Is the functional derivative a function or a functional

  • #1
Sonderval
234
11
I am confused whether the functional derivative ($\delta F[f]/\delta f$) is itself a functional or whether it is only a function

The Wikipedia article is not very rigorous
https://en.wikipedia.org/wiki/Functional_derivative
but from the examples (like Thomas-Fermi density), it seems as if the derivative of a functional is a function, for example


However, I would expect it to be a functional in itself (in the same way that the derivative of a function is a function)
 
Physics news on Phys.org
  • #2
A functional as I understand it is any function from a vector space into a field. Hence, every functional is always a function, but not vice versa.

What is a functional in your opinion?
What do you mean by functional derivative; or or simply the result in which case I'd ask what the variable is?
 
Last edited:
  • #3
I understand a functional to be a map from a space of functions to a number, as in my example above:

A Functional gets a function as input and gives a number.

The functional derivative should (if I understand things correctly, which I probably don't) produce a new object from a functional (so the second of your options above) in the same way the derivative of a function produces a new object (the derivative function in 1D or the gradient function in a vector space).

My question is exactly that: What kind of object is the derivative of a functional, i.e., if I apply the "operator"
to a functional , what is the result? A functional? An object that maps a function to a function (like a gradient maps a vector to a vector)?
 
  • #4
It depends. From the functional you create a new object, . Evaluating this at gives you the linear functional
Therefore is a linear operator which maps a functional to a function from the space of functions to the space of linear functionals.
 
  • #5
Thanks. But is this the same as ?
In analogy to functions, I would expect the first to be the equivalent of a total differential (in functional logic a variation) and the second to be equivalent to a derivative. I find the nomenclature quite confusing, to be honest.
 

Similar threads

Replies
1
Views
2K
Replies
6
Views
470
Replies
3
Views
1K
Replies
2
Views
2K
Replies
2
Views
2K
Back
Top