Calculus of Variations: Understanding functional derivative

Question

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) $

Answer 1

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..

Answer 2

For your reference, I would recommend you to read this book. It is a good introduction book for beginners in CoV.

To understand the derivative of functional, you only need to image that you are actually doing the derivative with respect to function itself, but not the argument of your function. For some examples, please just go ahead and read the reference I provided from the beginning and you will understand it. It is really a very friendly introduction book.