Functionals and functional derivatives

Functionals are mappings from function spaces to real or complex numbers. A general representation for a function $F$ is

$$F[g] = F_0 + \int dx F_1(x) g(x)$$ (131)

$$+ \int dx_1 \int dx_2 F_2(x_1,x_2) g(x_1)g(x_2)$$ (132)

$$+ \int dx_1 \int dx_2 \int dx_3 F_3(x_2,x_2,x_3) g(x_1)g(x_2)g(x_3) + \cdots$$ (133)

where the kernels $F_i$ are general functions.

Now let $g \rightarrow g+\Delta g$. To linear order in $\Delta g$ we obtain

$$F[g+\Delta g] = F[g] + \int dx F_1(x) \Delta g(x)$$ (134)

$$+ 2\int dx_1 \int dx F_2(x_1,x) g(x_1)\Delta g(x)$$ (135)

$$+ 3\int dx_1 \int dx_2 \int dx F_3(x_1,x_2,x)g(x_1)g(x_2)\Delta g(x) + \cdots$$ (136)

where we have assumed that the functions $F_i$ are symmetric functions of their arguments.

We can rewrite this equation as

$$F[g+\Delta g] = f[g] + \int dx \frac {\delta F[g]}{\delta g(x)} \Delta g(x)$$ (137)

where

$$\frac {\delta F[g]}{\delta g(x)} = F_1(x) + 2 \int dx_1 F_2(x_1,x) g(x_1)$$ (138)

$$+ 3\int dx_1 \int dx_2 F_3(x_1,x_2,x)g(x_1)g(x_2) + \cdots$$ (139)

In analogy, we find

$$\frac {\delta^2 F[g]}{\delta g(x)\delta g(x')} = 2 F_2(x,x') + 3\int dx_1 F_3(x_1,x,x')g(x_1) + \cdots$$ (140)