Functional derivative Definition and 27 Threads

  1. S

    A Is the functional derivative a function or a functional

    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...
  2. H

    Is My Understanding of Higher-Order Functional Derivatives Correct?

    (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...
  3. P

    I Definition of functional derivative

    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?
  4. J

    Doubt regarding functional derivative for the Thomas Fermi kinetic energy

    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...
  5. Q

    A Functional Derivatives in Q.F.T.

    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...
  6. E

    A Variational derivative and Euler-Poincare equations

    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...
  7. Q

    I What are the insights into the Total Derivative formula?

    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...
  8. Bishamonten

    Understanding functional derivative

    Homework Statement "The functional ## J[f] = \int [f(y)]^pφ(y)\, dy ## has a functional derivative with respect to ## f(x) ## given by: $$ \frac {δJ[f]} {δf(x)} = \lim_{ε \rightarrow 0} \frac 1 ε \left[ \int[f(y) + εδ(y-x)]^pφ(y)\, dy - \int [f(y)]^pφ(y)\, dy\right] $$ $$ =...
  9. vishal.ng

    A Taylor series expansion of functional

    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...
  10. naima

    B How do we obtain a Taylor expansion of a non-linear functional?

    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...
  11. binbagsss

    Chain rule / Taylor expansion / functional derivative

    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...
  12. E

    What is the solution for the attached equation?

    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.
  13. Ravendark

    Second functional derivative of fermion action

    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...
  14. A

    Functional Derivative: Evaluating & Understanding

    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...
  15. C

    Functional derivative of normal function

    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'}...
  16. D

    Partial or Regular Derivative for Functional Derivative Stationary Value of 0?

    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
  17. E

    Mastering Functional Derivatives in Quantum Field Theory

    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...
  18. D

    What is the proof for the functional derivative formula in DFT?

    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)}...
  19. S

    A simple functional derivative

    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...
  20. S

    What is meant by functional derivative?

    What is meant by functional derivative? Thanks in well advance.
  21. P

    Functional derivative of connection with respect to metric

    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...
  22. A

    Proving Functional Derivative for Current Research - Alice

    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...
  23. P

    Functional Derivative: Computing the d'Alembert Solution

    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)...
  24. J

    Chain rule with functional derivative

    Given that F = \int{f[h(s),s]ds} does \frac{\partial}{\partial h}ln(F)=\frac{1}{F}\frac{\delta F}{\delta h}=\frac{1}{F}\frac{\partial f}{\partial h} ?
  25. H

    Adjoint of functional derivative in superspace

    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)...
  26. J

    Chain rule with functional derivative

    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...
  27. C

    Functional derivative: chain rule

    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...
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