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Vector calculus construction
a time dependent vector field is a construction in vector calculus which generalizes the concept of vector fields. It can be thought of as a vector field
Time_dependent_vector_field
Assignment of a vector to each point in a subset of Euclidean space
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space R n {\displaystyle
Vector_field
Motion of particles in a fluid
one simple criterion is that the vector field F is compactly supported. In the case of time-dependent vector fields F : R n × R → R n {\displaystyle
Flow_(mathematics)
Vector field reconstruction is a method of creating a vector field from experimental or computer-generated data, usually with the goal of finding a differential
Vector_field_reconstruction
Measure of directional electromagnetic energy flux
the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area, per unit time) or power flow
Poynting_vector
inversion is smooth. Let X ( t , x ) {\displaystyle X(t,x)} be a time dependent vector field on M {\displaystyle M} (in C ∞ ( R , X ( M ) ) {\displaystyle
Convenient_vector_space
Algebraic structure in linear algebra
This means that for two vector spaces over a given field and with the same dimension, the properties that depend only on the vector-space structure are exactly
Vector_space
Function valued in a vector space; typically a real or complex one
of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector (that is, the dimension
Vector-valued_function
Technique for the generative modeling of a continuous probability distribution
deterministic flow along a time-dependent vector field, and the backward process is also a deterministic flow along the same vector field, but going backwards
Diffusion_model
Method to control electric motors
Field-oriented control (FOC), also called vector control, is a variable-frequency drive (VFD) control method in which the stator currents of a three-phase
Field-oriented_control
Physical field surrounding an electric charge
electric field between atoms is the force responsible for chemical bonding that result in molecules. The electric field is defined as a vector field that
Electric_field
Vector field representing a mass's effect on surrounding space
In physics, a gravitational field or gravitational acceleration field is a vector field used to explain the influences that a body extends into the space
Gravitational_field
Quantity in electromagnetism
electromagnetism, magnetic vector potential (often denoted A) is the vector quantity defined so that its curl is equal to the magnetic field, B: ∇ × A = B {\textstyle
Magnetic_vector_potential
Concept in differential geometry
{\displaystyle \sigma (t):=1-2\lambda t} and integrating the time-dependent vector field X ( t ) := 1 σ ( t ) V {\displaystyle X(t):={\frac {1}{\sigma
Ricci_soliton
Quantization giving rise to photons
are time-dependent vector fields that in vacuum depend on a third vector field A ( r , t ) {\displaystyle \mathbf {A} (\mathbf {r} ,t)} (the vector potential)
Quantization of the electromagnetic field
Quantization_of_the_electromagnetic_field
Trick relating differential forms
flows of a time-dependent vector field, i.e. of a smooth family { X t } t ∈ [ 0 , 1 ] {\displaystyle \{X_{t}\}_{t\in [0,1]}} of vector fields on M {\displaystyle
Moser's_trick
Property of space that quantifies the magnetic influence at a given location
magnetic field may vary with location, it is described mathematically by assigning a vector to each point of space, making it a vector field. There are
Magnetic_field
Simple quantum mechanical system
σ {\displaystyle {\boldsymbol {\sigma }}} is the vector of Pauli matrices. Solving the time dependent Schrödinger equation H ψ = i ℏ ∂ t ψ {\displaystyle
Two-state_quantum_system
constant vector fields). Let U ( x , t ) = u ( x , t ) ∂ ∂ x {\displaystyle U(x,t)=u(x,t){\frac {\partial }{\partial x}}} be a time-dependent vector field on
Hunter–Saxton_equation
Equation in fluid dynamics
{\displaystyle U(x,t)=u(x,t){\frac {\partial }{\partial x}}} be a time-dependent vector field on S 1 {\displaystyle S^{1}} , and let { φ t } {\displaystyle
Camassa–Holm_equation
Electric and magnetic fields produced by moving charged objects
field is a pair of vector fields consisting of one vector for the electric field and one for the magnetic field at each point in space. The vectors may
Electromagnetic_field
Vector in relativity
In special relativity, a four-vector (or 4-vector, sometimes Lorentz vector) is an element of a four-dimensional vector space object with four components
Four-vector
Equations in quantum field theory
temperature-dependent GL relaxation time of the order parameter; V {\displaystyle V} the electrochemical potential; A x {\displaystyle A_{x}} the magnetic vector
Time-dependent Ginzburg–Landau theory
Time-dependent_Ginzburg–Landau_theory
Geometric object that has length and direction
physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude
Euclidean_vector
Mathematical concept applicable to physics
property. In vector calculus, flux is a scalar quantity, defined as the surface integral of the perpendicular component of a vector field over a surface
Flux
Numerical analysis technique
in space and time for each electric and magnetic vector field component in Maxwell's curl equations. The descriptor "Finite-difference time-domain" and
Finite-difference time-domain method
Finite-difference_time-domain_method
Time rate of change of some physical quantity of a material element in a velocity field
macroscopic velocity is represented by the vector field u(x, t). The (total) derivative with respect to time of φ is expanded using the multivariate chain
Material_derivative
Modification of general relativity
metric and a unit timelike vector field named the aether. The aether in this theory is "a Lorentz-violating vector field" unrelated to older luminiferous
Einstein-aether_theory
Force acting on charged particles in electric and magnetic fields
dual to a vector which is the usual magnetic field vector. The relativistic velocity is given by the (time-like) changes in a time-position vector v = x ˙
Lorentz_force
Description of a quantum-mechanical system
(x,t)} as used above can be written as the inner product of a time-dependent state vector | Ψ ( t ) ⟩ {\displaystyle |\Psi (t)\rangle } with unphysical
Schrödinger_equation
Type of wave propagating in 3 dimensions
unit-length vector, and G ( d , t ) {\displaystyle G(d,t)} is a function that gives the field's value as dependent on only two real parameters: the time t {\displaystyle
Plane_wave
Type of signal in signal processing
noise in the theory of continuous-time signals, one must replace the concept of a random vector by a continuous-time random signal; that is, a random process
White_noise
Problem in quantum optics
Bloch equations, which define the dynamics of the pseudo-spin vector in an electric field: u ˙ = − δ v , {\displaystyle {\dot {u}}=-\delta v,} v ˙ = δ
Rabi_problem
Method for visualizing vector fields
are highly dependent on proper seed points. Texture-based methods, like LIC, avoid these problems since they depict the entire vector field at point-like
Line_integral_convolution
Mathematical operation on vectors in 3D space
product of the units of each vector. If two vectors are parallel or are anti-parallel (that is, they are linearly dependent), or if either one has zero
Cross_product
Formulation of electromagnetic potentials
potential ϕ {\displaystyle \phi } and the vector potential A {\displaystyle \mathbf {A} } which are used to find the fields as is commonly done. Considering cases
Hertz_vector
Local rate of change in potential with respect to displacement
field: − E = ∇ V . {\displaystyle -\mathbf {E} =\nabla V.\,\!} In electrodynamics, the E field is time dependent and induces a time-dependent B field
Potential_gradient
Concept in dynamical systems
\times \mathbb {R} ^{+};\mathbb {R} ^{n})} . We expand this time-dependent vector field in a Taylor series (in powers of ε {\displaystyle \varepsilon
Method_of_averaging
Specification of a derivative along a tangent vector of a manifold
presents an introduction to the covariant derivative of a vector field with respect to a vector field, both in a coordinate-free language and using a local
Covariant_derivative
Specialized notation for multivariable calculus
have an n-vector of dependent variables, or functions, of m independent variables we might consider the derivative of the dependent vector with respect
Matrix_calculus
potential, v(r,t), such as a time-varying electric field. The Runge–Gross theorem provides the formal foundation of time-dependent density functional theory
Runge–Gross_theorem
View of quantum mechanics
H_{\text{S}}t/\hbar }|\psi (0)\rangle } be the time-dependent state vector in the Schrödinger picture. A state vector in the interaction picture, | ψ I ( t )
Interaction_picture
Set of methods for supervised statistical learning
In machine learning, a support vector machine (SVM) or support vector network is a supervised max-margin model with associated learning algorithms that
Support_vector_machine
Method for numerically solving time-dependent incompressible fluid-flow problems
decomposition (sometimes called Helmholtz-Hodge decomposition) of any vector field into a solenoidal part and an irrotational part. Typically, the algorithm
Projection method (fluid dynamics)
Projection_method_(fluid_dynamics)
Application of Lagrangian mechanics to field theories
for vector fields, tensor fields, and spinor fields. In physics, fermions are described by spinor fields. Bosons are described by tensor fields, which
Lagrangian_(field_theory)
Basic law of electromagnetism
magnetic flux is defined as the surface integral of the magnetic field B over a time-dependent surface Σ(t), whose boundary is the wire loop: Φ B = ∬ Σ ( t
Faraday's_law_of_induction
Line integral of the electric field
electric field exerts force on a charged object, if the object has a positive charge, the force will be in the direction of the electric field vector at the
Electric_potential
Theorem in physics showing the conservation of energy for the electromagnetic field
the volume, given by the divergence of the Poynting vector S. J ⋅ E is the power density of the field doing work on charges (J is the current density corresponding
Poynting's_theorem
Algebra associated to any vector space
built from vector spaces, such as vector fields and functions whose domain is a vector space. Moreover, the field of scalars may be any field. More generally
Exterior_algebra
Property of waves that can oscillate with more than one orientation
as transverse waves, meaning that a plane wave's electric field vector E and magnetic field H are each in some direction perpendicular to (or "transverse"
Polarization_(waves)
Radiation description framework
the description of electromagnetic or gravitational radiation from time-dependent distributions of distant sources. These tools are applied to physical
Multipole_radiation
Complex vector of electromagnetic fields
ambiguously called the "electromagnetic field") is a complex vector that combines the electric field E and the magnetic field B. Heinrich Martin Weber published
Riemann–Silberstein_vector
Equations of electromagnetism
after Oleg D. Jefimenko) describe the electric field and magnetic fields generated by time-dependent distributions of electric charge and current. These
Jefimenko's_equations
Notation of differential calculus
\,\mathbf {A} } , of the vector field A is a vector, which is symbolically expressed by the cross product of ∇ and the vector A, curl A = ( ∂ A z ∂ y
Notation_for_differentiation
Vector field related to displacement current and flux density
In physics, the electric displacement field (denoted by D), also called electric flux density, is a vector field that appears in Maxwell's equations. It
Electric_displacement_field
Type of derivative in differential geometry
change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. This change is coordinate
Lie_derivative
Classical statement of gravity as force
vector quantity, and the right hand side is multiplied by the appropriate unit vector. Also, it can be seen that F12 = −F21. The gravitational field is
Newton's law of universal gravitation
Newton's_law_of_universal_gravitation
Method by which information is represented in the brain
correlate Neural decoding Neural oscillation Receptive field Sparse distributed memory Vector quantization Representational drift Brown EN, Kass RE, Mitra
Neural_coding
Equation in physics
the right side is the vector Laplacian, not Laplacian applied on scalar functions.) gives the wave equation for the electric field E: 1 c 2 ∂ 2 E ∂ t 2
Inhomogeneous electromagnetic wave equation
Inhomogeneous_electromagnetic_wave_equation
Speed and direction of a motion
physical objects. Velocity is a vector quantity, meaning that both magnitude and direction are needed to define it (velocity vector). The scalar absolute value
Velocity
Equations of light transmission and reflection
the position vector, ω is the angular frequency, t is time, and it is understood that the real part of the expression is the physical field. The value
Fresnel_equations
Polarization state
phase of the light as it travels through time and space. At any instant of time, the electric field vector of the wave indicates a point on a helix oriented
Circular_polarization
Electromagnetic quantum-mechanical effect in regions of zero magnetic and electric field
classical gravitational potential) and a stationary magnetic field as the curl of a vector potential (then a new concept – the idea of a scalar potential
Aharonov–Bohm_effect
Field lines in a fluid flow
are field lines in a fluid flow. They differ only when the flow changes with time, that is, when the flow is not steady. Considering a velocity vector field
Streamlines, streaklines, and pathlines
Streamlines,_streaklines,_and_pathlines
Process by which dust, particulates, etc. scatter light
manner: the electric field vector components in a volume of space are solved at a given instant in time; then the magnetic field vector components in the
Light_scattering_by_particles
Equations describing classical electromagnetism
magnetic field is a solenoidal vector field. The Maxwell–Faraday version of Faraday's law of induction describes how a time-varying magnetic field corresponds
Maxwell's_equations
Force in which the work done in moving an object depends only on its displacement
force field F, defined everywhere in space (or within a simply-connected volume of space), is called a conservative force or conservative vector field if
Conservative_force
Type of plane wave
plane wave is a special case of plane wave: a field whose value varies as a sinusoidal function of time and of the distance from some fixed plane. It
Sinusoidal_plane_wave
Type of potential in electrodynamics
these gives the retarded potentials below (all in SI units). For time-dependent fields, the retarded potentials are: φ ( r , t ) = 1 4 π ϵ 0 ∫ ρ ( r ′
Retarded_potential
F {\displaystyle F} can be a scalar or vector field and v {\displaystyle \mathbf {v} } is the velocity field. The first term on the right-hand side of
Semi-Lagrangian_scheme
Quantum field giving rise to gluons
In theoretical particle physics, the gluon field is a four-vector field characterizing the propagation of gluons in the strong interaction between quarks
Gluon_field
Type of neuroimaging
decoherence because of magnetic field inhomogeneity is not a true "relaxation" process; it is not random, but dependent on the location of the molecule
T2*-weighted_imaging
Hamiltonian of a bare electron bound in an atom interacting with a time-dependent electromagnetic field is given by the Pauli equation (the theoretical description
Magnetic_dipole_transition
Formulations of quantum mechanics
systems that evolve in time: the time-dependent nature of the system must be carried by some combination of the state vectors and the operators. For example
Dynamical_pictures
Multivariate derivative (mathematics)
In vector calculus, the gradient of a scalar-valued differentiable function f {\displaystyle f} of several variables is the vector field (or vector-valued
Gradient
Mathematical description of spacetime used in relativity
to form a four-vector. The 3-space electric field, E, combines with the 3-space magnetic field, B, to create a tensor in the four-vector formalism. This
Minkowski_spacetime
Gauge fixing of electro magnetic potential
the Lorenz condition is generally used in calculations of time-dependent electromagnetic fields through retarded potentials. The condition is ∂ μ A μ ≡
Lorenz_gauge_condition
Computer vision framework
Gradient vector flow (GVF), a computer vision framework introduced by Chenyang Xu and Jerry L. Prince, is the vector field that is produced by a process
Gradient_vector_flow
Physical phenomenon
longitudinal component of the total nuclear magnetic moment vector (parallel to the constant magnetic field) exponentially relaxes from a higher energy, non-equilibrium
Spin–lattice_relaxation
Physical quantity, density of magnetic moment per volume
In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic
Magnetization
Equation used to calculate the electromigration of ions in a fluid
Planck. The Nernst–Planck equation is a continuity equation for the time-dependent concentration c ( t , x ) {\displaystyle c(t,{\bf {x}})} of a chemical
Nernst–Planck_equation
Vector space with generalized dot product
product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner
Inner_product_space
Topics referred to by the same term
theory), use of a projection map in measure theory Vector projection, orthogonal projection of a vector onto a straight line Projection (relational algebra)
Projection
Concepts from linear algebra
algebra, an eigenvector (/ˈaɪɡən-/ EYE-gən-) or characteristic vector is a (nonzero) vector that has its direction unchanged (or reversed) by a given linear
Eigenvalues_and_eigenvectors
In mathematics, vector subspace
types of subspaces. If V is a vector space over a field K, a subset W of V is a linear subspace of V if it is a vector space over K for the operations
Linear_subspace
Branch of mathematics
vector spaces. Linear maps are mappings between vector spaces that preserve the vector-space structure. Given two vector spaces V and W over a field F
Linear_algebra
Physical theory with fields invariant under the action of local "gauge" Lie groups
there necessarily arises a corresponding field (usually a vector field) called the gauge field. Gauge fields are included in the Lagrangian to ensure
Gauge_theory
the motion field and the vector u is dependent both on the image position ( y 1 , y 2 ) {\displaystyle (y_{1},y_{2})} as well as on the time t. Similarly
Motion_field
Tensor index notation for tensor-based calculations
Tensor field Vector analysis While the raising and lowering of indices is dependent on a metric tensor, the covariant derivative is only dependent on the
Ricci_calculus
Electron microscopy technique
(2009) Burgers vector determination in deformed perovskite and post-perovskite of CaIrO3 using thickness fringes in weak-beam dark-field images, Ultramicroscopy
Weak-beam dark-field microscopy
Weak-beam_dark-field_microscopy
Pseudovector field describing the local rotation of a continuum near some point
In continuum mechanics, vorticity is a pseudovector (or axial vector) field that describes the local spinning motion of a continuum near some point (the
Vorticity
Non-tensorial representation of the spin group
to rotations: briefly, spinors respond to rotations in a path-dependent way, while vectors respond without seeing the path through which a rotation was
Spinor
Electromagnetic phenomenon
dipole moment, a vector quantity. Electric dipoles produce an electric field and experience forces and torques in an electric field that are proportional
Dipole
Extension of quantum field theory to curved spacetime
can be created by time-dependent gravitational fields (multigraviton pair production), or by time-independent gravitational fields that contain horizons
Quantum field theory in curved spacetime
Quantum_field_theory_in_curved_spacetime
Representation of a tensor in Euclidean space
a second order tensor field, again dependent on the position vector r and time t. For instance, the gradient of a vector field in two equivalent notations
Cartesian_tensor
Euclidean space without distance and angles
point, the zero vector is called the origin. Adding a fixed vector to the elements of a linear subspace (vector subspace) of a vector space produces an
Affine_space
Theory in neuropsychology
the input vector is calculated to continuous values with differential equations and is thus dependent on the length of time the input vector is presented
Adaptive_resonance_theory
Dynamical system whose system function is not directly dependent on time
a class of systems in the field of system analysis. The time-dependent system function is a function of the time-dependent input function. If this function
Time-invariant_system
Statistical modeling method
regression model assumes that the relationship between the dependent variable y and the vector of regressors x is linear. This relationship is modeled through
Linear_regression
Mathematical measure of how much a curve or surface deviates from flatness
where the curvature represents a field and a vector potential for the field is a quantity that is in general path-dependent: it may change if an observer
Curvature
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