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...

position vectors discretizing a trajectory. A vector may also result from the evaluation, at a particular instant, of a continuous vector-valued function (e...

(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...

An infinite-dimensional vector function is a function whose values lie in an infinite-dimensional topological vector space, such as a Hilbert space or...

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...

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...

differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through...

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...

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...

example, the function concerned are finite-dimension vector-valued function. In this case, the limit theorem for vector-valued function states that if...

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...

In vector calculus, the gradient of a scalar-valued differentiable function f {\displaystyle f} of several variables is the vector field (or vector-valued...

(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...

real-valued function with only some of the properties of a seminorm. Unlike seminorms, a sublinear function does not have to be nonnegative-valued and...

^{n}} to R {\displaystyle \mathbb {R} } for which there exists a vector valued function η {\displaystyle \eta } such that f ( x ) − f ( u ) ≥ η ( x , u...

learning algorithms, these functions produce a scalar output. Recent development of kernel methods for functions with vector-valued output is due, at least...

member of its domain. Real-valued functions of a real variable (commonly called real functions) and real-valued functions of several real variables are...

independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector. A function of a real variable f ( x...

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...

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...

properties of complex-valued functions (such as continuity) are nothing more than the corresponding properties of vector valued functions of two real variables...

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...

massless bosonic fields in string theory.) Scalar field theory Vector boson Vector-valued function Apostol, Tom (1969). Calculus. Vol. II (2nd ed.). Wiley....

functions can be defined for vector- or matrix-valued random variables, and can also be extended to more generic cases. The characteristic function always...

analysis, a differentiable vector-valued function from Euclidean space is a differentiable function valued in a topological vector space (TVS) whose domains...

moment-generating function, evaluated at 0. In addition to univariate real-valued distributions, moment-generating functions can also be defined for vector- or matrix-valued...

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...

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...

simplest form of the chain rule is for real-valued functions of one real variable. It states that if g is a function that is differentiable at a point c (i...

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...