Square of a differentiable functional

[LCSphysicist]

TL;DR Summary

Write a formula for the square functional's variation

I will consider first the case of $\left [ J \right ] = \int f(x,y,y')$, if it is right believe is easy to generalize...
$$ \Delta J $$
$$\int (f(x,y+h,y'+h'))^2 - (f(x,y,y'))^2 $$
$$\int \sim [f(x,y,y') + f_{y}(x,y,y')h + f_{y'}(x,y,y')h']^2 - [f(x,y,y')]^2$$
to first order: $$\int \sim 2f(x,y,y')[f_{y}h + f_{y'}h']$$

All is ok? Is this right?

 

[PeroK]

What sort of function are you talking about here, and why the integral?

 

[LCSphysicist]

What sort of function are you talking about here, and why the integral?

"What sort of function?" Sorry, i think i don't understand the question. It is a functional, and the integral, yes, there is another types of functional, but as this chapter just treat the integral cases, i follow it and consider this functional form. But if you are talking about f, i am restringing it to continuous function with first derivative functions too.

 

[PeroK]

"What sort of function?" Sorry, i think i don't understand the question. It is a functional, and the integral, yes, there is another types of functional, but as this chapter just treat the integral cases, i follow it and consider this functional form. But if you are talking about f, i am restringing it to continuous function with first derivative functions too.

You'll need to be careful about how things are defined here. Are you squaring the functional? Or some function, the integral of which is the functional? How is a functional derivative defined?

If you are trying to prove something you need a formal definition.

 

[Office_Shredder]

To Perok's point, here's what I expected to see for proving the square of a differentiable function is differentiable.

If f(x) is differentiable, let $g(x)=x^2$. Then g is differentiable, and by the chain rule g(f(x)) is differentiable.
Your proof looks, well, different, which suggests we are missing context. Some people may know what you're talking about, but a little extra guidance as to what the objects are you are dealing with will widen the net of who can help you.

 

[LCSphysicist]

Actually, the question can be divided in two parts, to prove it, and find an expression for the square functional's variation. My post was about the second, as it is in the summary, i believe it was easier to start