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Coordinate system used to represent certain spacetimes
(except in the trivial case of a locally flat manifold), the angular isotropic coordinates do not faithfully represent distances within the nested spheres
Isotropic_coordinates
Uniformity in all orientations
the direction of an isotropic vector is an isotropic line. Isotropic coordinates Isotropic coordinates are coordinates on an isotropic chart for Lorentzian
Isotropy
Exercise in general relativity
directions. Arthur Eddington gave alternative forms in isotropic coordinates. For isotropic spherical coordinates r 1 {\displaystyle r_{1}} , θ {\displaystyle
Derivation of the Schwarzschild solution
Derivation_of_the_Schwarzschild_solution
Method for specifying point positions
coordinates Gaussian polar coordinates Gullstrand–Painlevé coordinates Isotropic coordinates Kruskal–Szekeres coordinates Schwarzschild coordinates Woods
Coordinate_system
Coordinate system for the Schwarzschild geometry
Lemaître coordinates Eddington–Finkelstein coordinates Isotropic coordinates Gullstrand–Painlevé coordinates 't Hooft, Gerard (2019). "The Quantum Black
Kruskal–Szekeres_coordinates
Coordinates suitable for following a free-falling observer of a Schwarzchild black hole
the Painlevé solution. Isotropic coordinates Eddington–Finkelstein coordinates Kruskal–Szekeres coordinates Lemaître coordinates Paul Painlevé, "La mécanique
Gullstrand–Painlevé coordinates
Gullstrand–Painlevé_coordinates
State of balance between external forces on a fluid and internal pressure gradient
structure of a static, spherically symmetric relativistic star in isotropic coordinates: d P d r = − G M ( r ) ρ ( r ) r 2 ( 1 + P ( r ) ρ ( r ) c 2 ) (
Hydrostatic_equilibrium
Solution to the Einstein field equations
and pressure equations of a static and spherically symmetric body of isotropic material) Planck length Luminet, J.-P. (1979-05-01). "Image of a spherical
Schwarzschild_metric
Coordinate system in black hole physics
perfect fluids, isotropic coordinates, another popular chart for static spherically symmetric spacetimes, Gaussian polar coordinates, a less common alternative
Schwarzschild_coordinates
Riemannian manifolds with constant curvature. Isotropic manifolds Isotropic position Isotropic coordinates Keel, William C. (2007). The road to galaxy formation
Isotropic_vector_field
Measurement of ambient electromagnetic field
EMF are obtained using an E-field sensor or H-field sensor which can be isotropic or mono-axial, active or passive. A mono-axial, omnidirectional probe
EMF_measurement
Linear transformation of spacetime coordinates
standard Schwarzschild coordinates, isotropic coordinates, the Eddington-Finkelstein coordinates, and the Kruskal-Szekeres coordinates. Einstein's field equations
Biquaternion Lorentz transformation
Biquaternion_Lorentz_transformation
Approximation method for general relativity in physics
cosmological boundary conditions. Assume a homogeneous isotropic cosmology, with isotropic coordinates in the rest frame of the universe. A complete cosmological
Parameterized post-Newtonian formalism
Parameterized_post-Newtonian_formalism
Tensor that describes the 4D geometry of spacetime
metric, Isotropic coordinates, Lemaître–Tolman metric, Peres metric, Rindler coordinates, Weyl–Lewis–Papapetrou coordinates, Gödel metric. Some
Metric tensor (general relativity)
Metric_tensor_(general_relativity)
Measurement of distance
the expansion. Comoving coordinates assign constant spatial coordinate values to observers who perceive the universe as isotropic. Such observers are called
Comoving_and_proper_distances
Coordinate system
Static spherically symmetric perfect fluids Schwarzschild coordinates Isotropic coordinates Frame fields in general relativity for more about frame fields
Gaussian_polar_coordinates
Type of manifold in differential geometry
{\displaystyle \omega |_{N}} is a symplectic form on N {\displaystyle N} ; isotropic iff ω | N = 0 {\displaystyle \omega |_{N}=0} , equivalently, iff T p N
Symplectic_manifold
Metric based on the exact solution of Einstein's field equations of general relativity
/ˈfriːdmən ləˈmɛtrə ... /) is a metric that describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe that is path-connected
Friedmann–Lemaître–Robertson–Walker metric
Friedmann–Lemaître–Robertson–Walker_metric
Map projection system
are conformal, maps in UTM coordinates do not distort subtended angles or local shapes, and scale distortion is isotropic. Distortion at a specific point
Universal Transverse Mercator coordinate system
Universal_Transverse_Mercator_coordinate_system
Line along which a quadratic form applied to any two points' displacement is zero
surface, and we also call them isotropic lines. In the complex projective plane, points are represented by homogeneous coordinates ( x 1 , x 2 , x 3 ) {\displaystyle
Isotropic_line
Geometric system used in black hole physics
Schwarzschild coordinates Isotropic coordinates, in which light cones are round, and thus useful for studying null dusts. Gaussian polar coordinates, sometimes
Spherically symmetric spacetime
Spherically_symmetric_spacetime
Mathematical space
{\displaystyle Q(u,v)=0,\,\forall \,u,v\in w,} i.e., totally isotropic subspaces. Maximal isotropic Grassmannians with respect to a real or complex scalar product
Grassmannian
Solution of Einstein field equations
Hubble parameter H ( t ) = H 0 {\displaystyle H(t)=H_{0}} . In isotropic coordinates, the McVittie metric is given by d s 2 = − ( 1 − G M 2 c 2 a ( t
McVittie_metric
Physical quantity that expresses internal forces in a continuous material
surface's orientation. This type of stress may be called isotropic normal or just isotropic; if it is compressive, it is called hydrostatic pressure or
Stress_(mechanics)
Equations of motion for viscous fluids
fluid is assumed to be isotropic, as with gases and simple liquids, and consequently C {\textstyle \mathbf {C} } is an isotropic tensor; furthermore, since
Navier–Stokes_equations
Theory used to determine the stresses and deformations in thin plates
used to determine the shear forces in thin Kirchhoff-Love plates. For isotropic plates, these equations lead to Q α = − D ∂ ∂ x α ( ∇ 2 w 0 ) . {\displaystyle
Kirchhoff–Love_plate_theory
Class of exact solutions to the Einstein field equations
Schwarzschild coordinates or isotropic coordinates, 2004: Martin & Visser algorithm, another generating function method which uses Schwarzschild coordinates, 2004:
Static spherically symmetric perfect fluid
Static_spherically_symmetric_perfect_fluid
Geometric transformation
In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale
Scaling_(geometry)
Stress-strain relation in a linear elastic material
requirement that the stress derives from an elastic energy potential. For isotropic materials, the elasticity tensor has just two independent components,
Elasticity_tensor
Empirical study of systems in transformation
Tetrahedral mensuration also involves substituting what Fuller calls the "isotropic vector matrix" (IVM) for the standard XYZ coordinate system, as his principal
Synergetics_(Fuller)
Mathematical model of how solid objects deform
C_{ijkl}=C_{klij}=C_{jikl}=C_{ijlk}} . An elastostatic boundary value problem for an isotropic-homogeneous media is a system of 15 independent equations and equal number
Linear_elasticity
Type of crystal structure
space. Moreover, the diamond crystal as a network in space has a strong isotropic property. Namely, for any two vertices x, y of the crystal net, and for
Diamond_cubic
Coordinate system
equivalently the IVM (isotropic vector matrix) in Synergetics. Therefore, CCP ball centers all have non-negative integer coordinates. If one now calls this
Quadray_coordinates
Polynomial with all terms of degree two
only when all its variables are simultaneously zero; otherwise it is isotropic. Quadratic forms occupy a central place in various branches of mathematics
Quadratic_form
Brane in eleven-dimensional supergravity
solution is given by a metric and three-form gauge field which, in isotropic coordinates, can be written as d s M 2 2 = ( 1 + q r 6 ) − 2 3 d x μ d x ν η
M2-brane
Force needed to pull a spring grows linearly with distance
{k}{m}}}} Isotropic materials are characterized by properties which are independent of direction in space. Physical equations involving isotropic materials
Hooke's_law
Computer vision geometry concept
image coordinates can be transformed by means of an arbitrary 2D homography. This includes 2D translations and rotations as well as scaling (isotropic and
Camera_matrix
Fundamental space of geometry
being isotropic, that is, there is no property of the space that allows distinguishing between two points or two lines. Therefore, a more isotropic definition
Euclidean_space
Maximally symmetric Lorentzian manifold with a negative cosmological constant
symmetric, it is also possible to cast it in a spatially homogeneous and isotropic form like FRW spacetimes (see Friedmann–Lemaître–Robertson–Walker metric)
Anti-de_Sitter_space
Component of mechanical stress without shear
In continuum mechanics, hydrostatic stress, also known as isotropic stress or volumetric stress, is a component of stress which contains uniaxial stresses
Hydrostatic_stress
Matrix representing a Euclidean rotation
by a certain angle to bring the properties of the laminate closer to isotropic. We sometimes need to generate a uniformly distributed random rotation
Rotation_matrix
Geological concept
A transversely isotropic (also known as polar anisotropic) material is one with physical properties that are symmetric about an axis that is normal to
Transverse_isotropy
Partial differential equation describing the evolution of temperature in a region
phenomena, this equation describes the flow of heat in a homogeneous and isotropic medium, with u ( x , y , z , t ) {\displaystyle u(x,y,z,t)} being the
Heat_equation
Subspace defined by a polynomial of degree 2 over a field
algebraically closed field is rational. A quadric over a field k is called isotropic if it has a k-rational point. An example of an anisotropic quadric is
Quadric_(algebraic_geometry)
Everything in space and time
including general relativity, led to the modern view of an expanding, isotropic, homogeneous universe. Evidence accumulated supporting the Big Bang theory:
Universe
geometry, the circular points at infinity (also called cyclic points or isotropic points) are two special points at infinity in the complex projective plane
Circular_points_at_infinity
Lode coordinates ( z , r , θ ) {\displaystyle (z,r,\theta )} or Haigh–Westergaard coordinates ( ξ , ρ , θ ) {\displaystyle (\xi ,\rho ,\theta )} . are
Lode_coordinates
Light rays follow quickest paths
through a medium (a vacuum or some material, not necessarily homogeneous or isotropic), without action at a distance; During propagation, the influence of the
Fermat's_principle
Isotopic ratio Isotopic shift Isotopically pure diamond Isotropic coordinates Isotropic radiation Isotropic radiator Isovector Istituto Nazionale di Fisica Nucleare
Index_of_physics_articles_(I)
Mechanism of light transport
steps to scatter the light path further, hence the name "random walk". Isotropic scattering is simulated by picking random directions evenly along a sphere
Subsurface_scattering
Increase in distance between parts of the universe
universe at the largest scales is homogeneous (the same everywhere) and isotropic (the same in all directions), means that the universe is expanding uniformly
Expansion_of_the_universe
Mathematical model of the stresses within flat plates under loading
D_{\alpha \beta }:=\int _{-h}^{h}x_{3}^{2}~C_{\alpha \beta }~dx_{3}} For an isotropic and homogeneous plate, the stress–strain relations are [ σ 11 σ 22 σ 12
Plate_theory
Mathematical theory of the geometry of space and time
will give rise to momentum flow, so the i = j terms (green) represent isotropic pressure, and the i ≠ j terms (blue) represent shear stresses. One important
Curved_spacetime
Light motion in curved spacetime
in Newtonian limit of Schwarzschild field described by metric in isotropic coordinates they correspond to its passive gravitational mass equal to twice
Fermat's and energy variation principles in field theory
Fermat's_and_energy_variation_principles_in_field_theory
Quantum mechanical model
Cartesian harmonic oscillator and the two-dimensional isotropic harmonic oscillator in cylindrical coordinates have been treated in detail in the book of Müller-Kirsten
Quantum_harmonic_oscillator
German physicist (1873–1916)
incompressible fluids; the sun and stars viewed as a quasi-isotropic heated gas; and any homogeneous and isotropic distributed gas. Schwarzschild's first (spherically
Karl_Schwarzschild
Mathematical relationship describing the flow of groundwater through an aquifer
(-K\nabla h)-G.} Now if hydraulic conductivity (K) is spatially uniform and isotropic (rather than a tensor), it can be taken out of the spatial derivative
Groundwater_flow_equation
Raising and lowering operators in quantum mechanics
Physicists: Isotropic harmonic oscillator" (PDF). Weizmann Institute of Science. Retrieved 28 July 2021. Fradkin, D. M. (1965). "Three-dimensional isotropic harmonic
Ladder_operator
goniophotometer enables characterization of emitted light that is not isotropic. A goniophotometer can be used for various applications: Measurement of
Goniophotometer
Concept in relativity theory
frames and coordinates are defined from the outset so that space and time coordinates as well as slow clock-transport are described isotropically (see sections
One-way_speed_of_light
Tensor used in continuum mechanics
independent of the state of motion or stress in the fluid. If the fluid is isotropic as well as Newtonian, the viscosity tensor μ will have only three independent
Viscous_stress_tensor
Coordinates to capture characteristics of rotating frames of reference
its immediate vicinity clocks are synchronized and light propagates isotropically in space. But the experience when the observers try to synchronize their
Born_coordinates
Coordinate system for digital imaging
and column coordinates and are distinguished with a single binary coordinate. Hexagonal sampling is the optimal approach for isotropically band-limited
Hexagonal Efficient Coordinate System
Hexagonal_Efficient_Coordinate_System
Spin representations of the SO(3) group
they were needed in physics." Spinors can be constructed directly from isotropic vectors in 3-space without using the quaternionic construction. To motivate
Spinors_in_three_dimensions
Assumption that motions of nuclei and electrons can be separated
nucleus compared to an electron, the coordinates of the nuclei in a system are approximated as fixed, while the coordinates of the electrons are dynamic. The
Born–Oppenheimer approximation
Born–Oppenheimer_approximation
Spectroscopic technique
and named after Brazilian physicist Sergio Pereira da Silva Porto. For isotropic solutions, the Raman scattering from each mode either retains the polarization
Raman_spectroscopy
Physical model of propagating energy
will be interference consistent with wave properties. In homogeneous, isotropic media, electromagnetic radiation is a transverse wave, meaning that its
Electromagnetic_radiation
Mathematical concept
its own right. W is isotropic if W ⊆ W⊥. This is true if and only if ω restricts to 0 on W. Any one-dimensional subspace is isotropic. W is coisotropic
Symplectic_vector_space
Vector describing a wave; often its propagation direction
wavelength), and its direction is perpendicular to the wavefront. In isotropic media, this is also the direction of wave propagation. A closely related
Wave_vector
Soviet over-the-horizon early-warning radar system
Map all coordinates using OpenStreetMap Download coordinates as: KML GPX (all coordinates) GPX (primary coordinates) GPX (secondary coordinates) Duga (Russian:
Duga_radar
Non-tensorial representation of the spin group
Let W be a maximal isotropic subspace, i.e. a maximal subspace of V such that g|W = 0. If n = 2k is even, then let W′ be an isotropic subspace complementary
Spinor
Two geometries based on axioms closely related to those specifying Euclidean geometry
and elliptic geometry, the traditional non-Euclidean geometries. When isotropic quadratic forms are admitted, then there are affine planes associated
Non-Euclidean_geometry
Intersection of the three symmedian lines of a triangle
Kolar-Begović, Z.; Kolar-Šuper, R. (2013), "On Gergonne point of the triangle in isotropic plane", Rad Hrvatske Akademije Znanosti i Umjetnosti, 17: 95–106, MR 3100227
Lemoine_point
Equations of fluid dynamics
strain rate tensor or equivalently the velocity gradient. The fluid is isotropic. For a fluid at rest, ∇ ⋅ τ must be zero (so that hydrostatic pressure
Derivation of the Navier–Stokes equations
Derivation_of_the_Navier–Stokes_equations
Branch of geometry
\mathbb {R} ^{2n}} . Each 1-dimensional subspace V {\displaystyle V} is isotropic, and has a complementary coisotropic subspace V ω {\displaystyle V^{\omega
Contact_geometry
Model of shear deformation and bending effects
These parameters are not necessarily constants. For a linear elastic, isotropic, homogeneous beam of constant cross-section these two equations can be
Timoshenko–Ehrenfest beam theory
Timoshenko–Ehrenfest_beam_theory
Property of certain dynamical systems
The leaves of the foliation are totally isotropic with respect to the symplectic form and such a maximal isotropic foliation is called Lagrangian. All autonomous
Integrable_system
Spatial geometry with curvature
n-dimensional space can also be described with Riemannian geometry. An isotropic and homogeneous space can be described by the metric: d l 2 = e − λ (
Curved_space
Quantum mechanics concept for systems with central potentials, such as atoms
3D isotropic harmonic oscillator is V ( r ) = 1 2 m 0 ω 2 r 2 . {\displaystyle V(r)={\frac {1}{2}}m_{0}\omega ^{2}r^{2}.} An N-dimensional isotropic harmonic
Particle in a spherically symmetric potential
Particle_in_a_spherically_symmetric_potential
Reflector that has the shape of a paraboloid
parabolic reflectors can also be used to collimate radiation from an isotropic source into a parallel beam. In optics, parabolic mirrors are used to
Parabolic_reflector
symmetry must have a third one. Isotropic materials have an infinite number of planes of symmetry. Transversely isotropic materials are special orthotropic
Orthotropic_material
Turbulence modeling approach
by the mean flow. This change is balanced by the mean body force, the isotropic stress owing to the mean pressure field, the viscous stresses, and apparent
Reynolds-averaged Navier–Stokes equations
Reynolds-averaged_Navier–Stokes_equations
Equation explaining structure of a spherical body of isotropic material
equation constrains the structure of a spherically symmetric body of isotropic material which is in static gravitational equilibrium, as modeled by general
Tolman–Oppenheimer–Volkoff equation
Tolman–Oppenheimer–Volkoff_equation
Construction for n-dimensional noise functions
noise. Simplex noise has no noticeable directional artifacts (is visually isotropic), though noise generated for different dimensions is visually distinct
Simplex_noise
Measure of antenna performance
entire sphere, and weighted by the antenna's radiation pattern. Hence, an isotropic antenna would have a noise temperature that is the average of all temperatures
Antenna gain-to-noise-temperature
Antenna_gain-to-noise-temperature
according to different principles: In the first class, the sensors have an isotropic sensor surface that supplies continuous position data. The second class
Position_sensitive_device
Theorem in optics that explains light propagation in a medium
theorem in crystalline and isotropic media, respectively, in 1916 and 1915. Originally, the theorem applied to scattering by isotropic dielectric objects in
Ewald–Oseen extinction theorem
Ewald–Oseen_extinction_theorem
Complete reflection of a wave
transmitted wave (we assume isotropic media, but the transmitted wave is not yet assumed to be evanescent). In Cartesian coordinates (x, y, z), let the region
Total_internal_reflection
Electromagnetic phenomenon
{\sin ^{2}(\theta )}{r^{2}}}\mathbf {\hat {r}} } is not distributed isotropically, but concentrated around the directions lying perpendicular to the dipole
Dipole
Electromagnetic wave with oscillations perpendicular to the direction of travel
waveguide. Unguided electromagnetic waves in free space, or in a bulk isotropic dielectric, can be described as a superposition of plane waves; these
Transverse_mode
Type of mechanical vibration
J_{3}:=\int _{-h}^{h}x_{3}^{2}~\rho ~dx_{3}={\frac {2}{3}}\rho h^{3}\,.} For an isotropic and homogeneous plate, the stress-strain relations are [ σ 11 σ 22 σ 12
Vibration_of_plates
Elliptic partial differential equation
(describing charges brought from outside). Assuming the medium is linear, isotropic, and homogeneous (see polarization density), we have the constitutive
Poisson's_equation
Quantity in fracture mechanics; predicts stress intensity near a crack's tip
the Poisson's ratio of the material. The material is assumed to be an isotropic, homogeneous, and linear elastic. The crack has been assumed to extend
Stress_intensity_factor
Strain caused by an external load
originally straight and slender, and any taper is slight The material is isotropic (or orthotropic), linear elastic, and homogeneous across any cross section
Bending
Function in the theory of antennas
pattern obtained for an array of N {\displaystyle N} isotropic radiators located at coordinates r → n {\displaystyle {\vec {r}}_{n}} , as determined by:
Array_factor
Measure of directional electromagnetic energy flux
nondispersive (in which all frequency components travel at the same speed) and isotropic (for simplicity) materials, the constitutive relations can be written
Poynting_vector
Ratio of distance on a map to the corresponding distance on the ground
lines intersecting at point P. A conformal map has an isotropic scale factor. Conversely isotropic scale factors across the map imply a conformal projection
Scale_(map)
Atom of the element hydrogen
eigenfunctions (the orbitals) are not necessarily isotropic themselves, their dependence on the angular coordinates follows completely generally from this isotropy
Hydrogen_atom
Point in a triangle that can be seen as its middle under some criteria
for the definition. Some other names for dilation are uniform scaling, isotropic scaling, homothety, and homothecy. The study of triangle centers traditionally
Triangle_center
Association football tournament in South Africa
thermoplastic polyurethane-elastomer from Taiwan, ethylene vinyl acetate, isotropic polyester/cotton fabric, and glue and ink from China. Some football stars
2010_FIFA_World_Cup
ISOTROPIC COORDINATES
ISOTROPIC COORDINATES
ISOTROPIC COORDINATES
ISOTROPIC COORDINATES
Boy/Male
Bengali, Indian
Sky
Girl/Female
Hindu, Indian
Golden Creeper
Boy/Male
Arabic, Muslim
Sky
Boy/Male
Australian, Finnish, French, German, Greek, Hebrew, Spanish
Jehovah Increases; Spanish Form of Joseph; He Shall Add; Yahweh will Add-another Son
Boy/Male
Scandinavian Russian
Hero.
Girl/Female
Anglo, Christian, English
Beloved Loveing
Boy/Male
American, Australian
Will; Desire and Helmet; Protection
Boy/Male
Hindu, Indian
Mind; Cleaver; King; Sharp Mind
Boy/Male
Indian, Sanskrit
Having a Virtuous Character
Boy/Male
German, Polish
Spear Ruler; Ruler with a Spear
ISOTROPIC COORDINATES
ISOTROPIC COORDINATES
ISOTROPIC COORDINATES
ISOTROPIC COORDINATES
ISOTROPIC COORDINATES
a.
Of, pertaining to, or included by, two lines; as, bilinear coordinates.
a.
Anisotropic.
a.
Not isotropic; having different properties in different directions; thus, crystals of the isometric system are optically isotropic, but all other crystals are anisotropic.
a.
Relating to, or showing, geotropism.
a.
Of, pertaining to, or included by, three lines; as, trilinear coordinates.
n.
Isotropy.
a.
That can be passed over in a single course; -- said of a curve when the coordinates of the point on the curve can be expressed as rational algebraic functions of a single parameter /.
a.
Having equal entropy.
a.
Of equal value.
a.
Pertaining to, or designating, an acid obtained from atropine, and isomeric with cinnamic acid.
n.
Uniformity of physical properties in all directions in a body; absence of all kinds of polarity; specifically, equal elasticity in all directions.
n.
The mensuration of such phenomena of earthquakes as can be expressed in numbers, or by their relation to the coordinates of space.
a.
Having the same properties in all directions; specifically, equally elastic in all directions.
a.
Exhibiting differences of quality or property in different directions; not isotropic.
a.
Having or indicating, equal tones, or tension.
a.
Isotropic.
a.
Alt. of Anisotropic
n.
In the theory of gravitation, or of other forces acting in space, a function of the rectangular coordinates which determine the position of a point, such that its differential coefficients with respect to the coordinates are equal to the components of the force at the point considered; -- also called potential function, or force function. It is called also Newtonian potential when the force is directed to a fixed center and is inversely as the square of the distance from the center.
a.
Pertaining to, reckoned from, or having a common radiating point; as, polar coordinates.