What is Derivation: Definition and 999 Discussions

In cryptography, a key derivation function (KDF) is a cryptographic hash function that derives one or more secret keys from a secret value such as a main key, a password, or a passphrase using a pseudorandom function. KDFs can be used to stretch keys into longer keys or to obtain keys of a required format, such as converting a group element that is the result of a Diffie–Hellman key exchange into a symmetric key for use with AES. Keyed cryptographic hash functions are popular examples of pseudorandom functions used for key derivation.

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  1. H

    A Derivation of energy-momentum tensor in "QFT and the SM" by Schwartz

    My question is about this step in the derivation: When the ##\partial_\nu \mathcal L## in 3.33 moves under the ##\partial_\mu## in 3.34 and gets contracted, I'd expect it to become ##\delta_{\mu \nu} \mathcal L##. Why is it rather ##g_{\mu \nu} \mathcal L## in the 3.34? (In this text, ##g_{\mu...
  2. H

    A Derivation of QM limit of QFT in "QFT and the SM" by Schwartz

    In this derivation, a basis of one-particle states ##\langle x|=\langle \vec x,t|## is expressed with the field operator, $$\langle x|=\langle 0| \phi (\vec x, t)$$ "Then, a Schrodinger picture wavefunction is $$\psi (x)=\langle x| \psi \rangle$$ which satisfies $$i \partial _t \psi (x) = i...
  3. E

    Determine speed - derivatives

    I have been trying to solve this problem for hours using the mentioned equations but no matter what I do I cannot get the correct answer, that is v = 22.4 m/s. I thought that maybe if I could get an expression where v is a function of time I could solve the problem but I don't know how to do...
  4. Shauryafrom2006

    B I need help understanding the derivation of this 'Absolute Scale of Temperature'

    It is from [ Class 11th] SL Arora, pg no. 11.3, the heading is [11.7] Absolute Scale of Temperature.
  5. F

    B Deriving formula for sin(x-y)

    I was trying to show that ##sin(x-y) = sin(x)cos(y)-cos(x)sin(y)## using Pythagoras' theorem and ##cos(x-y)=cos(x)cos(y)+sin(x)sin(y)##. I have: $$sin^2(x-y)=1-cos^2(x-y)$$ $$sin^2(x-y)=1-(cos(x)cos(y)+sin(x)sin(y))^2$$...
  6. B

    Derivation for modified sine curve equations

    Hello, I am looking for a detailed derivation of the equations used to generate the modified sine curve. I found one in Cam design handbook by Harold A. Rothbart but I didn't understand how we get certain equations. My end goal is to combine the modified sine curve with constant velocity and get...
  7. PhysicsRock

    I Derivation of normal Zeeman-Effect

    I was / am trying to derive the energy shift resulting from the normal Zeeman-Effect by coupling the Hamiltonian to the external field ##\vec{A}##, that carries the information about the field ##\vec{B}## via ##\vec{B} = \nabla \times \vec{A}##. Let ##q = -e## be the charge of the electron and...
  8. L

    I Maxwell's equations in the presence of matter -- Derivation

    I want to calculate ##\int \vec{P}\left(\overrightarrow{r^{\prime}}\right) \cdot \vec{\nabla}_{\overrightarrow{r^{\prime}}} \frac{1}{\left|\vec{r}-\overrightarrow{r^{\prime}}\right|} d^{3} \overrightarrow{r^{\prime}}## with macroscopic polarization...
  9. printereater

    Formula derivation connecting vertical water flowrate & horizontal distance moved by a suspended sphere

    TL;DR Summary: I am Highschool student writing a 4000 word research paper on Bernoulli's principle and the coanda effect. I need help with derivation of a formula that connects flow rate of water and distance moved by the sphere in my experiment. I am a high school student writing a 4000 word...
  10. M

    A Question on Zeno time derivation

    Hi, I'm trying to follow the derivation of the Zeno time from two sources and am struggling. I think I'm missing some sort of algebraic trick and any tips would be appreciated. A bit more detail below. In the attached paper \citep{Facchi_2008}, the Zeno time (equation (6)) is derived from...
  11. A

    Derivation Of Torque On Current Loop Due To Uniform Magnetic Field

    I can derive it for a circular loop: $$dF=BI\sin\phi\ dl=BIr\sin\phi\ d\phi$$ Torque on quarter circle when field is parallel to plane of loop=$$\tau=\int^{(\pi/2)}_0 BI \ dl \sin\phi (r\sin\phi)$$$$=\int^{(\pi/2)}_0 BIr^2 \sin^2\phi\ d\phi$$ Net torque=##4\tau=BIA## If magnetic field is at any...
  12. Safinaz

    Dirac Delta Function identity

    I need help to understand how equation (27) in this paper has been derived. The definition of P(k) (I discarded in the question ##\eta## or the integration with respect for it) is given by (26) and the definition of h(k) and G(k) are given by Eq. (25) and Eq. (24) respectively. In my...
  13. J

    How to simplify \nabla (A.v) in the derivation of Lorentz force?

    I know that ##∇(A⋅v)=(A⋅∇)⋅v+(v⋅∇)⋅A+v×(∇×A)+A×(∇×v)## The third term ##v×(∇×A)## simplifies to ##v×B##. I'm just now sure how to "get rid" of the other terms. I tried checking for some vector identities but couldn't make any headways. Any guidance?
  14. M

    A Derivation of Generalized Khon-Sham Scheme

    Hello, I am following the paper: https://www.yumpu.com/en/document/read/42212557/exact-exchange-in-density-functional-calculations and I am confused on page 14 where the generalized Kohn-Sham equations are derived. I follow that the ground state energy is The minimization of this step leads...
  15. D

    Derivation of time period for physical pendula without calculus

    TL;DR Summary: I'm stuck trying to find the equation for time period T of a physical pendulum without any calculus using torque. Hello all. I am currently writing my IB Physics HL IA (high school physics lab report). I am investigating the effect of length on the time period of a uniform rod...
  16. M

    How to derive the associated Euler-Lagrange equation?

    a) We have ## S[y+\epsilon h]=\int_{1}^{2}[3(y'+\epsilon h')^2-2(y+\epsilon h)^2]dx ##. Note that ## \frac{d}{d\epsilon}S[y+\epsilon h]=\int_{1}^{2}[6(y'+\epsilon h')h'-4(y+\epsilon h)h]dx=2\int_{1}^{2}[3(y'+\epsilon h')h'-2(y+\epsilon h)]dx ##. Then ## \bigtriangleup S[y...
  17. B

    How do we derive the Earth's yearly mean temperature?

    I'm new to this. I would like to know how we derive the earth's yearly mean temp. I assume there are thermometers all over...
  18. L

    Several questions relating to work, electric potential energy, and potential difference

    I have several questions relating to electrostatics: first of all, in this derivation for the formula of the electric potential energy: work is being done against the electric field right, so the work should be negative, but in this case it's positive. I'm wondering if it's because the direction...
  19. TheMisterOdd

    How to derive the period of non-circular orbits?

    By conservation of mechanical energy: $$ E(r_0)=-\frac{GMm}{r_0}+\frac{1}{2}\mu \left (\dot{r_0}^2+r_0^2 \omega_0^2 \right) $$ where R0 =Rmax. Because our body is located at the apoapsis the radial velocity is 0. Hence: $$ E(r_0)=-\frac{GMm}{r_0}+\frac{1}{2}\mu (r_0\omega_0)^2 $$ By the...
  20. C

    Position vector for anti-clockwise circular motion derivation

    To derive ##\vec r (t)=(−Rsin(ωt),Rcos(ωt)) ## I start by integrating ##ω=\frac{dθ}{dt}## to get ##θ_f=θ_i+ωt##. Therefore since ##Δθ=θ## by definition since the angular displacement is always taken with respect to some initial reference line, then ##θ_f−θ_i=θ## , thus, ##\theta = \omega t##...
  21. Immer Tzaddi

    I Lorentz Transformation: Premises + Derivation

    Recollections of a late Spring semester's lesson describing the derivation of Lorentz's Transformation often solicit many unanswered questions. The textbook used has been secured; however, it is unknown. Whether, that secondary school instructor provided the premises used for the derivation from...
  22. S

    Derivation of uncertainty formula

    Let say we have two quantities, A and B. A = a ± Δa and B = b ± Δb, where a and b are the value of A and B and Δa and Δb are their absolute uncertainty respectively. Now we have a formula of C, where C = A + B. The absolute uncertainty is Δc = Δa + Δb. How to derive this formula? Is the...
  23. J

    I Derivation of ideal gas heat capacity relationship

    The text derives C_p-C_v=nR for ideal gasses. They start with $$H = U + PV = U + nRT$$ for ideal gas. Since U is only a function of temperature for an ideal gas, the right-hand side is only a function of temperature so $$\frac{dH}{dT} = \frac{dU}{dT} + nR$$. Now the text does something I...
  24. baby_1

    A Obtaining a variable value from a 5-th degree polynomial in the tangent form

    Hello, Please see this part of the article. I need to obtain the ##\rho (\phi)## value after obtaining the c0 to c5 constants of the ##\sigma (\phi)##. But as you can see after finding the coefficients, solving Eq.(1) could be a demanding job(I wasn't able to calculate the integral of Eq(1)...
  25. C

    Doppler effect derivation for moving observer and stationary source

    For this, Does someone please know where they got that ##f'## is number of waves fronts received per unit time from? Also could we write the equation highlighted as ##f' = \frac{n\lambda}{t}## where ##n## is the number of wavefronts in a time ##t##? I derived that from ##\frac{vt}{\lambda} =...
  26. C

    Deriving density formulae from first principles

    Can someone please help derive the relations below from first principles? Also does someone please know what happens when ##ρ_{object} = p_{fluid}##? Many thanks!
  27. P

    What does this expression involving Partial Derivatives mean?

    I already solved w x x/|x| For (w1,w2,w3) and (x1,x2,x3)
  28. C

    General form of Newton II -- Not understanding this step in the derivation

    For this, Does someone please know how do we derive equation 9.9 from 9.8? Do we take the limits as t approach's zero for both sides? Why not take limit as momentum goes to zero? Many thanks!
  29. C

    Help with derivation of Malus' Law

    I am trying to Derive Malus's Law. My textbook says that an electric field as an amplitude ##E## before passing thought the polarizer and reduce to ##E_{trans} = E\cos\theta##. I am trying to understand why this occurs my considering a vertically polarized light passing though a polarizer that...
  30. J

    I Fermi Golden Rule Derivation

    Dear Forum, I have a question about the derivation of the Fermi golden rule in Kenneth Krane's Introduction to Nuclear Physics. I understand everything up to equation 9.20. However, it is unclear how he goes directly to equation 9.21. Here is equation 9.20, ## d\lambda =...
  31. C

    I Absolute value bars in dot product derivation

    Dose someone please know why they have the absolute value bars in this derivation? many thanks!
  32. C

    I Derivation of Cauchy-Schwarz Inequality

    For this, I don't understand how they got from (1) to (2)? Dose someone please know what binary operation allows for that? I also don't understand how they algebraically got from line (2) to (3). Many thanks!
  33. K

    I Derivation for the indicial exponent in the Frobenius method

    I'm reading a book called Asymptotic Methods and Perturbation Theory, and I came across a derivation that I just couldn't follow. Maybe its simple and I am missing something. Equation 3.3.3b below. y(x) takes the form A(x)*(x-x0)^α and A(x) is expanded in a taylor series.
  34. E

    Solving Problem 2.4 in Ballentine: Nonnegativeness Derivation

    I am trying to solve Problem 2.4 in Ballentine: I note in my attempt below to what (2.6) and (2.7) refer. My attempt thus far is as follows: A ##2 \times 2## state operator can be represented in a particular orthonormal ##\beta = \{\phi_i\}## as below, where we have enforced trace...
  35. simonjech

    I Schwinger-Dyson equations derivation

    This is the part of Schwinger-Dyson equations derivation. I did not understand how can we obtain the commutator in the last line of the picture. I understand why the delta functions appeared from Heaviside functions but there is no minus sign in any term so how can we get the commutator...
  36. M

    I Derivation of two-electron density operator

    Hello, I am going over the derivation for two-electron density. I am having a hard time understanding how the second term in 2.11a seen below is derived. I know this term must eliminate the i=j products but can't seem to understand how. Thanks for the help.
  37. Twigg

    A Lame parameter mu = shear modulus derivation (rogue factor of 2)

    Hello, I am trying and failing to derive that the shear modulus ##G## is equal to the Lame parameter ##\mu##. I start with the linear, symmetric, isotropic stress-strain formula: $$\sigma = \lambda \mathrm{tr}(\epsilon) \mathrm{I} + 2\mu \epsilon$$ I then substitute a simple (symmetric) shear...
  38. F

    Looking for a particular function

    TL;DR Summary: I want to find a function with f'>0, f''<0 and takes the values 2, 2^2, 2^3, 2^4,..., 2^n Hello everyone. A professor explained the St. Petersburgh paradox in class and the concept of utility function U used to explain why someone won't play a betting game with an infinite...
  39. P

    I Derivation of SR's time-dilatation in 1d?

    Hey, I am looking for a derivation of time-dilatation or some trivially equivalent formulas (Lorentz-transformation, conservation of 4-distance (edit: invariance of spacetime interval) etc) in 1 dimension, using that c is observer independent. I only can find the one that uses a light-clock...
  40. C

    Derivation for capacitance of cylindrical capacitor

    I don't understand how they got from the previous step to the next step of the derivation circled in red: Many thanks!
  41. M

    The direction of flux vectors in derivation of conservation of mass

    In the derivation of the conservation law of the conservation of mass, the flux on one side enters and the flux on the other side leaves the control volume. I presume this is due to the assumption that the volume is infinitesimally small and hence v(x,y,z,t) will not change directions...
  42. P

    Lienard-Wiechert Potential derivation, chain rule

    I want to follow the Lienard-Wiechert potential derivation in Robert Wald's E-M book, page 179. I do not understand $$dX(t_\text{ret})/dt$$ on the right side. I assume the chain rule is applied, but I can't see how. $$ \frac{\partial[x'^i - X^i(t - |\mathbf x - \mathbf x'|/c)]}{\partial x'^j} =...
  43. PeterDonis

    A Landau's Maximum Mass Limit Derivation in Shapiro & Teukolsky (1983)

    In Section 3.4 of Shapiro & Teukolsky (1983), a simple derivation, due to Landau, of the maximum mass limit for white dwarfs and neutron stars is given. I will briefly describe it here and then pose my question. The basic method is to derive an expression for the total energy (excluding rest...
  44. J

    I Best Derivation of E=mc^2 - Fdx = V^2 dm

    F= V dm/dt = V (dm/dx)(dx/dt) = V^2 (dm/dx) Fdx = V^2 dm E = m v^2. ,mass flowing with constant velocity If velocity is changing rather than mass, then E = 1/2 m V^2 Ok, joking aside, what is the best derivation of E= mc^2 ?
  45. James1238765

    I Please help with derivation for Dirac spinors

    Could anyone help with some of the later parts of the derivation for Dirac spinors, please? I understand that an arbitrary vector ##\vec v## $$ \begin{bmatrix} x \\ y \\ z \end{bmatrix} $$ can be defined as an equivalent matrix V with the components $$ \begin{bmatrix} z & x - iy \\ x + iy...
  46. C

    I Annoying detail in derivation of Compton scattering

    In Compton's 1923 paper on X-rays scattering from light elements, he presents the following diagram: Here, ## h\nu_0/c ## is the momentum of the incident photon, ## h\nu_\theta/c ## is that of the scattered photon and ## mv/(1-\beta^2)^{1/2} ## is that of the recoiled electron. He uses this to...
  47. Halc

    I Einstein's Derivation of Elapsed Time for Remote Comoving Object

    This is a question on Einstein's 1907 paper first discussing equivalence principle and uniform acceleration. Picture a rigid accelerating object of length £ with a clock at each end. The rear accelerates for time τ (measured by the clock there) at a proper acceleration γ. The clock at the front...
  48. U

    A Question about dilaton monopole interaction derivation

    I am trying to understand how one derives the dilaton monopole interaction. In "Black holes and membranes in higher-dimensional theories with dilaton fields", Gibbons and Maeda mentioned that one could obtain the dilaton monopole interaction as such: where the action is given by However, I...
  49. M

    I Principle of Physics: Derivation of -dU/dx=F

    Hi, Everyone! This is the page(first image) from Principle of physics by resnik. I want to ask the definition of work(##W=F(x) \Delta x##) by variable force here is somewhat different from the usual integral version. I don't understand how is this valid definition? Secondly, how did they reach...
  50. mr_sparxx

    I Kepler's second law derivation from angular momentum conservation

    Many texts state that in an elliptic orbit you can find angular momentum magnitude as $$ L = r m v = m r^2 \frac {d \theta} {dt} $$ I wonder if $$ v = r \frac {d \theta} {dt} $$ is valid at every point. I understand this approximation in a circumference or radius r but what about an arc...