[[Calculus of variations MOC]]
# Functional derivative

Le $B$ be a [[Banach space|Banach subspace]] of $\Man^1(\Omega, \mathbb{K})$ where $\mathbb{K} = \mathbb{C}$ or $\mathbb{K} = \mathbb{R}$
and $F : B \to \mathbb{K}$ be a [[Functional]].
The **functional derivative** $\frac{\delta F}{\delta\rho}$ of $F$ at $\rho$ is a a function such that the [[functional differential]] is given by #m/def/anal/fun/var 
$$
\begin{align*}
\delta F[\rho;\phi] = \int _{\Omega} \frac{\delta F}{\delta\rho}(x) \,\phi(x) \, dx 
\end{align*}
$$


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