What is Functional derivative: Definition + 34 Threads
In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a Functional (a functional in this sense is a function that acts on functions) to a change in a function on which the functional depends.
In the calculus of variations, functionals are usually expressed in terms of an integral of functions, their arguments, and their derivatives. In an integral L of a functional, if a function f is varied by adding to it another function δf that is arbitrarily small, and the resulting integrand 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
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{\displaystyle J[f]=\int _{a}^{b}L(\,x,f(x),f\,'(x)\,)\,dx\ ,}
where f ′(x) ≡ df/dx. 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, then the change in the value of J to first order in δf can be expressed as follows:
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{\displaystyle \delta J=\int _{a}^{b}\left({\frac {\partial L}{\partial f}}\delta f(x)+{\frac {\partial L}{\partial f'}}{\frac {d}{dx}}\delta f(x)\right)\,dx\,=\int _{a}^{b}\left({\frac {\partial L}{\partial f}}-{\frac {d}{dx}}{\frac {\partial L}{\partial f'}}\right)\delta f(x)\,dx\,+\,{\frac {\partial L}{\partial f'}}(b)\delta f(b)\,-\,{\frac {\partial L}{\partial f'}}(a)\delta f(a)\,}
where the variation in the derivative, δf ′ was rewritten as the derivative of the variation (δf) ′, and integration by parts was used.
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...
(To moderators: although the question is mathematical, I post it in the physics forum because the definition and the notation are those used by physicists and because it comes from a QFT textbook; please move it if I'm wrong.)
My issue with this question is that the textbook has neither defined...
In the book Quantum Field Theory for the Gifted Amateur, they define the functional derivative as:
$$ \frac{\delta F}{\delta f(x))} = \lim_{\epsilon\to 0} \frac{F[f(x') + \delta(x'-x)) ] - F[ f(x') ]}{\epsilon} $$
Why do they use the delta function and not some other arbitrary function?
I have some doubts with respect on how the functional derivative for the kinetic energy in density functional theory is obtained.
I have been looking at this article in wikipedia: https://en.wikipedia.org/wiki/Functional_derivative
In particular, I'm interested in how to get the...
I'm can't seem to figure out how to functionally differentiate a functional such as Z(J)= e^{\frac{i}{2} \int \mathrm{d}^4y \int \mathrm{d}^4x J(y) G_F (x-y) J(x)}
with respect to J(x) . I know the answer is
\frac{\delta Z(J)}{\delta J(x)}= -i \int \mathrm{d}^4y J(y) G(x-y)
but I'm struggling...
Hi,
I'm trying to understand the Euler-Poincare equations, which reduce the Euler-Lagrange equations for certain Lagrangians on a Lie group. I'm reading Darryl Holm's "Geometric mechanics and symmetry", where he suddenly uses what seems to be a variational derivative, which I'm having a hard...
I’ve always been confused by the formula for the Total Derivative of a function. $$\frac{df(u,v)}{dx}= \frac{\partial f}{\partial x}+\frac{\partial f }{\partial u}\frac{\mathrm{d}u }{\mathrm{d} x}+\frac{\partial f}{\partial v}\frac{\mathrm{d}v }{\mathrm{d} x}$$
Any insight would be greatly...
Homework Statement
I am currently working on an exercise list where I need to calculate the second functional derivative with respect to Grassmann valued fields.
$$
\dfrac{\overrightarrow{\delta}}{\delta \psi_{\alpha} (-p)} \left( \int_{x} \widetilde{\bar{\psi}}_{\mu} (x) i \partial_{s}^{\mu...
I'm studying QFT in the path integral formalism, and got stuck in deriving the Schwinger Dyson equation for a real free scalar field,
L=½(∂φ)^2 - m^2 φ^2
in the equation,
S[φ]=∫ d4x L[φ]
∫ Dφ e^{i S[φ]} φ(x1) φ(x2) = ∫ Dφ e^{i S[φ']} φ'(x1) φ'(x2)
Particularly, it is in the Taylor series...
Hi PF
I try to understand how we get get a Taylor expansion of a non linear functional.
I found this good paper
here F maps functions to scalars. F[f] is defined. It has not scalars as arguments. I agree with A13 and A18.
In another paper (in french) skip to page 9
the fisrt term is ##\int dx...
Homework Statement
To show that ##\rho(p',s)>\rho(p',s') => (\frac{\partial\rho}{\partial s})_p\frac{ds}{dz}<0##
where ##p=p(z)##, ##p'=p(z+dz)##, ##s'=s(z+dz)##, ##s=s(z)##
Homework Equations
I have no idea how to approach this. I'm thinking functional derivatives, taylor expansions...
Good afternoon,
i was just wondering if this equation is possibly solvable where I(z) is a function of z. The equation is:
I(z)=cosh(1/2 ∫I(z)dz)
I know it looks stupid but is it possible? How would you approach this problem?
Thank you.
1./2. Homework Statement
In my QFT lecture we were introduced to the 1PI effective action ##\Gamma[\varphi]## for a scalar theory (in Euclidean space-time). In one-loop approximation we've found \Gamma^{(\text{1-loop})}[\varphi] = S[\varphi] + \frac{1}{2} \operatorname{Tr} \log D^{-1} where...
Homework Statement
[/B]
Consider the following action:
$$\begin{align}S = \int \mathrm{d}^4 z \; \bar\psi_i(z) \, (\mathrm{i} {\not{\!\partial}} - m)_{ij} \, \psi_j(z)\end{align}$$
where ##\psi_i## is a Dirac spinor with Dirac index ##i## (summation convention for repeated indices). Now I would...
In my textbook (see attached picture) there appears a functional derivative, but I honestly don't know how to evaluate a quantity like this. What should I do? I have tried to google but all I could find was how to take functional derivatives, where polynomials appeared under the integral, while...
I can't convince myself whether the following functional derivative is trivial or not:
##\frac \delta {\delta \psi(x)} \big[ \partial_x \psi(x)\big],##
where ##\partial_x## is a standard derivative with respect to ##x##.
One could argue that
## \partial_x \psi(x) = \int dx' [\partial_{x'}...
Hi everyone! I have a question on functional derivatives. I have a function defined as:
$$
F[\{u\}]=\int d^3r \sum_{i=1}^3 \frac{\partial u_i}{\partial r_i},
$$ where u_i(\vec r) is a function of the position. I need to compute its functional derivative. To do that I did the following:
$$...
Homework Statement
Dear all, Good day.
I am currently working on the phase field modeling of ferroelectrics. For this reason, I need to find functional derivative of an expression as presented in attached picture 1.
Then in picture 2, it shows the final form of equation that I am...
I'm self studying Chaikin's Principles of Condensed Matter Physics.
I'm trying to figure out how to go from (5.2.30) to (5.2.31).
Homework Statement
5.2.30 is the one-loop approx. to the free energy.
I'll denote G0^-1 from the book G
~ Integral(ln(G(phi(x)))
5.2.31 is (as far as...
Hi,
I have a question about a functional derivative. When determining the condition that the functional derivative have a stationary value of 0, do I use a partial derivative or a regular derivative? I would really appreciate the help. Thank you!
David
Alright, so I feel kind of dumb...but:
I have been working on some QFT material, specifically derivation of Feynman rules for some more simple models ( \phi^{4} for example), and I have been seriously failing with functional derivatives. Every time I try to use the definition I mess up...
Hi,
in their book ''Density-Functional Theory of Atoms and Molecules'' Parr and Yang state in Appendix A, Formula (A.33)
If F ist a functional that depends on a parameter \lambda, that is F[f(x,\lambda)] then:
\frac{\partial F}{\partial \lambda} = \int \frac{\delta F}{\delta f(x)}...
Hi!
I am doing some numerical calculations recently. I need to calculate the functional derivative. eg. functional :
n(\rho)=\int dr'r'\rho(r')f(r,r')
it need to calculate:
\frac{\delta n(r)}{\delta\rho(r')}
I think the...
I cannot work out the following functional derivative:
\frac{\delta}{\delta g_{\mu\nu}} \int d^4 x f^a_{\phantom{a}b} \nabla_a h^b
Where f is a tensor density f= \sqrt{\det g} \tilde{f} ( \tilde{f} is an ordinary tensor)
and should be consider as independent of g. In my opinion this is not...
Roger Penrose, in The Road to Reality, introduces the idea of what he calls a "functional derivative", "denoted by using \delta in place of \partial; "Carrying out a functional derivative in practice is essentially just applying the same rules as for ordinary calculus" (Vintage 2005, p. 487). He...
For my current research, I need to prove the following:
\int_0^1 \frac{dC(q(x) + k'(q'(x) - q(x)))}{dk'}\,dk' = \int_0^1 \int_L^U p(q(x) + k(q'(x) - q(x)))(q'(x)-q(x)) dx dk
where C(q(x)) = \int_0^1 \int_L^U p(kq(x)) q(x)\,dx\,dk
Here's what I've tried using the definition of functional...
In the literature (Ryder, path-integrals) I have found the following relation for the functional derivative with respect to a real scalar field \phi(x) :
i \dfrac{\delta}{\delta \phi(x)} e^{-i \int \mathrm{d}^{4} x \frac{1}{2} \phi(x) ( \square + m^2 ) \phi(x)} = ( \square + m^2 ) \phi(x)...
In the space of Riemannian metrics Riem(M), over a compact 3-manifold without boundary M, we have a pointwise (which means here "for each metric g") inner product, defined, for metric velocities k^1_{ab},k^2_{cd} (which are just symmetric two-covariant tensors over M)...
In the space of Riemannian metrics Riem(M), over a compact 3-manifold without boundary M, we have a pointwise (which means here "for each metric g") inner product, defined, for metric velocities k^1_{ab},k^2_{cd} (which are just symmetric two-covariant tensors over M)...
This is supposedly the chain rule with functional derivative:
\frac{\delta F}{\delta\psi(x)} = \int dy\; \frac{\delta F}{\delta\phi(y)}\frac{\delta\phi(y)}{\delta\psi(x)}
I have difficulty understanding what everything in this identity means. The functional derivative is usually a derivative...
Hmm, I've been working with functional derivatives lately, and some things aren't particularly clear.
I took the definition Wikipedia gives, but since I know little of distribution theory I don't fully get it all (I just read the bracket thing as a function inner product :)).
Anyway, I tried...