What is Variation: Definition and 574 Discussions

In music, variation is a formal technique where material is repeated in an altered form. The changes may involve melody, rhythm, harmony, counterpoint, timbre, orchestration or any combination of these.

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  1. Demystifier

    I A variation of the sleeping beauty problem

    Some time ago we had a discussion of the sleeping beauty problem https://www.physicsforums.com/threads/the-sleeping-beauty-problem-any-halfers-here.916459/ which is a well known problem in probability theory. In that thread, there was no consensus whether the probability of heads is 1/2 or 1/3...
  2. C

    I Variation of matter action under diffeomorphism (Carroll)

    Queries on Carroll's derivation of matter action ## S_M ## under a diffeomorphism: (Book B.23+4 Notes 5.35+6) ##\frac{\delta S_{M}}{\delta g_{\mu\nu}} \delta g_{\mu\nu} = \frac{\delta S_{M}} {\delta g_{\mu\nu}} \left( 2 \nabla _{(\mu}V_{\nu)} \right) =\left( 2 \right) \frac{\delta...
  3. C

    I Variation of Monty Hall problem

    There are three envelopes. One contains $50. Another contains $100. The third contains $500. You pick one but it is not opened. If the host reveals one of the other two and it contains $50 or $100 (both considered as goats), the answer is to switch as in Monty hall case. However, if he reveals...
  4. RChristenk

    Prove this variation: If a+b ∝ a-b, prove that a^2+b^2 ∝ ab

    ##a+b \varpropto a-b## means ##a+b = k_1(a-b)## ##(a+b)^2=k_1^2(a-b)^2## ##a^2+b^2+2ab=a^2k_1^2+b^2k_1^2-2abk_1^2## ##2ab(1+k_1^2)=a^2(k_1^2-1)+b^2(k_1^2-1)## ##2ab(k_1^2+1)=(a^2+b^2)(k_1^2-1)## ##a^2+b^2=ab(\dfrac{2k_1^2+2}{k_1^2-1})## Since ##a^2+b^2 \varpropto ab## means...
  5. A

    A Standard error of the coefficient of variation

    What is the standard error of the coefficient of variation in an exponential distribution?
  6. Lay1

    Voltage Variation w/ Variable Resistor: Correct?

    In this figure, I suppose the maximum voltage is when R2=1kohm and the minimum voltage is when R2=0kohm, which means R2 is a variable resistor. Is the way I think is correct or not? Please give me suggestions. Thank you.
  7. snoopies622

    I How to add time variation to a Schrodinger operator?

    I'm looking at Dirac's "Lectures on Quantum Field Theory" and I have a question about the basic mathematics of something that's part of ordinary quantum mechanics. On page 3, he says, The two pictures are connected in this way: any Schrodinger dynamical variable is connected with the...
  8. Conn_coord

    I Measurement accuracy or variation

    I find the values of the fundamental constants here https://physics.nist.gov/cuu/Constants/ In anticipation of the data of 2022, I want to clarify one point. Planck's constant is related by an exact expression to some other fundamental quantities: $$h = \frac{\alpha^2 m_e с^2}{2R_c}\qquad (1)$$...
  9. N

    I What is the expected (squared) coefficient of variation of a sample?

    I've been trying to figure this out, but am not getting anywhere and was hoping someone here might know. Say I have a distribution of which I know the variance and mean. I then take samples of n random variables from this distribution. Without knowing anything more about the distribution, can...
  10. P

    A Variation of Energy for Dielectrics (Zangwill's Electrodynamics)

    Hello PhysicsForums community, I have been reading through Zangwill's Modern Electrodynamics all on my own, and I've just joined here hoping I can post some questions that come up for me. To start, I am confused about something in section 6.7.1, concerning the variation of total energy U of a...
  11. Bernadette

    B A variation on the Fresnel Spot experiment

    Hello Fresnel discovered that the shadow of a small disc had a bright point in its center. Experiment that he carried out by letting sunlight pass through a small hole (quasi-point source). If, instead of looking at the shadow of the disk, we look (*) in the direction of the light source...
  12. C

    A Electron EM Field as Local Gauge Variation of Electron Matter Field

    The EM field seems to be required for for local gauge symmetry of the electron matter field under local phase variation. Following is a description (not my verbiage): There is a symmetry in physics which we might call the Local Phase Symmetry in quantum mechanics. In this symmetry we change...
  13. U

    Help with variation of the 3-dimensional ##\sigma##-model action

    Consider the following action $$S=\int\mathrm{d}^3x\sqrt{h}\left[R^{(3)}-\frac{1}{4}\mathrm{Tr}\left(\chi^{-1}\chi_{,i}\chi^{-1}\chi^{,i}\right)\right]$$ where ##h## is the determinant of the 3-dimensional metric tensor ##h_{ij}## and ##R## is the Ricci scalar. I want to get the equations of...
  14. T

    A Stationary Point of Variation of Action

    The action is the time integral of the Lagrangian - check. We take the variation of it, and set it to zero - check But how do we know it is a minimum and not a maximum? (Don't we formally have to deal with the second variation to be positive?) Or am I confused? Is it really just important to...
  15. Hubble_92

    I Variation of Four-Velocity Vector w/ Respect to Metric Tensor

    Hi everyone! I'm having some difficulty showing that the variation of the four-velocity, Uμ=dxμ/dτ with respect the metric tensor gαβ is δUμ=1/2 UμδgαβUαUβ Does anyone have any suggestion? Cheers, Rafael. PD: Thanks in advances for your answers; this is my first post! I think ill be...
  16. e2m2a

    A Variation of Catalan Conjecture

    I know that it has been proven that for the expression x^a -y^b = 1, only has this one integer solution, where x = 3, a =2, y =2, b = 3. I am interested in knowing if there is a proof for this expression: 2x^a - y^a =1 in which there are integer solutions for x,a, and y or if no integer...
  17. Shreya

    Variation of Electric Field at the centre of Spherical Shell

    My approach is thus: the shell will have induced charges if it's conducting resulting in E at the centre of shell(though flux at centre will be 0). For non conducting spheres there can be no induction only polarization of dipoles, therefore the E field at centre will remain 0. Is my approach...
  18. M

    Coupon Collection Problem Variation

    Hi, I was attempting the following problem but didn't quite understand the steps in the solution. The problem reminded me of the coupon problem, but the probabilities of the 'coupons' are different. Question: Call a “consecutive difference” the absolute value of the difference between two...
  19. Poetria

    Multivariable temperature variation while swimming in a hot spring

    I have computed ##T_x## and ##T_y## and evaluated it at the point (20, 20). ## \frac {-450*x}{x^2 + y^2 + 1)^(\frac {3} {2}} - \frac {420*(x + 10)} {(x + 10)^2 + (y - 20)^2 + 1)^(\frac {3} {2}}, \frac {-450*y}{x^2 + y^2 + 1)^(\frac {3} {2}} - \frac {420*(y - 20)} {(x + 10)^2 + (y - 20)^2 +...
  20. R

    Variation of the Speed of Sound in metals under tension

    In a block of metal, each metal has a characteristic speed of sound. When metal is under tension, such as a guitar string, the speed rises as the tension increases. How does the speed vary (in a block say) as a function of tension along each of the three axes? I am assuming that transverse...
  21. expn

    Exploring the Variation in Movement Capacity of Individuals

    Hello to all forum members, I came here in an attempt to find a solution to a problem that has been "haunting" me for some years. I have been trying for some time to understand mathematically the variation in the movement capacity of individuals, based on their effort. In other words, I would...
  22. Pouramat

    Variation principle -- looking for resources to read and understand

    Summary:: Can anyone introduce an informative resource with solved examples for learning variation principle? For example I cannot do the variation for the electromagnetic lagrangian when ##A_\mu J^\mu## added to the free lagrangian and also some other terms which are possible: $$ L =...
  23. duchuy

    Wave propation -- Speed variation in different indices

    Hi, I have this question about the variation of wavelength and frequency as light travels to an environment with a different index. As we have learned in class, celerity can change as light enters a different environment, however frequency and wavelenght are independent and remain constant...
  24. brotherbobby

    B Variation of air density with height

    Using the ideal gas equation ##PV = nRT\Rightarrow PV = \frac{m}{M} RT## where ##m,M## are the mass and molecular weights of the gas respectively. This yields ##\frac{m}{V} = \frac{PM}{RT} = \rho##, the density of the gas at a point with pressure ##P##. If only we can obtain the variation of...
  25. Addez123

    B Poisson distribution having variation coefficient = .5?

    Variation coefficient is calculated by And the very definition of poisson distribution is that $$\mu = \sigma $$ So how would any other value but 1 be a possible?
  26. E

    B A bit confused about this variation

    I came across a derivation of the Noether theorem with a step I don't understand; the transformation of the time and configuration space are written as$$\tau(t,\varepsilon) = t + \varepsilon \delta t, \quad Q^a(q,\varepsilon) = q^a + \varepsilon \delta q^a$$here ##\varepsilon## is an...
  27. I

    B Twin Paradox: Tom & Jerry Agree on Clock Time But Who is Older?

    I know this has been asked so many times, but would someone please answer why to this particular variation to the question. Tom is moving at constant velocity past Jerry on Earth (assume no acceleration). At the moment they pass each other they agree on the time seen on a clock. Tom thinks...
  28. J

    I Variation of Ricci scalar wrt derivative of metric

    I understand from the wiki entry on the Einstein-Hilbert action that: $$\frac{\delta R}{\delta g^{\mu\nu}}=R_{\mu\nu}$$ What is the following? $$\frac{\delta R}{\delta(\partial_\lambda g^{\mu\nu})}$$ Is there a place I could look up such GR expressions on the internet? Thanks
  29. aspodkfpo

    Variation of a graph requiring a proportionality to hold

    Do not understand the statement: Ben’s method requires that the voltage output be directly proportional to the intensity, which it is not. https://www.asi.edu.au/wp-content/uploads/2015/03/2010_Physiscs_solutions.pdf My thoughts are that by I= Io cos(theta)^2 We can relate voltage to theta...
  30. T

    Orbital speed variation as a planet orbits the Sun

    Summary:: At what distance from the Sun will the speed of the planet be equal to the average orbital speed? I'm not sure where to place this question, please move it in the right thread. [Mentor Note -- thread moved from the technical forums, so no Homework Template is shown] At what...
  31. A

    A Variation coefficient property

    For a random variable Ti, SD (Ti) / E (Ti) ≤ 1 with SD (Ti) = (Var (Ti))1/2 and E (Ti) the expectation of Ti and Var (Ti) the variance of Ti. My question now is whether the following property then also applies. For any variable T, SD (T) / E (T) ≤ 1 where T = T1 + T2 + ... + TN and where...
  32. M

    Weird condition describing symmetry transformation

    I'm a bit confused about the condition given in the description of the symmetry transformation of the filed. Usually, given any symmetry transformation ##x^\mu \mapsto \bar{x}^\mu##, we require $$\bar\phi (\bar x) = \phi(x),$$ i.e. we want the transformed field at the transformed coordinates to...
  33. mcastillo356

    Why does current need to vary in an inductor?

    In a power transformer, the variation of the current in one circuit induces a current in the other. This cause-effect relationship is two ways. If circuit A affects circuit B, then B circuit will affect A circuit; this is the cause of the concept of mutual inductance. However, it's not necessary...
  34. thaiqi

    I Variation sign and integral sign

    Hello, everyone. I know that it is feasible to exchange the order of one variation sign and one integral sign. But there gives a proof of this in one book. I wonder about a step in it. As below marked in the red rectangle: How can ##\delta y## and ##\delta y^\prime## be moved into the integral...
  35. alexm

    Motion of rotating rig, find the angle variation with control rod length

    Summary:: We have a rotating arm, offset from the centre of rotation by a certain length, which is controlled by varying the length of a control rod. Need the angle of the rotating arm in terms of length of the rod. The blue line is a fixed column structure. CE and BD form the rotational...
  36. T

    The variation of the information content of a large Einstein solid

    For ##q >> N ##. ##\Omega \approx \left( \frac{eq}{N} \right)^N \text{ } (2)## (Schroeder, An introduction to thermal physics (2.21)). Can we argue that: ##\Delta I = - \Delta S \text{ } (3)?## How large can ##\Delta N##, be? Thank you for your time.
  37. A

    I Quadrupole Moment Time Variation: Does Coordinate Choice Matter?

    [Moderator's note: Thread spun off from previous discussion due to topic change.] Does the observed quadrapole moment change over time when considering a relatively moving object, for certain choices of observer coordinates? My suspicion is that it does (Terrell-Penrose rotation implies...
  38. Q

    A Variation of Metric Tensor Under Coord Transf | 65 chars

    Under the coordinate transformation $\bar x=x+\varepsilon$, the variation of the metric $g^{\mu\nu}$ is: $$ \delta g^{\mu\nu}(x)=\bar g^{\mu\nu}(x)-g^{\mu\nu}(x)=-\frac{\partial{ g^{\mu\nu}}}{\partial x^{\alpha}}\varepsilon^{\alpha}+ g^{\mu\beta}\frac{\partial \varepsilon^{\nu}}{\partial...
  39. E

    What is the optimal power variation for a cyclist on a varied race course?

    I just thought up a little problem and wondered whether anyone could advise as to how to go about it! On a flat course, suppose a cyclist might be able to maintain 300W for around an hour. This gives a total allowed energy expenditure of ##1080t \text{ kJ}##, where ##t## is measured in hours...
  40. Chris Miller

    B Exploring Variations of the Collatz Conjecture: A Computational Approach

    The Collatz problem is perhaps the only unsolved math problem I actually understand. It "feels" like a proof would be trivial, though obviously it isn't. Been playing with different variations in hopes of understanding it better. Is it a set problem (proving there's no intersection between two...
  41. Pispi Choudhury

    Calculus Book suggestions and good lecture notes on the calculus of variation

    I need suggestions on books and good lecture notes on calculus of variation. I've previously studied vector calculus and multivariable calculus.
  42. Sonim

    Can Calorimeters Measure Enthalpy Variation of Fast Reactions?

    First of all this is a question that I had while reading some concepts of my book, so this isn't a homework question. I have started reading the thermochemistry chapter of my book, and it shows the story of Hess' Law and says that it was created is because a calorimeter can't be used to measure...
  43. Buzz Bloom

    I A variation of the Multiple World Interpretation

    The multiple worlds are not real, but instead they are contingent. This mean that each of the various possible combinations of all of the possible future measurements or relevant interactions that change the probabilities, that could possibly happen according the wave function of everything...
  44. balaustrada

    How to compute the variation of two covariant derivatives?

    I'm working with modfied gravity models and I need to consider the perturbation of field equations. I have problems with the term were I have two covariant derivatives, I'm not sure if I'm doing it right. I have: $$\delta(\nabla_\rho \nabla_\nu \left[F'(G)R_{\mu}^{\hphantom{\mu} \rho}\right])$$...
  45. R

    Sampling distribution's magnitude variation

    I have generated a distribution (normalized such that the sum is equal to 1) by using the code: M=500; % Number of samples z=1; SUM = 1; ns = rand(1,M).^z; % random numbers TOT = sum(ns); X = (ns/TOT)*SUM; % Re-scaling hist(X(1,:),100) For an exponent ##z=1##, the sampling distribution is...
  46. Physics lover

    Variation of electric field and potential along the axis of a cone

    Options are at the top of page as a) b) c) d) Answer may more than one. Now since 'a' is distance from the smaller surface of cone so as we move along the axis area will increase,So current charge density will decrease and as we know J=sigma E,E will decrease,but V will remain constant since...
  47. A

    I Variation of geometrical quantities under infinitesimal deformation

    This question is about 2-d surfaces embedded inR3It's easy to find information on how the metric tensor changes when $$x_{\mu}\rightarrow x_{\mu}+\varepsilon\xi(x)$$ So, what about the variation of the second fundamental form, the Gauss and the mean curvature? how they change? I found some...
  48. Arman777

    Proof that Variation of Integral is Equal to Integral of the Variation

    I actually don't know how to proceed. I tried something like this The left side of the equation equals to $$\delta(\int_a^b F(x)dx)=\delta f(x) |_{a}^{b}$$ where ##f'(x)=F(x)## However $$\delta f(x) |_{a}^{b}=f'(x)\delta x dx|_{a}^{b} = \delta (F(b)-F(a))$$ where ##f'(x)=F(x)##. For the...
  49. velvetmist

    Real life elastic collision and variation of kinetic energy

    How small should ##\Delta T## be in a collision to be considered elastic? In elastic collisions ##\Delta T =0##, but as far as I know, just atomic collisions are considered perfectly elastic. Then, which criterias are used to considere a collision between two objects elastic?
  50. M

    Calculating different "kinds" of variations

    Homework Statement Let ##x## and ##x'## be two points from the Minkowski space connected through a Poincare transformation such that ##x'^\mu =\Lambda_{\nu}^\mu x^\nu+a^\mu## and ##u:\mathcal{M}\to \mathbb{K}=\mathbb{R}## or ##\mathbb{C}##, ##\mathcal{M}## the Minkowski space. We define: $$...