= Functional Taylor expansion
{title2=$S[\phi+\eta f]=S[\phi]+\eta\int f\,\delta S/\delta\phi+O(\eta^2)$}
The first variation of a differentiable <action functional> gives its change under a small field deformation. With $E=\delta S/\delta\phi$, the change $\phi\mapsto\phi-\eta f$ generates $-\eta\int fE$ to first order. This is the basic cancellation behind the <equivalence theorem for field redefinitions>; the <functional Jacobian> and transformed operator insertions must also be treated consistently.
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