The variation of a functional is defined as $\delta F = F[f+\delta f]-F[f]$ where the variation of a function, $f(x)$ is $\delta f(x_1,...x_n)=\varepsilon\phi(x_1,...x_n).$ $\phi(x_1,...x_n)$ is an arbitrary function and $\varepsilon$ is an [[Infinitessimal number]]. ^abe9c3 # Evaluating a functional variation A [variation](Variation%20of%20a%20functional.md) may be expressed in terms of an arbitrary function $\phi$ multiplied by an infinitessimal value $\varepsilon$ such that $F[f+\delta f]=F[f+\varepsilon \phi].$ Therefore, we may treat $F[f+\varepsilon \phi]$ as an [ordinary](Analysis%20(index).md#Functions) [Continuous function](Continuous%20function.md) of $\varepsilon.$ This means we may use the [Taylor expansion](Analysis%20(index).md#Taylor%20Series) to evaluate $F[f+\varepsilon \phi]$ in terms of $\varepsilon$ such that $F[f+\varepsilon \phi] = F[f]+\frac{dF[f+\varepsilon \phi]}{d\varepsilon}\bigg|_{\varepsilon=0}\varepsilon+\frac{1}{2}\frac{d^nF[f+\varepsilon \phi]}{d\varepsilon^2}\bigg|_{\varepsilon=0}\varepsilon^2 + O(\varepsilon^3).$ Embedded in the first order expansion is the [functional derivative.](Variation%20of%20a%20functional.md#Relation%20to%20the%20functional%20derivative) # Relation to the functional derivative The derivative with respect to $\varepsilon$ of $F[f+\varepsilon \phi]$ that appears in the Taylor expansion of $\delta F$ is the definition of the [[Functional derivative]], which is given in its most general form by the expression  This expression naturally also gives the main condition for the [[Differentiability of a functional]]. ## variation in terms of an integral Given the relationship between [the variation and the functional derivative](Variation%20of%20a%20functional.md#Relation%20to%20the%20functional%20derivative) we may also define the variation as $\delta F[f]=\int dx'\,\frac{\delta F[f]}{\delta f(x')}\delta f(x')$ and we may rewrite our [Taylor expansion](Variation%20of%20a%20functional.md#Evaluating%20a%20functional%20variation) used to evaluate the [Variation of a functional](Variation%20of%20a%20functional.md) as $F[f+\delta f]=F[f+\varepsilon \phi]=F[f]+\int dx'\,\frac{\delta F}{\delta f(x')}\varepsilon \phi(x')+\int\int dx'dx''\,\frac{\delta^2 F}{\delta f(x')\delta f(x'')}\varepsilon^2\phi(x')\phi(x'')$$+O(\varepsilon^3)$ #MathematicalFoundations/Analysis/Functions/Functionals