Functional Taylor expansion ($S[\varphi +\eta f]=S[\varphi ]+\eta \int f\delta S/\delta \varphi +O({\eta }^2)$)

The first variation of a differentiable action functional gives its change under a small field deformation. With $E=\delta S/\delta \varphi$, the change $\varphi \mapsto \varphi -\eta f$ 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.