Also referred to as the _variational derivative_, the [functional](Functional.md) derivative is defined as $\frac{dF[f+\varepsilon \phi]}{d\varepsilon}\bigg|_{\varepsilon=0}=\int dx_1 \frac{\delta F[f]}{\delta f(x_1)}\phi(x_1),$ ^ffab81 which exists if the functional is [differentiable](Differentiability%20of%20a%20functional.md) Here, $\phi(x_1)$ is any arbitrary [function.](Analysis%20(index).md#Functions) # derivation The definition of the functional derivative may be derived from the 1st order term in the [Taylor expansion of a functional variation](Variation%20of%20a%20functional.md#Evaluating%20a%20functional%20variation). # derivatives of local functionals If we're given a [local functional](Functional.md#local%20functional) we may use a [Dirac delta distribution](Dirac%20delta%20function.md) as the arbitrary function in place of $\phi(x)$ then $\frac{dF[f+\varepsilon \phi]}{d\varepsilon}\bigg|_{\varepsilon=0}=\int dx_1 \frac{\delta F[f]}{\delta f(x_1)}\phi(x_1)=\frac{\delta F[f]}{\delta f(x_1)}$$= \lim_{\varepsilon\rightarrow 0}\frac{F[f(x')+\varepsilon \delta (x'-x)]-F[f(x')]}{\varepsilon}$ which should look familiar as being only slightly different from the definition of a [derivative](Analysis%20(index).md#Derivative) of an ordinary function. Thus for local functionals we define [[Functional derivative]] as $\frac{\delta F[f]}{\delta f(x_1)}=\lim_{\varepsilon\rightarrow 0}\frac{F[f(x')+\varepsilon \delta (x'-x)]-F[f(x')]}{\varepsilon}$ Thus we may employ the same derivative rules as for ordinary functions in evaluating functional derivatives. However, note the appearance of the [Dirac delta function](Dirac%20delta%20function.md) in the [examples](Functional%20derivative.md#Examples%20of%20local%20functional%20derivatives%20functional%2020derivative%20md%20local%2020functional%2020derivative) below. # Higher order functional derivatives based on the definition of [[Functional derivative]]s, higher order functional derivatives are defined as $\frac{d^n}{d\varepsilon^n}F[f+\varepsilon \phi]\bigg|_{\varepsilon=0}=\int dx_1...dx_n\,\frac{\delta^n F[f]}{\delta x_1 ... \delta_n}\phi(x_1)...\phi(x_n)$ # Differentiability of a functional --- # Proofs and Examples ## Examples of [derivatives of local functionals](Functional%20derivative.md#derivatives%20of%20local%20functionals) * $\frac{\delta}{\delta f(y)}\int dx\, f^2(x)=\int dx\, 2f(x)\delta(x-y)=2f(y)$ * $\frac{\delta}{\delta f(y)}\int dx\,(f''(x)-f(x))$ Here it's helpful to first rewrite the functions in the integral as functionals themselves. such that $f(x)=\int f(z)\delta(x-z)\;\;\;\; \mbox{and}$$f''(x)=\int dz\,f(z)\delta''(x-z).$ Plugging this in we obtain $\int\int dx dz\,\delta''(x-z)\frac{\delta f(z)}{\delta f(y)}-\int dx\,\frac{\delta f(x)}{\delta f(y)}=\int dx\,[\delta''(x-y)-\delta(x-y)]$ Note that the functional derivative of a functional of a constant is 0: * $\frac{\delta}{\delta f(y)}\int dx\,f(0)=0$ * $\frac{\delta}{\delta f(y)}\int dx\,f'(b)=\int dx\,\delta(b-y)\frac{d}{dx}f'(b)=0$ This is also true based on the fact that the functional derivative of functions are 0. --- # Recommended Reading #MathematicalFoundations/Analysis/Derivatives