Derivation Definition and 1000 Threads

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. N

    Lens Maker's Formula - Confusion in the derivation

    I was looking at the derivation of the lens maker's formula and I have a minor confusion which does not seem to go away. So it is derived from the same principle for the refraction of a light ray at a spherical convex/concave surface, except that it undergoes refraction twice due to two...
  2. R

    Deriving Maxwell's equation from Poynting Theorem

    Here is the solution: From vector identity, $$\nabla (\vec A \times \vec B) = \vec B \cdot \nabla \times A - \vec A \cdot \nabla \times \vec B $$ If ##\vec B = 0##, then $$\nabla \vec S = \nabla \frac{\vec E \times \vec B}{\mu_0} = \frac{1}{\mu_0} (\nabla \times E \cdot \vec B - \cdot E \cdot...
  3. smjchris

    Derivation process of Selection Rule of hydrogenic atom

    This page is Quantum mechanics by bransden. My homework is explain why there is no regulation of quantum number n in selection rule. Also explain that by solving that integral of radial part is always non-zero. ∫∞0[rRnl(r)]Rn′l′(r)r2dr (n is different with n') I tried to solve it by just...
  4. aliinuur

    I Where is the mistake in this derivation of Length Contraction?

    I tried to use square box with two perpendicularly oscillating photons to derive length contraction formula. Photons are emitted, reflected from box walls and absorbed simultaneously from the box's point of view. If the box have some velocity v, then assuming photons are still emitted and...
  5. L

    Issue deriving the third law of thermodynamics

    Okay, so we have that $$dU = \left( \frac{\partial U}{\partial V} \right)_S dV + \left( \frac{\partial U}{\partial S} \right)_V dS$$ And comparing that to the first law, we get that $$T=\left(\frac{\partial U}{\partial S}\right)_V$$. Comparing expressions of ##T##...
  6. domephilis

    Rod with Current on Rails in Magnetic Field

    I can't see what is the problem with my derivation but the answer is incorrect. Please help. We assume here that ##\omega_0 = 0## and ##v_0 = 0##, hence it immediately rolls without slipping without any transitional phase. Hence ##v = \omega R##. Thus, ##v(L) = R\omega(L)##. Since our...
  7. K

    I Is this derivation of MOND from General Relativity plausible?

    for over a century the weak field limit of general relativity is Newtonian, which when applied to galaxy rotation curves requires cold dark matter as Newtonian limit is too weak this recent paper arXiv:2408.00358 (gr-qc) [Submitted on 1 Aug 2024] Quasilocal Newtonian limit of general...
  8. Z

    Derive surface brightness formula

    Derive formula for surface brightness The formula in question we wanna derive is: S(mag/arcsec^2) = M⊙ + 21.572 -2.5 log_{10} S(L⊙/pc^2). Best regards.
  9. P

    Correct Derivation of the Adiabatic Condition? (PV Diagram)

    By the First Law, Definition of an Adiabatic Process, and Definition of Work: ##\Delta E = Q - W = - W = - P \Delta V ## (because ##Q = 0##) (Equation 1) By the Equipartition Theorem: ##\Delta E = \frac{3}{2} Nk \Delta T ## (Equation 2) By Setting Equation 1 equal to Equation 2 ## \Delta T...
  10. F

    Deriving a formula for KE (rolling + projection)

    I'm not sure where the equation E_k=(gmR^2)/4h comes from & I also don't really know where to start either :(
  11. B

    I Derivation of Fluid Pressure In A Gravitational Field

    Given that ##P = ρgh##, there's obviously a problem with the following derivation of fluid pressure under gravity. Can someone spot the flaw? $$W = mgh$$ $$W = ρVgh$$ $$F \cdot dh = ρVgh$$ $$F \cdot dh = ρ(Ah)gh$$ $$F \cdot dh = ρgAh^{2}$$ $$\frac{d(F \cdot dh)}{dh} = \frac{d(ρgAh^{2})}{dh}$$...
  12. D

    B An exact Paraxial equation derivation, 100% Cartesian

  13. H

    I Deriving the relativistic rocket equation with exhaust efficiency

    Hi, I'm looking at this relativistic rocket equation on Wikipedia. Something doesn't make sense here, and I can't find a derivation for this equation in the linked source, so I'm trying to derive it myself with limited success...
  14. L

    Why Is Deriving the Motion Formula for Hoops More Complex?

    I've worked out how to derive the formulas for a solid cylinder and a solid sphere rolling down a hill. E.g., for a cylinder: Emech = KE + PE mgh = 1/2 mv^2 + 1/2 Iw^2 gh = 1/2 v^2 + 1/2 (1/2r^2) v^2/r^2 gh = 3/4 v^2 v^2 = 4/3 gh I then performed a derivative with respect to time and found a...
  15. L

    Coulomb's law and energy - potential energy

    I know that the formula qqk/r applies to a system (two charges), but where is the flaw in my derivation? Thanks!
  16. H

    A Time-ordered products derivation in "QFT and the SM" by Schwartz

    This question is not crucial, but I'd like to understand better the equation (14.35) in this derivation: Here ##\Phi## is an eigenvalue of ##\hat \phi##, i.e., ##\hat \phi (\vec x ) |\Phi \rangle = \Phi (\vec x) |\Phi \rangle##. First, I think that there is a typo in (14.35): the Hamiltonian...
  17. 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...
  18. 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...
  19. E

    How Can Derivatives Determine Speed in Physics Problems?

    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...
  20. 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.
  21. F

    B How Do You Derive the 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: $$sin^2(x-y)=1-cos^2(x-y)$$ $$sin^2(x-y)=1-(cos(x)cos(y)+sin(x)sin(y))^2$$...
  22. 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...
  23. 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...
  24. 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...
  25. 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...
  26. 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...
  27. 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...
  28. Safinaz

    How Does the Dirac Delta Function Identity Apply in Equation (27) Derivation?

    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...
  29. 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?
  30. 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...
  31. 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...
  32. 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...
  33. 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...
  34. L

    Why Is Work Positive When Done Against the Electric Field?

    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...
  35. 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...
  36. M

    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##...
  37. 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...
  38. 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...
  39. 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...
  40. 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)...
  41. M

    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} =...
  42. M

    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!
  43. P

    What does this expression involving Partial Derivatives mean?

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

    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!
  45. M

    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...
  46. J

    I What is the derivation of the Fermi golden rule?

    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 =...
  47. M

    I Absolute value bars in dot product derivation

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

    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!
  49. 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.
  50. 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...
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