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Differential operator used in vector calculus
A vector operator is a differential operator used in vector calculus. Vector operators include: Gradient is a vector operator that operates on a scalar
Vector_operator
Circulation density in a vector field
In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional
Curl_(mathematics)
Vector differential operator
Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by ∇ (the nabla
Del
Differential operator in mathematics
The vector Laplace operator, also denoted by ∇ 2 {\displaystyle \nabla ^{2}} , is a differential operator defined over a vector field. The vector Laplacian
Laplace_operator
Vector operator in vector calculus
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the rate that the vector field alters
Divergence
Mathematical function, in linear algebra
mapping, or linear operator is a particular kind of function between vector spaces, which respects the basic operations of vector addition and scalar
Linear_map
Tensor operator generalizes the notion of operators which are scalars and vectors
a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply
Tensor_operator
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
Broad concept generalizing scalars in mathematics and physics
operator defined over a vector field Vector notation, common notation used when working with vectors Vector operator, a type of differential operator
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Function acting on function spaces
(physics) for other examples) The most basic operators are linear maps, which act on vector spaces. Linear operators refer to linear maps whose domain and range
Operator_(mathematics)
Exterior algebraic map taking tensors from p forms to n-p forms
mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a
Hodge_star_operator
Measure of the "size" of linear operators
defined on the space of bounded linear operators between two given normed vector spaces. Informally, the operator norm ‖ T ‖ {\displaystyle \|T\|} of a
Operator_norm
Calculus of vector-valued functions
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional
Vector_calculus
Operator generalizing the Laplacian in differential geometry
Euclidean space Rn, the Laplace operator (also known as the Laplacian) takes f to the divergence of its gradient vector field, which is the sum of the
Laplace–Beltrami_operator
Use of coordinates for representing vectors
Vector notation In mathematics and physics, vector notation is a commonly used notation for representing vectors, which may be Euclidean vectors, or more
Vector_notation
Kind of linear transformation
bounded. The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces. Outside of functional analysis
Bounded_operator
Mathematical operation on vectors in 3D space
product vector. As the cross product operator depends on the orientation of the space, in general the cross product of two vectors is not a "true" vector, but
Cross_product
Number of vectors in any basis of the vector space
in the ring. The dimension of a vector space may alternatively be characterized as the trace of the identity operator. For instance, tr id R 2 = tr
Dimension_(vector_space)
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
Algebraic structure in linear algebra
operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. Real vector spaces and complex vector spaces
Vector_space
Function acting on the space of physical states in physics
defined by the unit vector n ^ {\displaystyle {\hat {\boldsymbol {n}}}} and angle θ. If the transformation is infinitesimal, the operator action should be
Operator_(physics)
Operator shifting particles and fields by a certain amount in a certain direction
It is a special case of the shift operator from functional analysis. More specifically, for any displacement vector x {\displaystyle \mathbf {x} } , there
Translation operator (quantum mechanics)
Translation_operator_(quantum_mechanics)
Algebraic operation on coordinate vectors
numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the scalar product of two vectors is the dot product of their
Dot_product
Quantum mechanical operator related to rotational symmetry
constant. The orbital angular momentum operator is a vector operator, meaning it can be written in terms of its vector components L = ( L x , L y , L z )
Angular_momentum_operator
Theorem used in quantum mechanics for angular momentum calculations
is a theorem that tells how vector operators behave in a subspace. Within a given subspace, a component of a vector operator will behave in a way proportional
Wigner–Eckart_theorem
Discrete differentiation operator used in image processing
the result of the Prewitt operator is either the corresponding gradient vector or the norm of this vector. The Prewitt operator is based on convolving the
Prewitt_operator
Image edge detection algorithm
of the Sobel–Feldman operator is either the corresponding gradient vector or the norm of this vector. The Sobel–Feldman operator is based on convolving
Sobel_operator
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
Notation for quantum states
notation is a mathematical notation for linear algebra and linear operators on complex vector spaces together with their dual spaces both in the finite- and
Bra–ket_notation
Topics referred to by the same term
dictionary. Curl or CURL may refer to: Curl (mathematics), a vector operator that shows a vector field's rate of rotation Curl (programming language), an
Curl
Concept in linear algebra
The vector projection, also known as the vector component or vector resolution, of a vector a on (or onto) a non-zero vector b is the orthogonal projection
Vector_projection
Linear operator related to topological vector spaces
two topological vector spaces (TVSs). Throughout let X,Y, and Z be topological vector spaces (TVSs) and L : X → Y be a linear operator (no assumption of
Nuclear_operator
Vector used in astronomy
In classical mechanics, the Laplace–Runge–Lenz vector (LRL vector) is a vector used chiefly to describe the shape and orientation of the orbit of one
Laplace–Runge–Lenz_vector
Mathematical identities
following are important identities involving derivatives and integrals in vector calculus. For a function f ( x , y , z ) {\displaystyle f(x,y,z)} in three-dimensional
Vector_calculus_identities
Idempotent linear transformation from a vector space to itself
projection on points in the object. A projection on a vector space V {\displaystyle V} is a linear operator P : V → V {\displaystyle P\colon V\to V} such that
Projection_(linear_algebra)
Topics referred to by the same term
Curl (mathematics), known as rotor in some countries, a vector operator that shows a vector field's rate of rotation SC Rotor Volgograd, a Russian football
Rotor
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
Surjective bounded operator on a Hilbert space preserving the inner product
a unitary operator. Rotations in R2 are the simplest nontrivial example of unitary operators. Rotations do not change the length of a vector or the angle
Unitary_operator
Norm on a vector space of matrices
linear operator; then a matrix norm may describe how much the operator can stretch vectors. Such matrix norms induced by vector norms are called operator norms
Matrix_norm
Instantaneous rate of change of the function
instantaneous rate at which a function changes along a specified vector through a given point. If the vector is multiplied by a scalar, the corresponding directional
Directional_derivative
Algebra associated to any vector space
In mathematics, the exterior algebra or Grassmann algebra of a vector space V {\displaystyle V} is an associative algebra that contains V , {\displaystyle
Exterior_algebra
Mathematical operator
closure operator. Other examples include the linear span of a subset of a vector space, the convex hull or affine hull of a subset of a vector space or
Closure_operator
Software design pattern
Vector::operator[] out of range!"); } return elements[n]; } DoubleVector(const DoubleVector&) = delete; // disable copy construction DoubleVector& operator=(const
Iterator_pattern
Linear operator in mathematics
mathematics, the composition operator C ϕ {\displaystyle C_{\phi }} with symbol ϕ {\displaystyle \phi } is a linear operator defined by the rule C ϕ ( f
Composition_operator
Model of logic based on matrix algebra
executed by matrix operators. "Vector logic" has also been used to refer to the representation of classical propositional logic as a vector space, in which
Vector_logic
Function between topological vector spaces
continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed
Continuous_linear_operator
Topics referred to by the same term
village in the Älvdalen Municipality, Sweden Rot (mathematics), rotation vector operator Brain rot, slang for poor-quality digital content ROT (disambiguation)
Rot
Typically linear operator defined in terms of differentiation of functions
cotangent bundle). More generally, let E and F be vector bundles over a manifold X. Then the linear operator P : C ∞ ( E ) → C ∞ ( F ) {\displaystyle P:C^{\infty
Differential_operator
Result about when a matrix can be diagonalized
relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces
Spectral_theorem
matrix Hodge star operator Inverse function theorem Irrotational vector field Isoperimetry Jacobian matrix Lagrange multiplier Lamellar vector field Laplacian
List of multivariable calculus topics
List_of_multivariable_calculus_topics
Matrix whose only nonzero elements are on its main diagonal
the Hadamard product, and 1 is a constant vector with elements 1. The inverse matrix-to-vector diag operator is sometimes denoted by the identically named
Diagonal_matrix
Set of vectors used to define coordinates
In mathematics, a set B of elements of a vector space V is called a basis (pl.: bases) if every element of V can be written in a unique way as a finite
Basis_(linear_algebra)
Matrices important in quantum mechanics and the study of spin
Taking the dot product of any unit vector with the above formula generates the expression of any single qubit operator under any rotation. For example,
Pauli_matrices
Compact operator for which a finite trace can be defined
the special case of nuclear operators on Hilbert spaces and use the term "nuclear operator" in more general topological vector spaces (such as Banach spaces)
Trace_class
Description of a quantum-mechanical system
presented using quantities varying as functions of position, but as a vector-operator equation it has a valid representation in any arbitrary complete basis
Schrödinger_equation
Vector space with a notion of nearness
Sobolev spaces. Many topological vector spaces are spaces of functions, or linear operators acting on topological vector spaces, and the topology is often
Topological_vector_space
analysis, a hypercyclic operator on a topological vector space X is a continuous linear operator T: X → X such that there is a vector x ∈ X for which the
Hypercyclic_operator
Elliptic differential operators in geometry mathematics
{\displaystyle \Delta =\nabla ^{*}\nabla } which is a second order operator acting on sections of the vector bundle E. Note that the connection Laplacian and Bochner
Laplace operators in differential geometry
Laplace_operators_in_differential_geometry
Motion of a certain space that preserves at least one point
vector rotation presents many equivalent rotations about all points in the space. A motion that preserves the origin is the same as a linear operator
Rotation_(mathematics)
Branch of functional analysis
analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given
Operator_algebra
Algebra used in 2D conformal field theories and string theory
vertex operator algebra. It is "generated" by a single vector b, in the sense that by applying the coefficients of the field b(z) := Y(b,z) to the vector 1
Vertex_operator_algebra
Linear operator whose graph is closed
Definition: If X and Y are topological vector spaces (TVSs) then we call a linear map f : D(f) ⊆ X → Y a closed linear operator if its graph is closed in X × Y
Closed_linear_operator
Antisymmetric permutation object acting on tensors
product expression above, substituting components of the gradient vector operator (nabla). In any arbitrary curvilinear coordinate system and even in
Levi-Civita_symbol
In physics, a linear operator acting on a vector space of linear operators
In physics, a superoperator is a linear operator acting on a vector space of linear operators. Sometimes the term refers more specially to a completely
Superoperator
Algebraic object with geometric applications
transformation law for a contravariant vector, so that the action of a linear operator on a contravariant vector is represented in coordinates as the matrix
Tensor
Property of space that quantifies the magnetic influence at a given location
and can be calculated from the vector field they create using a well-understood vector operator. The divergence of a vector field A, ∇ · A is defined such
Magnetic_field
operator theory, an operator A {\displaystyle A} on an (infinite-dimensional) Banach space or Hilbert space H {\displaystyle H} has a cyclic vector f
Cyclic_vector
Topics referred to by the same term
on a tuple Bitwise rotation, a mathematical operator on bit patterns Curl (mathematics), a vector operator Differential rotation, objects rotating at different
Rotation_(disambiguation)
Special mathematical functions defined on the surface of a sphere
definition is that if the vector r {\displaystyle \mathbf {r} } is replaced by the quantum mechanical spin vector operator J {\displaystyle \mathbf {J}
Spherical_harmonics
Vectors mapped to 0 by a linear map
necessarily apply. If V and W are topological vector spaces such that W is finite-dimensional, then a linear operator L: V → W is continuous if and only if the
Kernel_(linear_algebra)
Linear operator acting on modular forms
Hecke operator, studied by Erich Hecke (1937a,1937b), is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces
Hecke_operator
Symbol used to indicate the del operator
mathematics of the vector differential operator Del in cylindrical and spherical coordinates Dirac operator grad, div, and curl, differential operators defined using
Nabla_symbol
Mathematical notation
linear transformation, it can be represented as a matrix. The hat operator takes a vector and transforms it into its equivalent matrix. a × b = a ^ b {\displaystyle
Hat_notation
Complex vector bundle on a complex manifold
{\mathcal {O}}(k)|_{X}} . Suppose E is a holomorphic vector bundle. Then there is a distinguished operator ∂ ¯ E {\displaystyle {\bar {\partial }}_{E}} defined
Holomorphic_vector_bundle
Matrix equal to its conjugate-transpose
only if it is equal to its conjugate transpose, that is, for any pair of vectors v , w {\displaystyle \mathbf {v} ,\mathbf {w} } , it satisfies ⟨ v
Hermitian_matrix
Subspace preserved by a linear mapping
every linear operator on a nonzero finite-dimensional complex vector space has an eigenvector. Therefore, every such linear operator in at least two
Invariant_subspace
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
Joining of strings in a programming language
from vector addition, depending on the language. String literal concatenation, in which adjacent strings are concatenated without any operator. Example
Concatenation
Bounded operators with sub-unit norm
In operator theory, a bounded operator T: X → Y between normed vector spaces X and Y is said to be a contraction if its operator norm ||T || ≤ 1. Every
Contraction_(operator_theory)
Concept in linear algebra
represent the spin operator when transforming the spin eigenstates into vector coordinates. Let B and C be two different bases of a vector space V, and let
Coordinate_vector
1865 physics paper by James Maxwell
what is now called the resistivity). ∇ {\displaystyle \nabla } is the vector operator del. Clarifications Maxwell did not consider completely general materials;
A Dynamical Theory of the Electromagnetic Field
A_Dynamical_Theory_of_the_Electromagnetic_Field
Mathematical function
the locally convex topological vector spaces (TVSs) X and Y. An integral linear operator is a continuous linear operator that arises in a canonical way
Integral_linear_operator
Ternary operation on vectors
algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products
Triple_product
Set of rules defining correctly structured programs
scan operators expect a dyadic function on their left, forming a monadic composite function applied to the vector on its right. The product operator "."
APL_syntax_and_symbols
Length in a vector space
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance
Norm_(mathematics)
Mathematical method in functional analysis
of graphs Continuous linear operator – Function between topological vector spaces Densely defined operator – Linear operator on dense subset of its apparent
Continuous_linear_extension
Mathematical result in differential geometry
local charts.) More generally, the symbol of a differential operator between two vector bundles E and F is a section of the pullback of the bundle Hom(E
Atiyah–Singer_index_theorem
Vector calculus formulas relating the bulk with the boundary of a region
set of three identities in vector calculus relating the bulk with the boundary of a region on which differential operators act. They are named after the
Green's_identities
Mathematical study of linear operators
underlying vector space on which the operator acts. A normal operator on a complex Hilbert space H {\displaystyle H} is a continuous linear operator N : H
Operator_theory
Operator on a Hilbert space that shifts basis vectors
In operator theory, the unilateral shift is a one-sided shift operator, that is, a shift operator acting on one-sided sequences or shift spaces. The term
Unilateral_shift_operator
Vector behavior under coordinate changes
It lives in the dual vector space, and represents a linear map from vectors to scalars. The dot product operator involving vectors is a good example of
Covariance and contravariance of vectors
Covariance_and_contravariance_of_vectors
Equation from stability analysis
equation the Bartels–Stewart algorithm can be used. Defining the vectorization operator vec ( A ) {\displaystyle \operatorname {vec} (A)} as stacking
Lyapunov_equation
Irreducible representation of the rotation group SO
SU(2) and SO(3). In quantum mechanics, these three operators are the components of a vector operator known as angular momentum. Examples are the angular
Wigner_D-matrix
Locally convex topology on function spaces
\epsilon )=\{T:\|Tx-T_{0}x\|<\epsilon \}} (where T0 is any bounded operator on H, x is any vector and ε is any positive real number). In concrete terms, this
Strong_operator_topology
Type of vector space in math
Locally convex topological vector space – Space with topology generated by convex sets Operator topologies – Topologies on operators on a Hilbert space Rigged
Hilbert_space
Generalization of finite-dimensional Euclidean spaces different from Hilbert spaces
In mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite-dimensional Euclidean spaces and share many
Nuclear_space
Canadian survey ship built in 1967
CCGS Vector is a hydrographic survey vessel in the Canadian Coast Guard. The ship was constructed in Canada and entered service in 1967 as a coastal research
CCGS_Vector
functional analysis, a discipline within mathematics, an operator space is a normed vector space (not necessarily a Banach space) "given together with
Operator_space
G-factor for electron with spin and orbital angular momentum
total magnetic moment μ → J {\displaystyle {\vec {\mu }}_{J}} , as a vector operator, does not lie on the direction of total angular momentum J → = L →
Landé_g-factor
Raising and lowering operators in quantum mechanics
momentum. For a general angular momentum vector J with components Jx, Jy and Jz one defines the two ladder operators J + = J x + i J y , J − = J x − i J y
Ladder_operator
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