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Concept in integration theory
In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates
Volume_element
Term used in composite materials theory
representative elementary volume (REV) (also called the representative volume element (RVE) or the unit cell) is the smallest volume over which a measurement
Representative elementary volume
Representative_elementary_volume
Set of points equidistant from a center
has area element d A = r 2 sin θ d θ d φ {\displaystyle dA=r^{2}\sin \theta \,d\theta \,d\varphi } . This can be found from the volume element in spherical
Sphere
Differential form
{\displaystyle M} of dimension n {\displaystyle n} , a volume form is an n {\displaystyle n} -form. It is an element of the space of sections of the line bundle
Volume_form
Size of a mathematical ball
proof of the volume formula. The volume of the n-ball V n ( R ) {\displaystyle V_{n}(R)} can be computed by integrating the volume element in spherical
Volume_of_an_n-ball
Type of three-dimensional shape
theorem). A representative disc is a three-dimensional volume element of a solid of revolution. The element is created by rotating a line segment (of length
Solid_of_revolution
Quantity of a three-dimensional space
three-dimensional bodies. A 'unit' of infinitesimally small volume in integral calculus is the volume element; this formulation is useful when working with different
Volume
Representing a 3D-modeled object or dataset as a 2D projection
pattern. This is an example of a regular volumetric grid, with each volume element, or voxel represented by a single value that is obtained by sampling
Volume_rendering
Equation of statistical mechanics
particle occupies a given very small region of space (mathematically the volume element d 3 r {\displaystyle d^{3}\mathbf {r} } ) centered at the position r
Boltzmann_equation
Metric tensor describing constant negative (hyperbolic) curvature
{y^{2}}{|cz+d|^{4}}}}={\frac {dz\,d{\overline {z}}}{y^{2}}}.} The invariant volume element is given by d μ = d x d y y 2 . {\displaystyle d\mu ={\frac {dx\,dy}{y^{2}}}
Poincaré_metric
Matrix of partial derivatives of a vector-valued function
the volume of the spherical differential volume element. Unlike rectangular differential volume element's volume, this differential volume element's volume
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Analysis and solving of problems that involve fluid flows
the volume of the control volume element, and A {\displaystyle \mathbf {A} } is the surface area of the control volume element. The finite element method
Computational_fluid_dynamics
Concept in physics
stresses do not exert a torque on the volume element, the resultant force must lead through the center of the volume element. The line of action of the inertia
Balance_of_angular_momentum
State of balance between external forces on a fluid and internal pressure gradient
Finally, the weight of the volume element causes a force downwards. If the density is ρ, the volume is V and g the standard gravity, then:
Hydrostatic_equilibrium
Numerical method for solving physical or engineering problems
Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical
Finite_element_method
Integral over a 3-D domain
corresponding density function. Often the volume integral is represented in terms of a differential volume element d V = d x d y d z {\displaystyle dV=dx\
Volume_integral
Coordinates comprising a distance and two angles
\theta ,\\r^{2}&=ax^{2}+by^{2}+cz^{2}.\end{aligned}}} An infinitesimal volume element is given by d V = | ∂ ( x , y , z ) ∂ ( r , θ , φ ) | d r d θ d φ =
Spherical_coordinate_system
Primitive quantum mechanical model of electronic structure
one volume element to the next. For a small volume element ΔV, and for the atom in its ground state, we can fill out a spherical momentum-space volume VF
Thomas–Fermi_model
Low energy photon scattering off charged particles
depending on where an observer is located, the light scattered from a small volume element may appear to be more or less polarized. In the diagram, everything
Thomson_scattering
Branch of theoretical physics
{\displaystyle \mathbf {r} -\mathbf {r'} } is the vector that points from the volume element d 3 r ′ {\displaystyle \mathrm {d^{3}} \mathbf {r'} } to the point in
Classical_electromagnetism
Chemical element with atomic number 80 (Hg)
Mercury is a chemical element; it has symbol Hg and atomic number 80. It is commonly known as quicksilver. A heavy, silvery d-block element, mercury is the
Mercury_(element)
Sum of a scalar and vector in Clifford algebra
_{3}\mathbf {e} _{3}=-1.} Moreover, the volume element e 123 {\displaystyle \mathbf {e} _{123}} commutes with any other element of the C ℓ ( 3 ) {\displaystyle
Paravector
Line segment of infinitesimally small length
In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in
Line_element
Electric charge per unit length, area or volume
{\displaystyle \sigma _{q}={\frac {dQ}{dS}}\,,} and the volume charge density uses a volume element dV ρ q = d Q d V , {\displaystyle \rho _{q}={\frac {dQ}{dV}}\
Charge_density
Integration over a non-flat region in 3D space
Area element Divergence theorem Stokes' theorem Line integral Line element Volume element Volume integral Cartesian coordinate system Volume and surface
Surface_integral
Vector operator in vector calculus
unit of volume (a volume element) as it flows with the vector field. On a pseudo-Riemannian manifold, the divergence with respect to the volume can be
Divergence
Topics referred to by the same term
average of the orientation of the long molecular axis within a small volume element of liquid crystal Director (military), a device that continuously calculates
Director
Generalized sphere of dimension n (mathematics)
^{2}\left(\varphi _{m}\right)\right)d\varphi _{k}^{2}} To express the volume element of n {\displaystyle n} -dimensional Euclidean space in terms of spherical
N-sphere
an abbreviated term for a "surface element," analogous to a "voxel" (volume element) or a "pixel" (picture element). In 3D computer graphics, the use
Surfel
Mathematical model for describing material deformation under stress
{F} \,\!} . The corresponding formula for the transformation of the volume element is d v = J d V {\displaystyle dv=J~dV} Derivation of Nanson's relation
Finite_strain_theory
— including multiphysics simulation, finite-element, finite-volume, finite difference, boundary element, riemann solver, dissipative particle dynamics
List of computational fluid dynamics software
List_of_computational_fluid_dynamics_software
Measurable property of a material or system
x_{2}\cdots x_{n}\right)} , then Differential The differential n-space volume element is d n x ≡ d V n ≡ d x 1 d x 2 ⋯ d x n {\displaystyle \mathrm {d} ^{n}x\equiv
Physical_quantity
Special point in the density of states of a crystal
{L^{3}}{(2\pi )^{3}}}\,d^{3}k} where d 3 k {\displaystyle d^{3}k} is a volume element in k-space, and which, for electrons, will need to be multiplied by
Van_Hove_singularity
Analysis of composite materials
based on the concept of the representative volume element (RVE). An RVE is understood to be a sub-volume of an inhomogeneous medium that is of sufficient
Micromechanics
Force which acts throughout the volume of a body
_{V}\mathbf {f} (\mathbf {r} )\mathrm {d} V\,,} where dV is an infinitesimal volume element, and f is the external body force density field acting on the system
Body_force
Classification in abstract algebra
the pseudoscalars (degree n elements) as well. After rescaling the volume element by a nonzero complex scalar if necessary, one may choose a normalized
Classification of Clifford algebras
Classification_of_Clifford_algebras
Intersection of a sphere and cone emanating from its center
subtended by a cap area of A = r2. The volume can be calculated by integrating the differential volume element d V = ρ 2 sin ϕ d ρ d ϕ d θ {\displaystyle
Spherical_sector
Distribution of an uncertain quantity
conditions (as many particles as flow out of a volume element also flow in steadily, so that the situation in the element appears static), i.e., independent of
Prior_probability
Branch of physics
molecular length scale. Fluid properties can vary continuously from one volume element to another and are average values of the molecular properties. The continuum
Fluid_mechanics
State of thermodynamic systems where no net flow of matter or energy occurs
material in any small volume element of the system can be interchanged with the material of any other geometrically congruent volume element of the system, and
Thermodynamic_equilibrium
Energy related to Earth's gravity
where ρ2 = ρ(x, y, z) is the mass density at the volume element and of the direction from the volume element to point mass 1. u {\displaystyle u} [clarification
Geopotential
Branch of thermodynamics
follows. The assumptions have the effect of making each very small volume element of the system effectively homogeneous, or well-mixed, or without an
Non-equilibrium thermodynamics
Non-equilibrium_thermodynamics
Method of solving linear partial differential equations
The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations (PDEs) arising in engineering and
Boundary_element_method
Standard for describing objects for additive manufacturing
single <mesh> element. The mesh is defined using one <vertices> element and one or more <volume> elements. The required <vertices> element lists all vertices
Additive manufacturing file format
Additive_manufacturing_file_format
Topics referred to by the same term
(call sign UVE) Uve Sabumei, Papua New Guinean rugby coach Uncorrelated volume element, a term used in the theory of composite materials Unión Velocipédica
Uve
Body of matter in a state of internal equilibrium
conceptual basis of thermodynamics requires some type of wall: a small volume element of a continuous medium subject to a thermodynamic change will exchange
Thermodynamic_system
Equations describing classical electromagnetism
{\displaystyle Q=\iiint _{\Omega }\rho \ \mathrm {d} V,} where dV is the volume element. The net magnetic flux ΦB is the surface integral of the magnetic field
Maxwell's_equations
Statistical model used in machine learning
{1}}'\end{bmatrix}}} . To define U {\displaystyle U} , the differential volume element at the transformation input ( p ∈ Δ n − 1 {\displaystyle \mathbf {p}
Flow-based_generative_model
Force acting on charged particles in electric and magnetic fields
\mathbf {B} \right)} Dividing both sides by the volume d V {\displaystyle \mathrm {d} V} of the charge element gives the force density f = ρ ( E + v × B )
Lorentz_force
Law of classical electromagnetism
{\displaystyle dV} is the volume element, and J {\displaystyle \mathbf {J} } is the current density vector in that volume (in SI in units of A/m2). In
Biot–Savart_law
\theta &-\rho \sin \theta &0\end{pmatrix}}\end{aligned}}} So for the volume element: d x d y d z = det ∂ ( x , y , z ) ∂ ( ρ , θ , φ ) d ρ d θ d φ = ρ 2
List of common coordinate transformations
List_of_common_coordinate_transformations
Technique to solve geological problems by computational simulation
include the finite element, finite difference, or finite volume method that subdivide the object of interest into smaller pieces (element) by mesh. These
Numerical_modeling_(geology)
Type of physical quantity
manifolds, one cannot define a volume form globally due to the non-orientability, but one can define a volume element, which is formally a density, and
Pseudotensor
Astronomical phenomenon
^{2}\rho ,} where m {\displaystyle m} is mass (in this case of a small volume element of the star), Ω {\displaystyle \Omega } is the angular velocity, and
Gravity_darkening
incident field is not greatly altered within one particle so that each volume element is considered to be illuminated by an intensity and phase determined
Rayleigh–Gans_approximation
Mathematical concept applicable to physics
|^{2}.} So the probability of finding a particle in a differential volume element d3r is d P = | ψ | 2 d 3 r . {\displaystyle dP=|\psi |^{2}\,d^{3}\mathbf
Flux
Coordinates comprising two distances and an angle
to know the line and volume elements; these are used in integration to solve problems involving paths and volumes. The line element is d r = d ρ ρ ^ + ρ
Cylindrical_coordinate_system
Irreducible parts of a load-bearing structural system
simple elements (each bearing a structural load). Within a structure, an element cannot be broken down (decomposed) into parts of different kinds (e.g.
Structural_element
Formula for the average value of a function over its domain
{\displaystyle dV} are, respectively, the domain volume and volume element (or generalizations thereof, e.g., volume form). The above generalizes the arithmetic
Mean_of_a_function
Three-dimensional coordinate system
\right)\left(\nu -\mu \right)}{S(\nu )}}}} Hence, the infinitesimal volume element equals d V = ( λ − μ ) ( λ − ν ) ( μ − ν ) 8 − S ( λ ) S ( μ ) S ( ν
Ellipsoidal_coordinates
Definite integral of a scalar or vector field along a path
Methods of contour integration Nachbin's theorem Line element Surface integral Volume element Volume integral Kwong-Tin Tang (30 November 2006). Mathematical
Line_integral
Mass per unit volume
relevant to buoyancy, purity and packaging. Osmium is the densest known element at standard conditions for temperature and pressure. To simplify comparisons
Density
Subfield of materials science
nodes connected by segments. This is similar to a mesh used in finite element modelling. Then, the forces on each of the nodes of the dislocation are
Computational materials science
Computational_materials_science
Imaging by sections or sectioning using a penetrative wave
pattern. This is an example of a regular volumetric grid, with each volume element, or voxel represented by a single value that is obtained by sampling
Tomography
Study of still or slow electric charges
\mathrm {d} ^{3}r=\mathrm {d} x\ \mathrm {d} y\ \mathrm {d} z} is a volume element. If the charge is distributed over a surface or along a line, replace
Electrostatics
As of 2025[update], the most expensive non-synthetic element by mass is rhodium, and by volume, iridium. It is followed by rhodium, caesium, iridium
Prices_of_chemical_elements
{\displaystyle \mathbf {f} =-\nabla p} . The net force on a differential volume element dV of the fluid is: d F = f d V {\displaystyle d\mathbf {F} =\mathbf
Force_density
Chemical element with atomic number 2 (He)
(from Ancient Greek: ἥλιος, romanized: helios, lit. 'sun') is a chemical element; it has symbol He and atomic number 2. It is a colorless, odorless, non-toxic
Helium
Branch of physics which studies the behavior of materials modeled as continuous media
finer resolution than the size of the representative volume element (RVE), a statistical volume element (SVE) is employed, which results in random continuum
Continuum_mechanics
Mathematical function
fields[clarification needed] in which the key role is played by a statistical volume element (SVE), which is a spatial box over which properties can be averaged;
Random_field
Phenomenon in physics
distribution of speeds both toward and away from the observer in any volume element of the radiating body, the net effect will be to broaden the observed
Doppler_broadening
Key result in Hamiltonian mechanics and statistical mechanics
of phase space volume and Liouville's theorem". Retrieved January 6, 2014. A rigorous proof based on how the Jacobian volume element transforms under
Liouville's theorem (Hamiltonian)
Liouville's_theorem_(Hamiltonian)
Rate of separation of infinitesimally close trajectories
justification. If the system is conservative (i.e., there is no dissipation), a volume element of the phase space will stay the same along a trajectory. Thus the sum
Lyapunov_exponent
The bilinear quadrilateral element, also known as the Q4 element, is a type of element used in finite element analysis which is used to approximate in
Bilinear quadrilateral element
Bilinear_quadrilateral_element
Topics referred to by the same term
Surface element may refer to An infinitesimal portion of a 2D surface, as used in a surface integral in a 3D space. The volume form of a 2D manifold Surfel
Surface_element
Three-dimensional orthogonal coordinate system
{\displaystyle h_{\phi }=a\cosh \mu \ \cos \nu } Consequently, an infinitesimal volume element equals d V = a 3 cosh μ cos ν ( sinh 2 μ + sin 2 ν ) d μ
Oblate_spheroidal_coordinates
Interval boundary element method is classical boundary element method with the interval parameters. Boundary element method is based on the following integral
Interval boundary element method
Interval_boundary_element_method
Method of studying partial differential equations
material), is known as the "Representative Volume Element" in homogenization and micromechanics. This element contains enough statistical information about
Asymptotic_homogenization
One of the four classical elements
four triangles and contains the least volume with the greatest surface area. This also makes fire the element with the smallest number of sides, and
Fire_(classical_element)
Chemical element with atomic number 5 (B)
Company first popularized and produced them in volume at low cost. Boron was not recognized as an element until it was isolated by Sir Humphry Davy and
Boron
Degree of concentration of countable objects
= dx dy dz is a volume element. If each object possesses the same mass m0, the total mass m of all the objects in the volume V can be expressed as m
Number_density
Theorem in calculus
{g(x',0)}}\,dx'={\sqrt {g_{\partial \Omega }(x')}}\,dx'=dS} is the volume element on ∂ Ω {\displaystyle \partial \Omega } and the above formula reads
Divergence_theorem
Graphics created using computers
pattern. This is an example of a regular volumetric grid, with each volume element, or voxel represented by a single value that is obtained by sampling
Computer_graphics
Chemical element with atomic number 86 (Rn)
Radon is a chemical element; it has symbol Rn and atomic number 86. It is a radioactive noble gas and is colorless and odorless. Of the three naturally
Radon
Rare chemical element in organism
In biochemistry, an ultratrace element is a chemical element that normally comprises less than one microgram per gram of a given organism (i.e. less than
Ultratrace_element
Equation for Gibbs free energy of solvation
{E}}|={\frac {ze}{4\pi \varepsilon _{0}\varepsilon _{r}r^{2}}}} and the volume element d V {\displaystyle dV} can be expressed as d V = 4 π r 2 d r {\displaystyle
Born_equation
Three-dimensional coordinate system
)d\varphi ^{2}\right].\end{aligned}}} Consequently, an infinitesimal volume element equals d V = a 3 sinh μ sin ν ( sinh 2 μ + sin 2 ν ) d μ d ν
Prolate spheroidal coordinates
Prolate_spheroidal_coordinates
Simulation of multiple aspects of physics
with discretization methods such as the finite element method, finite difference method, or finite volume method. Multiphysics simulations can be performed
Multiphysics_simulation
Method for representing and evaluating partial differential equations
Cross, M.; Taylor, G. A. (2000-06-01). "Comparison of finite element and finite volume methods application in geometrically nonlinear stress analysis"
Finite_volume_method
DC Comics character
Encyclopedia of Comic Book Heroes, Volume Three: Superman. DC Comics. p. 62. ISBN 978-1-4012-1389-3. Companik, Chris. "Element Lad & Shvaughn Erin". Gay League
Element_Lad
Foundational law of classical magnetism
states that for each volume element in space, there are exactly the same number of "magnetic field lines" entering and exiting the volume. No total "magnetic
Gauss's_law_for_magnetism
Chemical element with atomic number 1 (H)
Hydrogen is a chemical element; it has the symbol H and atomic number 1. It is the lightest and most abundant chemical element in the universe, constituting
Hydrogen
In mathematics, an element of a *-algebra is called positive if it is the sum of elements of the form a ∗ a {\displaystyle a^{*}a} . Let A {\displaystyle
Positive_element
Fundamental study of potential theory
= ρ(r) dv(r), where dv(r) is the Euclidean volume element, then the gravitational potential is the volume integral V ( x ) = − ∫ R 3 G ‖ x − r ‖ ρ ( r
Gravitational_potential
Chemical element with atomic number 74 (W)
Tungsten (also called wolfram) is a chemical element; it has symbol W (from German: Wolfram) and atomic number 74. It is a metal found naturally on Earth
Tungsten
Direction of no double refraction
average of the orientation of the long molecular axis within a small volume element of material demonstrating a mesophase. Electrical manipulation of the
Optic_axis_of_a_crystal
Any of the fifteen lanthanides plus scandium and yttrium
plentiful in the entire Earth's crust (cerium being the 25th-most-abundant element at 68 parts per million, more abundant than copper), but in practice they
Rare-earth_element
Interdisciplinary branch of science concerned with presenting scientific data visually
pattern. This is an example of a regular volumetric grid, with each volume element, or voxel represented by a single value that is obtained by sampling
Scientific_visualization
Vector field describing the density of electric dipole moments in a dielectric material
neutral in charge, yet an electric dipole moment forms. For a certain volume element Δ V {\displaystyle \Delta V} in the material, which carries a dipole
Polarization_density
Two-dimensional orthogonal coordinate system
h_{\tau }} are the same as in the two-dimensional case. The infinitesimal volume element is then d V = h σ h τ h φ d σ d τ d φ = σ τ ( σ 2 + τ 2 ) d σ d τ d
Parabolic_coordinates
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