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Function valued in a vector space; typically a real or complex one
A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional
Vector-valued_function
Assignment of a vector to each point in a subset of Euclidean space
(which represents the rotation of a flow). A vector field is a special case of a vector-valued function, whose domain's dimension has no relation to the
Vector_field
Association of one output to each input
example, the function that associates to each point of a fluid its velocity vector is a vector-valued function. Some vector-valued functions are defined
Function_(mathematics)
Whose values lie in an infinite-dimensional vector space
An infinite-dimensional vector function is a function whose values lie in an infinite-dimensional topological vector space, such as a Hilbert space or
Infinite-dimensional vector function
Infinite-dimensional_vector_function
Point to which functions converge in analysis
example, the function concerned are finite-dimension vector-valued function. In this case, the limit theorem for vector-valued function states that if
Limit_of_a_function
Matrix of partial derivatives of a vector-valued function
In vector calculus, the Jacobian matrix (/dʒəˈkoʊbiən/, /dʒɪ-, jɪ-/) of a vector-valued function of several variables is the matrix of all its first-order
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Broad concept generalizing scalars in mathematics and physics
scientific disciplines, notably physics and engineering. Vector-valued functions, where the output is a vector, are scrutinized using calculus to derive essential
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Multivariate derivative (mathematics)
In vector calculus, the gradient ∇f of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) whose
Gradient
Theorem in mathematics
situations to which the mean value theorem is applicable in the one dimensional case: Theorem—For a continuous vector-valued function f : [ a , b ] → R k {\displaystyle
Mean_value_theorem
Matrix of second derivatives
second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix
Hessian_matrix
Mathematical function that outputs real values
member of its domain. Real-valued functions of a real variable (commonly called real functions) and real-valued functions of several real variables are
Real-valued_function
Point where function's value is zero
(also sometimes called a root) of a real-, complex-, or generally vector-valued function f {\displaystyle f} , is a member x {\displaystyle x} of the domain
Zero_of_a_function
Instantaneous rate of change (mathematics)
object's jerk. A vector-valued function 𝐲 of a real variable sends real numbers to vectors in some vector space ℝn. A vector-valued function can be split
Derivative
Instantaneous rate of change of the function
measures the 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_derivative
Branch of mathematics studying functions of a complex variable
{R} ^{2}\to \mathbb {R} .} A complex function is continuous if and only if its associated vector-valued function of two variables is also continuous.
Complex_analysis
Geometric object that has length and direction
length) and direction. Euclidean vectors can be added and scaled to form a vector space. A vector quantity is a vector-valued physical quantity, including
Euclidean_vector
Assignment of numbers to points in space
massless bosonic fields in string theory.) Scalar field theory Vector boson Vector-valued function Apostol, Tom (1969). Calculus. Vol. II (2nd ed.). Wiley.
Scalar_field
Derivative defined on normed spaces
generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to
Fréchet_derivative
Physical quantity that is a vector
natural sciences, a vector quantity (also known as a vector physical quantity, physical vector, or simply vector) is a vector-valued physical quantity.
Vector_quantity
Type of derivative in mathematics
property is generalized to define the derivative of a vector-valued function or function of a vector argument. Sometimes called the total derivative, in
Derivative (multivariable calculus)
Derivative_(multivariable_calculus)
Concept in probability theory and statistics
generating function, evaluated at 0. In addition to univariate real-valued distributions, moment generating functions can also be defined for vector- or matrix-valued
Moment_generating_function
Linear map or polynomial function of degree one
polynomial functions of degree 0 or 1 are the scalar-valued affine maps. In linear algebra, a linear function is a map f {\displaystyle f} from a vector space
Linear_function
Fourier transform of the probability density function
distributions, characteristic functions can be defined for multivariate probability distributions of vector- or matrix-valued random variables, and can also
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Theorem in mathematics
0 {\displaystyle a=b=0} . By the mean value theorem for vector-valued functions, for a differentiable function u : [ 0 , 1 ] → R m {\displaystyle u:[0
Inverse_function_theorem
Mathematical description of quantum state
the entries, and the wave function is a complex vector-valued function of space and time only. All values of the wave function, not only for discrete but
Wave_function
assumed that the functions are sufficiently smooth that derivatives can be taken. Let f(v) be a real valued function of the vector v. Then the derivative
Tensor derivative (continuum mechanics)
Tensor_derivative_(continuum_mechanics)
In vector calculus, an invex function is a differentiable function f:ℝn→ℝ for which there exists a vector-valued function η such that f(x)−f(u)≥η(x,u)⋅∇f(u)
Invex_function
Mathematics concept
is a generalization of the notion of monotonicity to the case of vector-valued function. Let ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } denote
Cyclical_monotonicity
Generalization of finite measure to Banach spaces
In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization
Vector_measure
Concept in differential topology
In mathematics, a vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a
Vector-valued differential form
Vector-valued_differential_form
that an elastic body experiences when it is deformed by a given vector-valued function. The inequality is therefore an important tool as an a priori estimate
Korn's_inequality
Mathematical function
real-valued functions, the restriction of the codomain to the real numbers, and allowing complex values. If f(x) is such a complex valued function, it
Function_of_a_real_variable
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)
Function computable with bounded loops
functions are the basic functions and those obtained from the basic functions by applying these operations a finite number of times. A (vector-valued)
Primitive_recursive_function
Differentiable function in functional analysis
analysis, a differentiable vector-valued function from Euclidean space is a differentiable function valued in a topological vector space (TVS) whose domains
Differentiable vector-valued functions from Euclidean space
Differentiable_vector-valued_functions_from_Euclidean_space
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
Initial result in using test functions to find extremum
coordinate separately, or treats the vector-valued case from the beginning. If a continuous multivariable function f on an open set Ω ⊂ R d {\displaystyle
Fundamental lemma of the calculus of variations
Fundamental_lemma_of_the_calculus_of_variations
Function with a multiplicative scaling behaviour
whose domain and codomain are vector spaces over a field F: a function f : V → W {\displaystyle f:V\to W} between two F-vector spaces is homogeneous of degree
Homogeneous_function
Inequality between integrals in Lp spaces
\infty ]} with 1/p + 1/q = 1. Then for all measurable real- or complex-valued functions f and g on S, ‖ f g ‖ 1 ≤ ‖ f ‖ p ‖ g ‖ q . {\displaystyle \|fg\|_{1}\leq
Hölder's_inequality
Derivative of a function with multiple variables
In this case, it is said that f is a C1 function. This can be used to generalize for vector valued functions, f:U→ℝm, by carefully using a component-wise
Partial_derivative
Function returning one of only two values
1\}^{k}\to \{0,1\}^{m}} with m > 1 {\displaystyle m>1} is a vectorial or vector-valued Boolean function (an S-box in symmetric cryptography). There are 2 2 k
Boolean_function
Holomorphic function: complex-valued function of a complex variable which is differentiable at every point in its domain. Meromorphic function: complex-valued function
List_of_types_of_functions
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
learning algorithms, these functions produce a scalar output. Recent development of kernel methods for functions with vector-valued output is due, at least
Kernel methods for vector output
Kernel_methods_for_vector_output
Representation of a curve by a function of a parameter
length Parametric derivative Parametric estimating Position vector Vector-valued function Weisstein, Eric W. "Parametric Equations". MathWorld. Kreyszig
Parametric_equation
Distance from zero to a number
absolute value for real numbers can be used, with a slight modification, to generalise the notion to an arbitrary vector space. A real-valued function on a
Absolute_value
Mathematical function with multiple real-number arguments
another real number, the value of the function, commonly denoted f(x1, x2, …, xn). For simplicity, in this article a real-valued function of several real variables
Function of several real variables
Function_of_several_real_variables
Degree of differentiability of a function or map
manifolds. For complex-valued functions, one may still speak of Ck or C∞ smoothness by regarding the function as a map between real vector spaces. This should
Smoothness
Concept in mathematics
into more abstract spaces, vector-valued functions, and operator spaces. Examples of such extensions include vector-valued Laplace transforms and abstract
Bochner_integral
Theorem of convex functions
in Perlman, Michael D. (1974). "Jensen's Inequality for a Convex Vector-Valued Function on an Infinite-Dimensional Space". Journal of Multivariate Analysis
Jensen's_inequality
This is a list of some well-known periodic functions. The constant function f (x) = c, where c is independent of x, is periodic with any period, but lacks
List_of_periodic_functions
Mathematical function defined piecewise by polynomials
{\displaystyle f(t_{i}^{})} are the values of the function at the ith knot. For a given interval [a,b] and a given extended knot vector on that interval, the splines
Spline_(mathematics)
Functions in mathematics
the study of cohomology. Also, it is possible to define harmonic vector-valued functions, or harmonic maps of two Riemannian manifolds, which are critical
Harmonic_function
Functions such that f(–x) equals f(x) or –f(x)
and all vector spaces. Thus, for example, a real function could be odd or even (or neither), as could a complex-valued function of a vector variable
Even_and_odd_functions
Extension of the scalar spherical harmonics for use with vector fields
fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors. Several conventions have been used to define
Vector_spherical_harmonics
Fundamental construction of differential calculus
denotes the function t↦f′(x)⋅t. In multivariable calculus, in the context of differential equations defined by a vector valued function Rn to Rm, the
Generalizations of the derivative
Generalizations_of_the_derivative
Type of function in linear algebra
sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm, on a vector space is a real-valued function with
Sublinear_function
Real function with secant line between points above the graph itself
mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the
Convex_function
Solving multiple machine learning tasks at the same time
within the context of RKHSvv (a complete inner product space of vector-valued functions equipped with a reproducing kernel). In particular, recent focus
Multi-task_learning
S-shaped curve
immediately generalizes to more alternatives as the softmax function, which is a vector-valued function whose i-th coordinate is e x i / ∑ i = 0 n e x i {\textstyle
Logistic_function
Mathematical measure of how much a curve or surface deviates from flatness
associated with increasing parameter values. A curve that is parametrized by arc length is a vector-valued function that is denoted by the Greek letter
Curvature
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
Particular representation of a signal
complex-valued function that has no negative frequency components. The real and imaginary parts of an analytic signal are real-valued functions related
Analytic_signal
Mathematical parametrization of vector spaces by another space
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space
Vector_bundle
Process by which a quantum system takes on a definitive state
interpretations of quantum mechanics, wave function collapse, also called reduction of the state vector, occurs when a wave function—initially in a superposition of
Wave_function_collapse
Vector field that is the gradient of some function
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property
Conservative_vector_field
Gradient of the likelihood function
gradient of the log-likelihood function with respect to the parameter vector. Evaluated at a particular value of the parameter vector, the score indicates the
Informant_function
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
Normed vector space that is complete
respective extensions of the map sending the tensor f⊗y to the vector-valued function s∈K→f(s)y∈Y. Let X be a Banach space. The tensor product X′⊗^εX
Banach_space
Mathematical identities
and integrals in vector calculus. For a function f(x,y,z) in three-dimensional Cartesian coordinate variables, the gradient is the vector field: grad(f)=∇f=(∂∂x
Vector_calculus_identities
Mathematical relation consisting of a multi-variable function equal to zero
f(x) involving the multi-valued implicit function f. Not every equation R(x, y) = 0 implies a graph of a single-valued function, the circle equation being
Implicit_function
Quadratic form related to curvatures of surfaces
regular parametrization of a surface in R3, where r is a smooth vector-valued function of two variables. It is common to denote the partial derivatives
Second_fundamental_form
Approximation of a function by a polynomial
physics. Taylor's theorem also generalizes to multivariate and vector valued functions. It provided the mathematical basis for some landmark early computing
Taylor's_theorem
Mathematical function, in linear algebra
linear operator is a particular kind of function between vector spaces, which respects the basic operations of vector addition and scalar multiplication.
Linear_map
Vector operator in vector calculus
vector fields F and G and all real numbers a and b. There is a product rule of the following type: if φ is a scalar-valued function and F is a vector
Divergence
Similarity measure for number sequences
can then be used to define angular similarity function bounded between 0 and 1, inclusive. When the vector elements may be positive or negative: angular
Cosine_similarity
Map from a vector space to its underlying field
continuous real-valued functions over the interval [a, b], as it linearly maps each function to a real number. In contrast, the norm of a vector on a normed
Functional_(mathematics)
Definition of integral for regulated functions
closed, bounded interval in the real line R. A real-valued function φ : [a, b] → R is called a step function if there exists a finite partition Π = { a = t
Regulated_integral
In functional analysis, a Hilbert space
provide the framework for real-valued Hilbert spaces. The theory can be easily extended to spaces of complex-valued functions and hence include the many important
Reproducing kernel Hilbert space
Reproducing_kernel_Hilbert_space
Defines a notion of parallel transport on a bundle
made there apply to all vector bundles). Let M be a differentiable manifold, such as Euclidean space. A vector-valued function M → R n {\displaystyle M\to
Connection_(vector_bundle)
Integration over a non-flat region in 3D space
(that is, a function of position which returns a scalar as a value), or a vector field (that is, a function which returns a vector as value). If a region
Surface_integral
Space curve that winds around a line
helix has constant non-zero curvature and torsion. A helix is the vector-valued function r = a cos t i + a sin t j + b t k v = − a sin t i + a cos
Helix
Statistical model
constraints into Gaussian processes already exists: Consider the (vector valued) output function f(x) which is known to obey the linear constraint (i.e. ℱX is
Gaussian_process
Smooth approximation of one-hot arg max
one-hot version of the arg max function: the function whose vector component values are all zero, except that it has a value of 1 corresponding to the location
Softmax_function
Physical field surrounding an electric charge
electric field at point 𝐫0 due to the point charge q1; it is a vector-valued function equal to the Coulomb force per unit charge that a positive point
Electric_field
Form of the utility function
economics and finance, exponential utility is a specific form of the utility function, used in some contexts because of its convenience when risk (sometimes
Exponential_utility
Vector space with a notion of nearness
multiplication) are also continuous functions. Such a topology is called a vector topology and every topological vector space has a uniform topological structure
Topological_vector_space
Circulation density in a vector field
vector of a function F at a point is explicitly as the limiting value of a vector-valued surface integral around a shell enclosing p divided by the volume
Curl_(mathematics)
Coordinates comprising a distance and two angles
Jacobian matrix and determinant – Matrix of partial derivatives of a vector-valued function List of canonical coordinate transformations Sphere – Set of points
Spherical_coordinate_system
Mathematical function whose derivative exists
or complex function of a single variable is differentiable if its derivative exists at each point in its domain. For real-valued functions of a real variable
Differentiable_function
Matrix decomposition
case is similar; there f(𝐱)=𝐱∗𝐌𝐱 is a real-valued function of 2n real variables. Singular values are similar in that they can be described algebraically
Singular_value_decomposition
Theorem in optimal transport
Yann (1991). "Polar factorization and monotone rearrangement of vector-valued functions". Communications on Pure and Applied Mathematics. 44 (4): 375–417
Brenier's_theorem
Distance from origin of tangent hyperplanes
A\}.} The interpretation of the support function is most intuitive when x {\displaystyle x} is a unit vector. By definition, the convex set A {\displaystyle
Support_function
American mathematician
known for his work in functional analysis, namely integration of vector valued functions, ergodic theory, and linear operators. The Dunford decomposition
Nelson_Dunford
James Pettis, extends the definition of the Lebesgue integral to vector-valued functions on a measure space, by exploiting duality. The integral was introduced
Pettis_integral
In mathematics, invariant of square matrices
In mathematics, the determinant is a scalar-valued function of the entries of a square matrix that has many properties which make it fundamental for the
Determinant
Vector-valued function of multiple vectors, linear in each argument
is a linear function of v i {\displaystyle v_{i}} . One way to visualize this is to imagine two orthogonal vectors; if one of these vectors is scaled by
Multilinear_map
A measurable function on a probability space is usually referred to as a random variable (or random vector if it takes values in a vector space such as
Weakly_measurable_function
Term in mathematics
trajectories or orbits. Suppose that F is a static vector field, that is, a vector-valued function with components (F1,F2,...,Fn) in a Cartesian coordinate
Integral_curve
Mathematical function with convex lower level sets
In mathematics, a quasiconvex function is a real-valued function defined on a convex subset of a real vector space, such that for any real number y, the
Quasiconvex_function
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VECTOR VALUED-FUNCTION
VECTOR VALUED-FUNCTION
VECTOR VALUED-FUNCTION
VECTOR VALUED-FUNCTION
VECTOR VALUED-FUNCTION
VECTOR VALUED-FUNCTION
VECTOR VALUED-FUNCTION
VECTOR VALUED-FUNCTION
VECTOR VALUED-FUNCTION
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