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I need to evaluate the following limit which intuitively I know its equal to 0 but I can't really prove it, so I need some help:

$\lim_{\epsilon \to 0}{\frac{F[\rho + \epsilon\rho' + \epsilon^2\rho'']-F[\rho + \epsilon\rho']}{\epsilon}}$

where $\epsilon$ is a real number, $F$ is a functional and $\rho$, $\rho'$ and $\rho''$ are functions in some function space.

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    The exchange correlation functionals have the general form $E_{xc}[\rho]=\int{\rho(\mathbf{r})f(\rho(\mathbf{r}),\nabla\rho(\mathbf{r}))d \mathbf{r}}$ where $f$ is an unknown function.2012-09-12

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Here is a very, very rough sketch of a possible proof...

Let $\bar{\rho} (\epsilon) := \rho + \epsilon \rho'$. The limit can then be written in the form

$\displaystyle\lim_{\epsilon \to 0} \frac{1}{\epsilon}\left[ F (\bar{\rho} (\epsilon) + \epsilon^2 \rho'') - F (\bar{\rho} (\epsilon))\right]$

The "Taylor expansion" of $F$ is the following

$F (\bar{\rho} (\epsilon) + \epsilon^2 \rho'') = F (\bar{\rho} (\epsilon)) + \langle \nabla F (\bar{\rho} (\epsilon)), \epsilon^2 \rho''\rangle + \omicron (\epsilon^4)$

where $\nabla F (\bar{\rho} (\epsilon))$ is the "functional gradient" of $F$. Then, we have that

$\displaystyle\lim_{\epsilon \to 0} \frac{1}{\epsilon}\left[ F (\bar{\rho} (\epsilon) + \epsilon^2 \rho'') - F (\bar{\rho} (\epsilon))\right] = \lim_{\epsilon \to 0} \left[\frac{1}{\epsilon} \langle \nabla F (\bar{\rho} (\epsilon)), \epsilon^2 \rho''\rangle + \omicron (\epsilon^3)\right]$

If we can show that

$\frac{1}{\epsilon} \langle \nabla F (\bar{\rho} (\epsilon)), \epsilon^2 \rho''\rangle = \langle \nabla F (\bar{\rho} (\epsilon)), \epsilon \rho''\rangle$

then the limit becomes

$\displaystyle\lim_{\epsilon \to 0} \frac{1}{\epsilon}\left[ F (\bar{\rho} (\epsilon) + \epsilon^2 \rho'') - F (\bar{\rho} (\epsilon))\right] = \lim_{\epsilon \to 0} \left[\langle \nabla F (\bar{\rho} (\epsilon)), \epsilon \rho''\rangle + \omicron (\epsilon^3)\right] = 0$

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    @Manuel: This is a good introductory overview of calculus of variations: http://www.math.umn.edu/~olver/am_/cvz.pdf2012-09-12