Functional derivative

In mathematics and theoretical physics, the functional derivative is a generalization of the usual derivative that occurs in the calculus of variations. In a functional derivative, instead of differentiating a function with respect to a variable, one differentiates a functional with respect to a function.

Definition

Two possible, restricted definitions suitable for certain computations are given here. There are more general definitions of functional derivatives. The first one is the Gâuteaux derivative (which generalizes the concept of directional derivative to infinite-dimensional Banach spaces):

For any functional ${displaystyle scriptstyle F}$ application of functions ${displaystyle scriptstyle varphi:{mathcal {M}}to mathbb {R} ^{n}$ defined on a variety ${displaystyle scriptstyle {mathcal {M}}}}$ (note that the set of these functions forms a Banach space), then the functional derivative in the sense of Gâteaux is such a distribution that for all test functions (test) f:

(1)${displaystyle leftlangle {frac {delta F[varphi (x)]}{delta varphi (x)}},f(x)rightrangle =left.{frac {d}{depsilon }}}F[varphi +epsilon f]rightrelated_{epsilon =0}. !$

The previous limit does not have to exist, worse still, even when the limit exists it can depend on the function f chosen, so the previous definition should be understood rather as a generalization of the previous one. directional derivative, rather than as a generalization of the concept of a differentiable function.

Alternate definition

You can also define the functional derivative in terms of a limit involving the Dirac delta, δ:

(2)${displaystyle {frac {delta F[phi (x)]}{delta phi (y)}}}=lim _{varepsilon to 0}{frac {F[phi (x)+varepsilon delta (x-y)]-F[phi (x)]}{varepsilon }}}}. !$

The definitions (1) and (2) are not equivalent, since the result of the first is always another functional, while the result of the second definition is a distribution. This is illustrated by the following example:

${displaystyle Eval_{x_{0}}[phi ]mapsto phi (x_{0})left[=int _{mathbb {R} }delta (x-x_{0})phi (x)dxright}}}$

The derivatives in the two previous senses are given by:

${displaystyle left({frac {delta Eval_{x_{0}}}{delta phi }right)_{[f]}=f(x_{0})})qquad {frac {delta Eval_{x_{0}}}}{phi(x)}{delta phi(yx}=$

Functional differentiability

If considered a uniparametric family of soft functions ${displaystyle psi _{lambda }:{mathcal {M}}longrightarrow mathbb {R} ^{n}}$ and is considered the function of a real variable built from a functional one:

${displaystyle S_{lambda }=S[psi _{lambda }];}$

And assuming that for every family like the previous one that satisfies certain conditions, the following derivatives exist:

${displaystyle delta psi =left.{frac {dpsi _{lambda }}{dlambda }}{right presumpda = 0}qquad left.{frac {dS_{lambda }}{dlambda }{lambda }{lambda = }{$

Under these conditions, the functional derivative is defined as:

(♪)${displaystyle left.{frac {delta S_{lambda }{delta psi}}{right structured_{psi _{0}}}=chi }$

Note that the previous equation is satisfied then the linear functional allows to approximate until first order to the original functional ${displaystyle S;}$. Note also that the expression (♪) is a Jacobean linear application that generalizes in the concept of Jacobin matrix.