Talk:Functional derivative

message: > Another definition is in terms of a limit and the Dirac delta function, δ:
> \frac{\delta F[\phi(x)]}{\delta \phi(y)}=\lim_{\varepsilon\to 0}\frac{F[\phi(x)+\varepsilon\delta(x-y)]-F[\phi(y)]}{\varepsilon}.
But in general, F is a functional which is only defined over continuous/smooth functions. φ(x)+εδ(x-y) does not even count as a function. It's a distribution. Phys 22:03, 3 Dec 2004 (UTC)
thread_title:  Definition with Dirac delta 

message: From wikipedia:<BR>
"the functional derivative is a generalization of the usual derivative that arises in the calculus of variations. In a functional derivative, instead of differentiating a function with respect to a variable, one differentiates a functional with respect to a function."
From MathWorld (http://mathworld.wolfram.com/FunctionalDerivative.html):<BR>
"The functional derivative is a generalization of the usual derivative that arises in the calculus of variations. In a functional derivative, instead of differentiating a function with respect to a variable, one differentiates a functional with respect to a function."
I don't know how different things need to be in order to avoid looking like someone was just copying their site.  http://mathworld.wolfram.com/about/faq.html#copyright --anon
: Thanks. I removed that text, just in case. Oleg Alexandrov (talk) 23:35, 23 October 2005 (UTC)
replies: text: This text was added by User:Kumkee on 13 feb 2004. It was his first (non-anonymous) edit ever; he's only made some half-dozen non-user-space edits since. linas 23:39, 24 October 2005 (UTC)
thread_title:  Is this too close to Wolfram's description? 

message: In the first example in the Examples section, there seems to be a sign error when the "Partial integration of second term" happens.  I also think that the text within the math is not good style.  Since this is my first time editing a math-related article, I'm not too confident about either of these points.  I've changed:
:
\begin{matrix}
\left\langle \delta F[\rho], \phi \right\rangle 
& = & \frac{d}{d\epsilon} \left. \int f( \mathbf{r}, \rho + \epsilon \phi, \nabla\rho+\epsilon\nabla\phi ) d^3r \right|_{\epsilon=0} \\
& = & \int \left( \frac{\partial f}{\partial\rho} \phi + \frac{\partial f}{\partial\nabla\rho} \cdot \nabla\phi \right) d^3r \\
& = & [\mbox{partial integration of second term}] \\
& = & \int \left( \frac{\partial f}{\partial\rho} \phi + \nabla \cdot \frac{\partial f}{\partial\nabla\rho} \phi \right) d^3r \\
& = & \left\langle \frac{\partial f}{\partial\rho} + \nabla \cdot \frac{\partial f}{\partial\nabla\rho}, \phi \right\rangle 
\end{matrix}
to
:
\begin{matrix}
\left\langle \delta F[\rho], \phi \right\rangle 
& = & \frac{d}{d\epsilon} \left. \int f( \mathbf{r}, \rho + \epsilon \phi, \nabla\rho+\epsilon\nabla\phi )\, d^3r \right|_{\epsilon=0} \\
& = & \int \left( \frac{\partial f}{\partial\rho} \phi + \frac{\partial f}{\partial\nabla\rho} \cdot \nabla\phi \right) d^3r \\
& = & \int \left( \frac{\partial f}{\partial\rho} \phi + \nabla \cdot \left[ \frac{\partial f}{\partial\nabla\rho} \phi \right] - \left[ \nabla \cdot \frac{\partial f}{\partial\nabla\rho} \right] \phi \right) d^3r \\
& = & \int \left( \frac{\partial f}{\partial\rho} \phi - \left[ \nabla \cdot \frac{\partial f}{\partial\nabla\rho} \right] \phi \right) d^3r \\
& = & \left\langle \frac{\partial f}{\partial\rho} - \nabla \cdot \frac{\partial f}{\partial\nabla\rho}\,, \phi \right\rangle 
\end{matrix}
,
Eubene 22:13, 18 September 2006 (UTC)
replies: text: You're correct; I had made an error. Good change! Md2perpe 20:58, 26 September 2006 (UTC)
thread_title:  Partial Integration? 

message: Hi, when I follow the steps to the Weizsäcker derivative the first, eg. (1/8), term has a negative sign, as it comes from deriving 1/rho. Then, in the last equality, it is positive, and in accordance with all texts I know. Is it there a typo somewhere? Also, is it nabla^2 (eg. the sum of the second derivatives) or nabla.nabla ?
I've read the paper:
T Gál, Á Nagy. A method to get an analytical expression for the non-interacting kinetic energy density functional. J. Mol. Struct. (Theochem) 1280, 167 - 171 (2000).  
and I think I got confused.  —Preceding unsigned comment added by 83.53.150.6 (talk) 04:00, 21 August 2008 (UTC)
thread_title:  Weizsäcker kinetic energy functional 

message: I have proposed to merge the entry First variation into the Functional derivative entry, since they are the same concept: any standard text on the calculus of variation reports the definition of the first functional derivative, calling it "first variation". Also, the concept of functional derivative has its origins in the systematization of the calculus of variation in framework of functional analysis as done by Vito Volterra, i.e. as a concept in the theory of functionals, linear and non-linear. Daniele.tampieri (talk) 15:20, 4 July 2009 (UTC)
replies: text: I don't actually disagree with you, but some authors do make a distinction between the two terms. In Gelfand and Formin's "Caluculus of Variations", The functional derivative (if I remember correctly) is essentially the left hand side of the Euler-Lagrange equations for the functional.--129.69.21.117 (talk) 09:48, 30 January 2012 (UTC)

text: As a practical matter, I think this should remain a separate topic.  There are plenty of practical cases where the calculational machinery of the functional derivative can be applied without need for recourse to the First variation.  — Preceding unsigned comment added by 68.232.116.79 (talk) 02:14, 11 February 2012 (UTC)
thread_title:  Merge proposal 

message: I think all the definitions that use delta functions are based on historical, but generally invalid, methods. In particular, since the delta function is not generally in the space of functions being varied, the limit in question doesn't even make sense. 
In reality, two different things are happening. First, we're looking for a first approximation to the functional in question near a given function:
\delta F[\phi][\eta] = F[\phi + \eta] - F[\phi] + O(\eta^2)
Here, η is any "small" test function.
On the other hand, physicists like to define objects via integral kernels. Hence, we look for a function (or, more generally, a distribution) \frac{\delta F}{\delta \phi[x]} such that:
\delta F[\phi][\eta] = \int \frac{\delta F}{\delta \phi[x]} \eta(x) dx
This gives the right results in all cases that I've ever checked, and is much less confusing. If it were up to me, I would write \frac{\delta F}{\delta \phi[x]} as \frac{\delta F}{\delta \phi}(x), which I consider much clearer. 
Note that, generally, when a second variable is present, it's part of the definition of the functional and should be clearly shown as such, e.g., by making it a subscript on the functional, etc.
For example, in the case given in the article, \rho(r) = \int \rho(r') \delta(r-r') dr', what we really have is a functional F_r, where F_r [\rho] = \int \rho(r') \delta(r-r') dr' = \rho(r) for a fixed r - this is just the Dirac delta distribution, centered at r. 
Then: 
\delta F_r[\rho][\eta] = F_r [\rho + \eta] - F_r [\rho] + O(\eta^2) = \int \eta(r') \delta(r-r') dr' = \int \eta(r') \frac{\delta F_r}{\delta \rho[r']} dr', so \frac{\delta F_r}{\delta \rho[r']} = \delta(r-r'), as expected.
As a matter of philosophy, one has to ask when Wikipedia should push a point of view and when it should just reflect what exists. I don't know a good answer to this. But I'd love people's thoughts on whether we couldn't incorporate this way of looking at functional derivatives into this article.
Rwilsker (talk) 19:36, 18 July 2010 (UTC)
replies: text: Thank you so much for trying to make this stuff rigorous. It's ubiquitous in physics, but it's really hard to find any formal description in the literature that's also usable. --130.149.215.94 (talk) 10:13, 19 April 2015 (UTC)
replies: text: Use any mathematical textbook on Lagrangian mechanics? Seriously. 81.225.32.185 (talk) 16:04, 6 June 2022 (UTC)
thread_title:  A different way to look at this 

message: Hi!
There are some examples - which is good. What is completely missing though is a section with "Properties" of the functional derivative.
I am thinking about the "product rule", the "chain rule" and things like that. (These rules, by the way, are quite easy to derive, especially if one "cheats" and use the physics treatment.)
I suggest a new section labelled "Properties" where properties of the functional derivative are listed. YohanN7 (talk) 17:36, 20 August 2012 (UTC)
thread_title:  Properties 

message: I added a computational step to the definition that would have saved me a lot of frustration and time had someone put it there for me. I didn't know this stuff before coming to this page, and I had to do a lot of digging to unravel what you're doing. Do we want people to learn, or are we going for brevity? I understand if you want to delete it (actually I don't), but you threw in the epsilon out of nowhere, and didn't clarify it further down the page. I know the traditional job of an epsilon, but coming at it blind, especially when you aren't familiar with functional derivative notation, most people who don't know it already won't automatically deduce that step, and so right off the bat you crush outsiders who are fluent in the language but not the mechanics.
173.25.54.191 (talk) 22:07, 16 February 2013 (UTC)
replies: text: Re "but you threw in the epsilon out of nowhere, and didn't clarify it further down the page." — Thanks for pointing that out, although that was done according to the source. I just now added clarification of ε after the equations and deleted the right hand side that you added. Regarding ε, not clear why you didn't "clarify it further down the page" as you thought was needed, instead of making an edit that is  just creating a right hand side of an equation by adding factors 1/ε and ε to the left hand side, which didn't seem useful. --Bob K31416 (talk) 15:42, 24 February 2013 (UTC)
thread_title:  Computational step 

message: Some observations as above, related to the reference section and referencing.
*"Undid revision 541704933 by Daniele.tampieri (talk) per §9.64 of Chicago Style Manual": unfortunately, the WP:CT page, reporting all accepted citation templates and examples, does not cite the Chicago Style Manual as a unique source of the true style, nor it forces an editor to drop a digit (!) to follow it: as you can see, every example in those page does not follow such practice. Therefore I reverted your removal of the "2" in the relevant reference. Also, I did not find anything on such topic in the Manual at the paragraph you stated: did you mean chapter 9, paragraph 64? I have access to the 2010 edition: is it the edition you refer to? 
— Preceding unsigned comment added by  Daniele.tampieri  (talk • contribs)   20:49, 3 March 2013‎  (UTC)
::Since you insist about the "2", that's OK. I don't think that Wikipedia guidelines has stated a preferred style.  
::"§9.64 of Chicago Style Manual" was referring to the 15th edition of 2003. If you're interested, it was in Chapter 9 "Numbers", "inclusive Numbers", §9.64 Abbreviating, or condensing, inclusive numbers. --Bob K31416 (talk) 17:05, 5 March 2013 (UTC)
thread_title:  On referencing and References section 

message: See my recent edit.  "Thomas–Fermi" required an en-dash, not a hyphen.  But Springer-Verlag requires a hyphen, not an en-dash.  It's not one person or thing named Springer and another named Verlag.  Rather, "Verlag" means "publishing company".  That way of using a hyphen is standard in German. 174.53.163.119 (talk) 00:15, 16 April 2013 (UTC)
replies: text: En_dash#Relationships_and_connections --Bob K31416 (talk) 23:15, 18 April 2013 (UTC)
thread_title:  hyphens and en-dashes 

message: In the second paragraph, we have "In an integral L of a functional, if a function f is varied..." which doesn't make sense to me.  L is not an integral, it is a function which is part of the integrand which is integrated to form the functional.  Maybe replace the text with "In a functional, if a function f is varied...", or something better. Ian Hinder (talk) 14:42, 27 January 2018 (UTC)
thread_title: Integral L of a functional

message: The current article contains the phrase:
"If  is varied by adding to it a function , and the resulting integrand  is expanded in powers of "
As I understand the context, \delta  is real variable, so expanding a function in powers of \delta  would indicate a Taylor series expansion in powers of \delta.  However f is a given function. What procedure is used to perform an expansion "in powers of  \delta f "?  
  Tashiro~enwiki (talk) 14:06, 21 June 2019 (UTC)
thread_title: Powers of {{math|''δf''}} ? 

message: This article about a mathematical topic seems to use the appendix of a quantum chemistry book as a main source...Where is the rigorous mathematical information in this article? — Preceding unsigned comment added by 2001:638:911:103:134:109:144:1 (talk) 22:38, 5 June 2020 (UTC)
thread_title:  what is the mathematical background 

message: :\displaystyle\frac{\delta F[G[\rho]] }{\delta\rho(y)}  = \int dx \frac{\delta F[G]}{\delta G(x)}_{G = G[\rho]}\cdot\frac {\delta G[\rho](x)} {\delta\rho(y)} \ . 
The notation used isn't defined anywhere else in the article; and G can't be a functional. --Svennik (talk) 21:15, 5 January 2021 (UTC)
thread_title:  Chain rule doesn't make sense 

message: The functional derivative is (in a way) a special case of the directional derivative (sometimes known as the Gateaux derivative) where the direction vector is a delta function. In other words,
\frac{\delta F[\phi(x)]}{\delta \phi(y)}=\lim_{\varepsilon\to 0}\frac{F[\phi(x)+\varepsilon\delta(x-y)]-F[\phi(y)]}{\varepsilon}.
It is certainly not the Gateaux or Frechet derivative, and this claim should be removed. --Svennik (talk) 09:30, 7 January 2021 (UTC)
replies: text: Done. --Svennik (talk) 09:33, 7 January 2021 (UTC)
thread_title:  Dispute claim that the functional derivative is the Frechet / Gateaux derivative 

message: Essentially, I propose using the analogy:
: Gradient : Directional derivative = Functional derivative : Fréchet derivative
Essentially, the functional derivative generalises the gradient to infinite dimensions, so that if \nabla F(x) denotes the functional derivative at the function x, we have that \langle \nabla F(x), v\rangle = \lim_{\epsilon \to 0} \frac{F(x + \epsilon v) - F(x)}{\epsilon}, where \langle-,-\rangle denotes the inner product. What do people think of this? --Svennik (talk) 11:22, 7 January 2021 (UTC)
replies: text: What kind of inner product do you have in a general Banach space, matey? 81.225.32.185 (talk) 16:02, 6 June 2022 (UTC)
replies: text: Well, the functional derivative isn't defined for general Banach spaces, is it? Only for spaces of functions where the integral int(f*g) is at least a bilinear form, so I dont think that the analogy is that far from truth. What do you think? JackozeeHakkiuz (talk) 21:32, 4 August 2023 (UTC)
thread_title:  Idea for a better introduction 

message: Since the article goes into some length explaining why the "physicist's" definition is ill-defined perhaps an example can be included that shows this. I.e. somewhere where the "mathematician's" definition is well-defined where the "physicist's" is not.  — Preceding unsigned comment added by 129.104.107.227 (talk) 14:11, 11 April 2022 (UTC)
replies: text: The delta "function" is not a function, so how can a *functional* map it to anything? Functionals are defined on function spaces. 81.225.32.185 (talk) 02:02, 6 June 2022 (UTC)
thread_title:  Example where delta function test function fails 

message: This article is in critical need of rewriting, right now it's from the applied physicist's point of view and have grave mathematical errors.
This article shouldn't exist in its current form.  — Preceding unsigned comment added by 81.225.32.185 (talk) 00:35, 6 June 2022 (UTC)
thread_title:  This article is seriously flawed and doesn't make sense 

message: Hello. As mentioned previously in my reply to Rwilsker's post above (titled "A different way to look at this") I made some changes that I think address some of the worries expressed by previous editors, in particular the remarks mentioned that some people define the functional differential as the Fréchet derivative and some as the Gateaux derivative but tried to keep the easy-to-use definition of the differential quotient that was already present. Also flipped the appearance of the functional differential and the functional derivative in the article. Hope it's better like that. Comments are welcome! JackozeeHakkiuz (talk) 10:43, 4 August 2023 (UTC)
thread_title:  Flipped order of differential and derivative and added references