= Solution
Use the <Minkowski metric> $\eta=\operatorname{diag}(-1,1,1,1)$, $\hbar=c=1$, and $\Box=\partial_a\partial^a=-\partial_t^2+\nabla^2$. The <Proca field> has nonzero mass $m$. Vary its <action>, using the fact that the <gauge field strength> is an <antisymmetric second-rank tensor>:
$$
\delta S=\int d^4x\left[-F^{ab}\partial_a\delta A_b-m^2A^b\delta A_b\right]
=\int d^4x\left(\partial_aF^{ab}-m^2A^b\right)\delta A_b.
$$
The <integration by parts> discards a boundary term, with the variation fixed at the boundary. Thus the <Euler-Lagrange field equation> is the <Proca equation>
$$
\boxed{\partial_aF^{ab}-m^2A^b=0},\qquad
(\Box-m^2)A^b-\partial^b(\partial_aA^a)=0.
$$
The minus sign on $m^2$ is required by the chosen <Minkowski metric>; after imposing the constraint below, the <plane waves> have positive <energy> $E^2=|\boldsymbol q|^2+m^2$.
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