Reference Request - Functional Derivatives I am looking for some good references that introduce functional derivatives in a quick but rigorous way. Any suggestions?
In addition, I saw somewhere that functional derivatives are related to Frechét derivatives. Is this accurate? How are they related?
 A: Meanwhile someone could add something about the references, let me tell you how the Frechét technique is used to get the famous Euler-Lagrange condition with a classical example.
Let us suppose the you have a integral functional like 
$$A=\int_I{\cal L}(x,y,y')dx$$
where $y=y(x)$. 
Now to mimic Frechét procedure we take
$$A_{\varepsilon}=\int_I{\cal L}(x,y+\varepsilon h,y'+\varepsilon h')dx.$$
Employ expansions to get
$${\cal L}(x,y+\varepsilon h,y'+\varepsilon h')={\cal L}(x,y,y')+\varepsilon h\frac{\partial{\cal L}}{\partial y}+\varepsilon h'\frac{\partial{\cal L}}{\partial y'}+R(\varepsilon^2,x).
$$
Upon some basic manipulation you will get
$$\frac{A_{\varepsilon}-A}{\varepsilon}=
\int_I h\left(\frac{\partial{\cal L}}{\partial y}-\frac{d}{dx}\frac{\partial{\cal L}}{\partial y'}\right)dx+\frac{1}{\varepsilon}\int_IR(\varepsilon^2,x)dx,$$
So, under some mild conditions, in the limit we arrive at
$$\lim_{\varepsilon\to0}\frac{A_{\varepsilon}-A}{\varepsilon}=
\int_I h\left(\frac{\partial{\cal L}}{\partial y}-\frac{d}{dx}\frac{\partial{\cal L}}{\partial y'}\right)dx,$$
for all function $h=h(x)$, which if they obey $h|_{\partial I}=0$ then, asking for
$\lim_{\varepsilon\to0}\frac{A_{\varepsilon}-A}{\varepsilon}=0$ in order to be a detector of extreme data $(x,y,y')$, one is led to the fact that
$$\int_I h\left(\frac{\partial{\cal L}}{\partial y}-\frac{d}{dx}\frac{\partial{\cal L}}{\partial y'}\right)dx=0,$$
for all suitable $h$, then is needed that 
$$\frac{\partial{\cal L}}{\partial y}-\frac{d}{dx}\frac{\partial{\cal L}}{\partial y'}=0,$$ to manipulate and find $y$ the extremizing function.
