 12 \partial_{\mu} \frac{\partial (\partial_{\nu}h\partial^{\nu}h)}{\partial(\partial_{\mu}h)}, $$ which seems to be $$\partial_{\mu}\partial^{\mu} h $$
Why $$ \frac 12 \partial_{\mu} \frac{\partial (\partial_{\nu}h\partial^{\nu}h)}{\partial(\partial_{\mu}h)} = 0 $$ for $\mu \neq \nu$ ?
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