Calculus of Variations: Understanding functional derivative

I am trying to understand the basics of the Calculus of Variations and the first thing to understand is the functional derivative. I failed to find a good introductory material, so I am trying to make sense out of various sources I found on the internet.

Now, $F[y]$ is a functional, dependending on the function $y(x)$ on an interval $[a,b]$. The most explanatory resources I have came across builds the functional derivative from the definitions of multivariable calculus, so we divide the interval $[a,b]$ into $N$ subintervals and assume that $F$ depends on values of $y$ at such interval points. So we have a multivariable function: $F(y_0,y_1,y_2,\dots, y_N)$ where $y_i = y(a + i(\dfrac{b-a}{N}))$. Then we make a small displacement $\epsilon \vec{d}$ from the point $\{y_0,\dots,y_N\}$ and obtain: $$F(y_0 + \epsilon d_0 ,y_1 + \epsilon d_1,y_2+ \epsilon d_2,\dots, y_N+ \epsilon d_N) = F(y_0,y_1,y_2,\dots, y_N) + \epsilon\sum_{i=0}^N\frac{\partial F}{\partial y_i}d_i + O(\epsilon^2)$$ $O(\epsilon^2)$ shows that the residue is on the order of $\epsilon^2$.

Now, how can I approach from here and obtain the formula for the functional derivative? What I have in mind is to refine the number of subintervals on $[a,b]$; make denser and denser meshes which results in taking the limit $N \to \infty$. But I don't know how to apply this idea in a correct way. The term $\epsilon\sum_{i=0}^N\frac{\partial F}{\partial y_i}d_i$ should turn into a definite integral if I take this limit, but $\frac{\partial F}{\partial y_i}d_i$ is not a proper continuous function to begin with. So I need help in understanding this derivation.

Edit: The end result I am trying to reach is $F [y (x) + \epsilon n (x)] = F [y (x)] + \epsilon \int \dfrac {\delta F}{\delta y (x)}n (x) dx +O (\epsilon^2)$

• I made an edit to my post; added the result I am trying to get. – Ufuk Can Bicici Jan 24 '15 at 15:18
• Why is $F_{y_i} d_i$ not continous? – mvw Jan 24 '15 at 18:45
• I found a helpful introduction to functionals and functional derivatives posted here by Professor benhamin Svetitsky at Tel Aviv University: julian.tau.ac.il/~bqs/functionals.pdf – RobertF Jan 15 '20 at 20:00

For general function $F$ and general functions $y : I=[a,b] \to \mathbb{R}$ I have problems to justify $$F[y] = F[Y] \approx F[Y_N]$$ where $Y$ is the exact graph of $y$ and $Y_N$ the discretized graph.
Also $$\lim_{\epsilon\to 0} \frac{F[Y+\epsilon E_N] - F[Y_N]}{\epsilon} = \sum_{i = 0}^N \frac{1}{h} \frac{\partial F}{\partial y_i}\, \eta_i \, h \quad (*)$$ just looks similar to $$\lim_{\epsilon\to 0} \frac{F[y+\epsilon \eta] - F[y]}{\epsilon} = \int\limits_a^b \frac{\delta F}{\delta y}\, \eta \, dx$$ but I do not know how to use $(*)$ to actually calculate values, especially how to come up with a good choice for $\partial F/\partial y_i$ in the discretization..