 Functional Taylor expansion

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.