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...

represented as a vector-valued function that associates an n-tuple of real numbers to each point of the domain. This representation of a vector field depends...

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...

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

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

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...

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

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...

the inverse function. In multivariable calculus, this theorem can be generalized to any continuously differentiable, vector-valued function whose Jacobian...

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...

(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...

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

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...

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...

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

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...

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...

^{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...

natural sciences, a vector quantity (also known as a vector physical quantity, physical vector, or simply vector) is a vector-valued physical quantity....

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...

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

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...

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...

Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional...

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...

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

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...

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

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

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...