Isotropic Definition and 123 Threads

Isotropy is uniformity in all orientations; it is derived from the Greek isos (ἴσος, "equal") and tropos (τρόπος, "way"). Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix an, hence anisotropy. Anisotropy is also used to describe situations where properties vary systematically, dependent on direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented.

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  1. cianfa72

    I Spatial homogeneity condition for a free particle Lagrangian

    Hi, reading "Mechanics" book by Landau-Lifshitz, they derive from spatial homogeneity that the Lagrangian ##L## of a free particle cannot explicitly depend on spatial coordinates ##q## in an inertial frame. However my point is as follows: suppose to consider the Lagrangian ##L= \frac 1 2...
  2. ergospherical

    Integrals of isotropic tensors, for expansion over spherical harmonics

    Consider an expansion for the density ##\rho(t,\mathbf{x})## of the form$$\rho(t,\mathbf{x}) = \sum_{l=0}^{\infty} a_{i_1 i_2 \dots i_{\mathscr{l}}}(t,r) \hat{x}_{i_1} \hat{x}_{i_2} \dots \hat{x}_{i_{\mathscr{l}}}$$where ##r = |\mathbf{x}|## and ##\hat{x}_i = x_i/r##. Also, ##a_{i_1 i_2 \dots...
  3. cianfa72

    I SR flat Lorentzian manifold and Anderson simultaneity convention

    Hi, I was thinking about the following. From a mathematical point of view, SR assumes the following postulate: spacetime is a flat Lorentzian smooth manifold. From the above and a minimal interpretation (i.e. a minimal set of "rules" to define the correspondence between mathematical objects...
  4. milkism

    Non-linear isotropic dielectric capacitor

    Question: Solution first part: Have I done it right? I don't know how to begin with second part since the dielectric is non-lineair, and most formulas like $$ D=\epsilon E$$ and $$P= \epsilon_0 \xhi_e E$$, only apply for lineair dielectrics. What to do?
  5. P

    A De-Sitter Spacetime: Is it Homogeneous & Isotropic?

    The question is in the title. I believe the answer is yes.
  6. C

    I Is the Isotropic Universe Truly Centerless?

    In an isotropic universe, every observer sees themself as being at the center. But consider 3 observers, A, B, and C who are 5 billion light years apart and all lined-up in a straight line with B at the center. B knows this to be true because A is in one direction and C is in exactly the...
  7. anonymous99

    Why does pressure need to be constant in all directions to maintain equilibrium?

    I don't understand how pressure must be constant in all directions to balance out the force? Arent the forces in each direction independent, so that pressure forces in the x direction and y direction and z direction can all be different to each other, as long as they are balanced in that...
  8. LuccaP4

    I General Form of Fourth Rank Isotropic Tensor: A Scientific Inquiry

    I have this statement: Find the most general form of the fourth rank isotropic tensor. In order to do so: - Perform rotations in ## \pi ## around any of the axes. Note that to maintain isotropy conditions some elements must necessarily be null. - Using rotations in ## \pi / 2 ## analyze the...
  9. S

    A hypothetical isotropic antenna

    At 100m: (a) 0.03315 W/m (b)4166 W Since E is inversely proportional to 1/r^2, then E at 150m is 2.22 V/m. (a) 2.22/377= 0.00654 W/m (b) 4*pi*r^2*Wrad= 1665 W Is this reasoning correct?
  10. A

    I Is it true that isotropic biaxial strain does not lower C2 symmetry?

    Hello, My question is simple. I have read that isotropic biaxial strain does not lower C2 symmetry, but no proof whatsoever was provided. I would like to know if it is actually true and have a solid proof. If someone can provide it, that would be wonderful. But also explaining me how to start...
  11. Shahi

    Unraveling Landau's Mechanics: Why is Space Isotropic?

    Hi I am reading Landau's mechanics So in the first chapter page 5 It reads : since space is isotropic, the lagrangian must also be independent of the direction of v , and is therefore a function only of it's magnitude ... I can't understand why , I think Landau's book has many fans in this...
  12. Diracobama2181

    A Volume Element for Isotropic Harmonic oscillator

    I am currently having trouble deriving the volume element for the first octant of an isotropic 3D harmonic oscillator. I know the answer I should get is $$dV=\frac{1}{2}k^{2}dk$$. What I currently have is $$dxdydz=dV$$ and $$k=x+y+z. But from that point on, I'm stuck. Any hints or reference...
  13. G

    Potential generated by a point charge in a isotropic medium

    Homework Statement When a point charge is positioned at the origin = 0 in an isotropic material, a separation of charge occurs around it, the Coulomb field of the point charge is screened, and the electrostatic potential takes the form \phi(r) = \frac{A}{r} \exp\left( -\frac{r}{\lambda}...
  14. CharlieCW

    2D isotropic quantum harmonic oscillator: polar coordinates

    Homework Statement Find the eigenfunctions and eigenvalues of the isotropic bidimensional harmonic oscillator in polar coordinates. Homework Equations $$H=-\frac{\hbar}{2m}(\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial...
  15. L

    I Shock crossing probability for isotropic particle flux

    Hi there, I am currently trying to understand the theoretical frame work of diffusive shock acceleration. I am having trouble understanding a step in the derivation given by drury 1983 (http://www.oa.uj.edu.pl/user/mio/Ast-Wys-En/Literatura/drury.pdf). In the derivation of eq. 2.47 it is stated...
  16. D

    I Killing vectors in isotropic space-times

    I've been reading up on Killing vectors, and have got on to the topics of homogeneous, isotropic and maximally symmetric space-times. I've read that for an isotropic spacetime, one can construct a set of Killing vector fields ##K^{(i)}##, such that, at some point ##p\in M## (where ##M## is the...
  17. JuanC97

    I SU(2) invariance implies isotropy?

    Hello guys, I've came up with three statements in a discussion with a friend where we were trying to check if we had a clear vision of what isotropy and group invariance would imply in an arbitrary theory of gravity at the level of its matter lagrangian. We got stuck at some point so I came here...
  18. BookWei

    I Spacetime is homogeneous and isotropic

    I read the Special Theory of Relativity in Jackson's textbook, Classical Electrodynamics 3rd edition. Consider the wave front reaches a point ##(x,y,z)## in the frame ##K## at a time t given by the equation, $$c^{2}t^{2}-(x^{2}+y^{2}+z^{2})=0 --- (1)$$ Similarly, in the frame ##K^{'}## the wave...
  19. binbagsss

    I FRW metric derivation: constraints from isotropic and homoge

    I don't understand the reasoning for any of the three constraints imposed. why would ##dtdx^i## terms indicate a preferred direction? what if there was identical terms for each ##x^i## would there still be a specified or preferred direction? (or is it that in this case we could rename ##t## to...
  20. binbagsss

    I Are spherically symmetric and isotropic the same

    If space-time is isotropic does this imply it is spherically symmetric? why doesn't it need to be both isotropic and homogeneous?
  21. F

    What causes pressure and is it isotropic in a moving fluid?

    I'm studying fluid dynamics and we just had a lecture about the momentum equation. We started the lecture by talking about pressure in terms of molecules moving across a hypothetical surface element and carrying their momentum with them (in both directions). There are 2 things confusing me about...
  22. J

    Defining Multi-Linear Isotropic Stress-Strain Curve Ansys WB

    In order to use Solid65 in Ansys workbench for simulating concrete, we shall define the multi-linear isotropic stress-strain curve as well. I have the concrete compression stress-strain data in excel. I would like to ask that how could I get the multi-linear isotropic stress-strain curve in...
  23. davidge

    I Static, Isotropic Metric: Dependence on x & dx

    In Weinberg's book it is said that a Static, Isotropic metric should depend on ##x## and ##dx## only through the "rotational invariants" ##dx^2, x \cdot dx, x^2## and functions of ##r \equiv (x \cdot x)^{1/2}##. It's clear from the definition of ##r## that ##x \cdot dx## and ##x^2## don't...
  24. Q

    Dielectric tensor in an isotropic media

    In the lecture notes http://top.electricalandcomputerengineering.dal.ca/PDFs/Web%20Page%20PDFs/ECED6400%20Lecture%20Notes.pdf at page 15 eq. (2.46) it says that the dielectric tensor in an isotropic media can be represented by: δi j A(k,ω) + ki kj B(k,ω) I understood that in the case of I. M...
  25. binbagsss

    General Relativity, identity isotropic, Ricci tensor

    Homework Statement Attached Homework EquationsThe Attempt at a Solution So the question says 'some point'. So just a single point of space-time to be isotropic is enough for this identity hold? I don't quite understand by what is meant by 'these vectors give preferred directions'. Can...
  26. alan

    Isotropic material fitted by Ornstein-Zernike form

    I have known what Ornstein-Zernike equation is. I try to plug in the form as follow to the isotropic materials: Still, I cannot show the pair correlation function as follow. Can anyone know what I have missed?
  27. Spinnor

    I Harmonically forcing a drum membrane -- are the waves isotropic?

    Suppose I apply a pair of equal and opposite harmonically varying forces perpendicular to an infinite drum membrane. Consider the following forcing functions at two nearby points,(x=0,y=a) and (x=0,y=-a), separated by a distance 2a, F(t,0,a) = Acos(ωt), F(t,0,-a) = -Acos(ωt) Let the forcing...
  28. R

    How does a hollow fiber membrane work to filter water?

    What is the process? And can/cannot it filter out?
  29. Leonardo Machado

    A Non static and isotropic solution for Einstein Field Eq

    Hello dear friends, today's question is: In a non static and spherically simetric solution for Einstein field equation, will i get a non diagonal term on Ricci tensor ? A R[r][/t] term ? I'm getting it, but not sure if it is right. Thanks.
  30. Jonathan Scott

    A Isotropic metric and circumference of sphere

    Schwarzschild coordinates for the Schwarzschild black hole solution become very weird near the event horizon because the radial coordinate is based on the proper circumference of a sphere but that has a minimum at the event horizon. This is easy to see in isotropic coordinates, where the...
  31. Cocoleia

    Find gravitational potent. energy - isotropic distribution

    Homework Statement I am told that the gravitational force of a mass m located inside an isotropic distribution of spherical radius R and total mass M is given by Fg = -GmM(r)/r^2 where r is the distance between m and the center of distribution and M (r) is the mass contained below the distance...
  32. S

    A Cosmological perturbations in homogeneous and isotropic spac

    It is common is cosmology to study density fluctuations in the early universe. However, it is also common to assume that the background space is homogeneous and isotropic and use the FRW metric. I do not see how density fluctuations can be possible in a homogeneous and isotropic space. Can you...
  33. D

    I Confusion about derivation for isotropic fluids

    In Woodhouse's 'General Relativity' he finds an expression for the energy-momentum tensor of an isotropic fluid. If W^a is the rest-velocity of the fluid and \rho is the rest density then the tensor can be written as T^{ab} = \rho W^aW^b - p(g^{ab} -W^aW^b) for a scalar field p. The...
  34. J

    Calculating Turbine Work Using Ammonia - Reality Check

    I'm doing some back of the envelope calculations for the potential of a turbine thermal generator using ammonia as a working fluid. I've never done thermodynamics before so I'm looking for a reality check. Isotropic turbine work done by a unit mass is given as h2 - h1 or simply dh between the...
  35. JulienB

    Equation of motion for isotropic harmonic oscillator

    Homework Statement Hi everybody! I'm a bit stuck in this problem, hopefully someone can help me to make progress there: A mass point ##m## is under the influence of a central force ##\vec{F} = - k \cdot \vec{x}## with ##x > 0##. a) Determine the equation of motion ##r = r(\varphi)## for the...
  36. Jonathan Scott

    A Event horizon vicinity in isotropic coordinates

    The Schwarzschild radial coordinate ##r## is defined in such a way that the proper circumference of a sphere at radial coordinate ##r## is ##2\pi r##. This simplifies some maths but creates some rather odd side-effects, so to get a more physical picture I like to use isotropic coordinates...
  37. W

    The elasticity/stiffness tensor for an isotropic materials

    Hi PF, As you may know, is the the elasticity/stiffness tensor for isotropic and homogeneous materials characterized by two independant material parameters (λ and μ) and is given by the bellow representation. C_{ijkl} = \lambda\delta_{ij}\delta_{kl} + \mu(\delta_{ik}\delta_{jl} +...
  38. W

    Pressure tensor reduces to scalar pressure for isotropic dis

    1. Does anyone know why for an isotropic distribution function, pressure tensor reduces to a scalar pressure? For instance, for a Maxwellian distribution P=A ∫ vx vy exp-(vx2 + vy2 + vz2) dvx dvy dvz is not zero. I think everybody should realize how bogus some of the authors are. Google...
  39. E

    What is the difference between standard and isotropic metrics?

    The metric $$ds^2=-R_1(r)dt^2+R_2(r)dr^2+R_3(r)r^2(d\theta^2+sin^2d\phi^2)$$ when changed to $$ds^2=-R_1(r)dt^2+R_2(r)(dr^2+r^2d\Omega^2)$$ upon setting ##R_2(r)=R_3(r)##, the later metric holds the name of isotropic metric. My question what is the difference between the first and the second...
  40. N

    Is Turbulence at Point A Isotropic? Calculation Help

    Homework Statement Hello everyone, I am having a problem whether or not a turbulence at a specific location (let's say A) is isotropic or not. I have calculated the two root mean square values of velocity fluctuations measured at the point A in a fully developed turbulent pipe flow. the first...
  41. H

    F = (1/4)(n2/n1)2[1-{(n1-n2)/(n1+n2)}2] ?

    Hi，I was wondering if someone could tell me the name of this equation, where does the equation come from? “If light is isotropically generated in a medium then the fraction transmitted to the outside world is given by: F = (1/4)(n2/n1)2[1-{(n1-n2)/(n1+n2)}2]” Thank you so much :)
  42. S

    Degeneracy of a 2-dimensional isotropic Harmonic Oscillator

    Homework Statement The Hamiltonian is given by: H = \frac{1}{2} \sum_{i=1,2}[p_i^2 + q_i^2] We define the following operators: J = \frac{1}{2} (a_1^+ a_1 + a_2^+ a_2) J_1 = \frac{1}{2} (a_2^+ a_1 + a_1^+ a_2) J = \frac{i}{2} (a_2^+ a_1 - a_1^+ a_2) J = \frac{1}{2} (a_1^+ a_1 - a_2^+...
  43. P

    An Explanation of the Effective Area of Isotropic Antenna

    Hey all, I realize a question on this topic has been asked elsewhere, but the links to references they use seem to be dead, so I'll press on! I'm reading some introduction to antenna theory and I've often puzzled on the equation: A_{eff} = \frac{\lambda^2}{4\pi} which relates the effective...
  44. Aafia

    What is isotropic medium and homogeneous medium

    Can anybody give me a simple and easy example to understand it
  45. Ganesh Ujwal

    Why the space is isotropic in the vector particle's decay?

    I come cross one proof the Landau-Yang Theorem, which states that a ##J^P=1^+## particle cannot decay into two photons, in this paper (page 4). The basic idea is, the photon's wavefunction should be symmetric under exchange, however the spin part is anti-symmetric and the space part is...
  46. AdityaDev

    Radiation pressure from light source

    The energy of photon is $$E=\frac{hc}{\lambda}$$ Now if we have an isotropic point light source of power P, Number of photons $$N=\frac{P}{E} = \frac{P \lambda}{hc}$$ Hence one can find the change in momentum and hence the force exerted by a beam or light sources. But let's say we keep an...
  47. G

    Example of a homogeneous, but not isotropic system

    Hi, I have some trouble understanding if linear momentum and angular momentum (and their conservation laws) are completely independent or not. For example, one can calculate the angular momentum of a uniformly moving body with respect to a fixed point in space and show that it is indeed...
  48. M

    Proof? Kronecker delta is the only isotropic second rank tensor

    It is pretty straight forward to prove that the Kronecker delta \delta_{ij} is an isotropic tensor, i.e. rotationally invariant. But how can I show that it is indeed the only isotropic second order tensor? I.e., such that for any isotropic second order tensor T_{ij} we can write T_{ij} =...
  49. carllacan

    Perturbation of a degenerate isotropic 2D harmonic oscillator

    Homework Statement A two-dimensional isotropic harmonic oscillator of mass μ has an energy of 2hω. It experiments a perturbation V = xy. What are its energies and eigenkets to first order? Homework Equations The energy operator / Hamiltonian: H = -h²/2μ(Px² + Py²) + μω(x² + y²) The...
  50. C

    Mechanistic analysis of lambertian and isotropic radiators

    Hello Physics Forums I need a little help wrapping my head around the concept of lambertian emitters as compared to isotropic emitters. As I understand it a lambertian emitter emits less and less photons as the angle of emission gets further and further away from the surface normal. An...
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