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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
Multivariate derivative (mathematics)
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) ∇f whose
Gradient
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
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)
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
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)
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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 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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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)
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
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
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
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)
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
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
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
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
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
Kind of mathematical function
measurable functions as exclusively real-valued ones with respect to the Borel algebra. If the values of the function lie in an infinite-dimensional vector space
Measurable_function
Function that is continuous everywhere but differentiable nowhere
nowhere-differentiable real-valued functions on [0, 1] is comeager in the vector space C([0, 1]; ℝ) of all continuous real-valued functions on [0, 1] with the
Weierstrass function (nowhere-differentiable function)
Weierstrass_function_(nowhere-differentiable_function)
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
Mathematical function
bijective proper map followed by a coercive map is coercive. An (extended valued) function f : R n → R ∪ { − ∞ , + ∞ } {\displaystyle f:\mathbb {R} ^{n}\to \mathbb
Coercive_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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