Talk:Functional derivative

Definition with Dirac delta

> 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)

Yes, I think that it should be noted that some physicians write the definition as above, but that it isn't a mathematically correct definition. Also, the correct definition (the first one) misses $f$ on the left side. I'll do these corrections. --Md2perpe 22:53, 30 July 2006 (UTC)

Could it be that due to the facts that

• physics problems most often find their mathematical statement with respect to a Hilbert space
• where on its dual space the Riesz representation theorem applies and
• due to the scalar product's symmetry
• the delta distribution is very often mistaken as the Dirac delta function

physicists (although this is mathematically not correct) calculate the gradient in the direction of the delta "function", yielding the physicist's version? If so, this could be pointed out more clearly. Greetings--Alexnullnullsieben (talk) 21:22, 5 November 2011 (UTC)

"where the arrow on the right handside, inside the functional F, indicates a function definition"--what arrow on the RHS? Steve Avery (talk) 02:18, 25 November 2007 (UTC)

Is this too close to Wolfram's description?

From wikipedia:
"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):
"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)

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)

Partial Integration?

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:

${\begin{matrix}\left\langle \delta F[\rho ],\phi \right\rangle &=&{\frac {d}{d\epsilon }}\left.\int f(\mathbf {r} ,\rho +\epsilon \phi ,\nabla \rho +\epsilon \nabla \phi )d^{3}r\right|_{\epsilon =0}\\&=&\int \left({\frac {\partial f}{\partial \rho }}\phi +{\frac {\partial f}{\partial \nabla \rho }}\cdot \nabla \phi \right)d^{3}r\\&=&[{\mbox{partial integration of second term}}]\\&=&\int \left({\frac {\partial f}{\partial \rho }}\phi +\nabla \cdot {\frac {\partial f}{\partial \nabla \rho }}\phi \right)d^{3}r\\&=&\left\langle {\frac {\partial f}{\partial \rho }}+\nabla \cdot {\frac {\partial f}{\partial \nabla \rho }},\phi \right\rangle \end{matrix}}$

to

${\begin{matrix}\left\langle \delta F[\rho ],\phi \right\rangle &=&{\frac {d}{d\epsilon }}\left.\int f(\mathbf {r} ,\rho +\epsilon \phi ,\nabla \rho +\epsilon \nabla \phi )\,d^{3}r\right|_{\epsilon =0}\\&=&\int \left({\frac {\partial f}{\partial \rho }}\phi +{\frac {\partial f}{\partial \nabla \rho }}\cdot \nabla \phi \right)d^{3}r\\&=&\int \left({\frac {\partial f}{\partial \rho }}\phi +\nabla \cdot \left[{\frac {\partial f}{\partial \nabla \rho }}\phi \right]-\left[\nabla \cdot {\frac {\partial f}{\partial \nabla \rho }}\right]\phi \right)d^{3}r\\&=&\int \left({\frac {\partial f}{\partial \rho }}\phi -\left[\nabla \cdot {\frac {\partial f}{\partial \nabla \rho }}\right]\phi \right)d^{3}r\\&=&\left\langle {\frac {\partial f}{\partial \rho }}-\nabla \cdot {\frac {\partial f}{\partial \nabla \rho }}\,,\phi \right\rangle \end{matrix}}$,

Eubene 22:13, 18 September 2006 (UTC)

You're correct; I had made an error. Good change! Md2perpe 20:58, 26 September 2006 (UTC)

Weizsäcker kinetic energy functional

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)

Merge proposal

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.

The result was do not merge into Functional derivative. -- DarkCrow^Caw 19:14, 8 May 2012 (UTC)

---

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)

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)

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)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

A different way to look at this

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)

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)

Use any mathematical textbook on Lagrangian mechanics? Seriously. 81.225.32.185 (talk) 16:04, 6 June 2022 (UTC)

I'm thinking of modifying the page to flip the order of appearance of the subsections "Functional derivative" and "Functional differential" in a hope to make clear that the definition of the differential

$F[\rho +\eta ]=F[\rho ]+\delta F[\rho ,\eta ]+O(\eta ^{2})$

comes first and only sometimes can $\delta F[\rho ,\eta ]$ be written as integration against some distribution, which is then called the functional derivative of $F$ at $\rho$. The current phrasing makes one think that this distribution (the functional derivative) is more fundamental, and then "the functional differential is defined in terms of the functional derivative". This is just backwards.

Also, why does the subsection "Functional derivative" start by mentionning manifolds? Maybe whoever wrote that part had in mind a Banach space or some sort of locally convex topological vector space, but for general manifolds in general the sum $\rho +\varepsilon \phi$ doesn't make sense.

I will leave this here for a while before proceeding with the edits. -- JackozeeHakkiuz (talk) 22:37, 31 July 2023 (UTC)

Properties

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)

Computational step

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)

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)

Leading section of the entry

I apologize for writing this note so late, but I've been busy and forgot it completely. However, here there is a very short list of the reasons why I changed the lead section.

• According to the manual of style (WP:MOS), precisely the section WP:BOLDTITLE, the first sentence should give a short explanation of what the concept/item is, not where it is used/to what field it belongs. This was the main reason that made me beginning editing the lead section.
• The changes I made in the lines following the first sentences of the heading section are motivated by the following reasons, however not so compelling as the preceding one. It is inexact that functionals are usually expressed as integrals: this is true for some functionals, not for all (a trivial example: Dirac delta function has not the structure of an integral). However, the idea of using an integral functional in order to give an example of the concept is nice (and also such integral formalism is customary in a variety of useful contexts), so I tried (perhaps in a too verbose way) to precise the hypotheses under which the example works. This is also the reason for which I added the exact expression on the functional derivative as the integral of the Euler-Lagrange operator multiplied by the increment, which can be found in every text on the calculus of variation (it is exactly the step before the use of DuBois-Reymond lemma in deriving the Euler-Lagrange equation, 1-dimensional case).

Well, as a matter of fact, this is all. Daniele.tampieri (talk) 19:10, 24 February 2013 (UTC)

1) Thanks for pointing out a more preferred style for the first sentence. Also according to WP:BOLDTITLE, we should display the title as early as possible in the first sentence. Here is a rewrite of the original version of the first paragraph that accomplishes that. I thought it was better to rewrite the original, rather than the present one, because it states the concept more simply.

"A functional derivative (or variational derivative) relates a change in a functional to a change in a function that the functional depends on. Functional derivatives are used in the calculus of variations, which is a field of mathematical analysis."

This is the first installment in response to your above message and changes. Regards, --Bob K31416 (talk) 21:10, 24 February 2013 (UTC)

2) Re "It is inexact that functionals are usually expressed as integrals" — You implemented this idea in the following part of your version: "The concept can be easily understand by considering the common case encountered in the classical calculus of variation". Using the idea in this suggested version of yours, I rewrote the corresponding sentence in the original version:

"In the calculus of variations, functionals are usually expressed in terms of an integral of functions, their arguments, and their derivatives."

Then I restored the rest of the corresponding part of the lead before your discussion of Euler-Lagrange, since you haven't explained your other changes in that part and those changes don't appear to be an improvement. The result for that part of the lead is:

In the calculus of variations, functionals are usually expressed in terms of an integral of functions, their arguments, and their derivatives. In an integrand L of a functional, if a function f is varied by adding to it another function δf that is arbitrarily small, and the resulting L is expanded in powers of δf, the coefficient of δf in the first order term is called the functional derivative.

For example, consider the functional

$J[f]=\int \limits _{a}^{b}L[\,x,f(x),f\,'(x)\,]\,dx\ ,$

where f ′(x) ≡ df/dx.

This is the second installment in response to your above message and changes. Regards, --Bob K31416 (talk) 11:59, 25 February 2013 (UTC)

I partly agree with the changes you made: however, I cannot tell you now precisely where I do not agree since, as I told you on my talk page, I'm busy. You'll have to wait the weekend, and I'll answer here with a complete biography and links. Daniele.tampieri (talk) 16:58, 25 February 2013 (UTC)

Hi Bob. I can now precise why not all the changes you made are improvements, and why I consequently modified it:

• You have not read completely WP:BOLDTITLE since, in the first sentence, you should also provide context, as stated precisely in the subsection WP:CONTEXTLINK:

In an article about a technical or jargon term, the opening sentence or paragraph should normally contain a link to the field of study that the term comes from.

In heraldry, tinctures are the colours used to emblazon a coat of arms.

therefore, I changed the first sentence in order to comply this accepted, simpler and clearer practice.

— Preceding unsigned comment added by Daniele.tampieri (talk • contribs) 20:49, 3 March 2013 (UTC)

I agree re first sentence and context. --Bob K31416 (talk) 17:12, 5 March 2013 (UTC)

On referencing and References section

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)

hyphens and en-dashes

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)

En_dash#Relationships_and_connections --Bob K31416 (talk) 23:15, 18 April 2013 (UTC)

Integral L of a functional

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)

Powers of δf ?

The current article contains the phrase: "If f is varied by adding to it a function δf, and the resulting integrand L(x, f +δf, f '+δf ′) is expanded in powers of δf"

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)

what is the mathematical background

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)

Chain rule doesn't make sense

$\displaystyle {\frac {\delta F[G[\rho ]]}{\delta \rho (y)}}=\int dx{\frac {\delta F[G]}{\delta G(x)}}_{G=G[\rho ]}\cdot {\frac {\delta G[\rho ](x)}{\delta \rho (y)}}\ .$

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)

Dispute claim that the functional derivative is the Frechet / Gateaux derivative

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)

Done. --Svennik (talk) 09:33, 7 January 2021 (UTC)

Idea for a better introduction

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)

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)

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)

Example where delta function test function fails

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)

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)

This article is seriously flawed and doesn't make sense

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)

I agree that this article is leaving much to wish for when it comes to mathematics, and that it obfuscates much of the clarity found in a standard mathematical textbook. These "errors" don't however mean the article is wrong. It just means the article is bad at explaining a mathematical concept properly. Whether or not this should be the case is up for debate. I think it explains it clearly enough for the applications of the functional derivative, but not if you are here to learn what it really is. 84.216.129.69 (talk) 12:20, 6 June 2022 (UTC)

Flipped order of differential and derivative and added references

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)