Functional derivative ($\delta F/\delta \varphi$)

A functional derivative represents the linear first-order response of a functional to a perturbation of its input function. With a specified integration measure and admissible perturbations,For $F[\varphi ]=\int \varphi (x{)}^{2}dx$, direct expansion gives $\delta F/\delta \varphi =2\varphi$. For $F[\varphi ]=\frac{1}{2}\int |{\varphi }^{\prime }{|}^{2}dx$, integration by parts gives $-{\varphi }^{\prime \prime }$ when the boundary term vanishes under the chosen perturbation conditions. The boundary conditions and function space are therefore part of identifying the derivative, rather than optional details.

Let $S$ be a Fréchet differentiable functional on a Banach space. Its first-order expansion isfor fixed $f$ as $\eta \to 0$. Differentiability alone gives this little-o remainder; it does not imply the $O({\eta }^2)$ remainder displayed in the usual second-order-bounded form.

For that stronger bound, suppose the derivative is locally Lipschitz in operator norm with constant $L$. The fundamental theorem of calculus along the line segment givesso the remainder has magnitude at most $L\Vert h{\Vert }^2/2$. A locally bounded continuous second Fréchet derivative is a sufficient hypothesis. Where a functional derivative represents $D{S}_{\varphi }[f]$ by an integral, the corresponding coefficient is $\int f\delta S/\delta \varphi$.

Taking $h=-\eta f$ therefore produces the linear change $-\eta \int f\delta S/\delta \varphi$. That is the cancellation used in the equivalence theorem for field redefinitions. The functional Jacobian and transformed operator insertions still need consistent treatment; the Taylor estimate is a statement about the specified function space and admissible deformation.

Along a differentiable path of fields $\varphi (x,t)$, a differentiable functional obeysIntegrating in time expresses the functional change as the line integral of its functional derivative along the path.