Vector function

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A function of an argument whose values belong to a vector space .

A vector function with values in a finite-dimensional (-dimensional) vector space is completely determined by its components , , with respect to some basis of :


A vector function is said to be continuous, differentiable, etc. (at a point or in a domain) if all functions are continuous, differentiable, etc. The following formulas are valid for a function of one variable:


(Taylor's formula).

The set of vectors (starting at zero in ) is called the hodograph of the vector function. The first derivative of a vector function of one real variable is a vector in tangent to the hodograph at the point . If describes the motion of a point mass, where denotes the time, then is the instantaneous velocity vector of the point at the time . The second derivative is the acceleration vector of the point.

Partial derivatives and multiple integrals of vector functions of several variables are defined in analogy with formulas (2) and (3). See Vector analysis; Gradient; Divergence; Curl, for the concepts of vector analysis for vector functions.

In an infinite-dimensional normed vector space with a basis, the representation of a vector function in the form (1) is an infinite series, and a coordinate-wise definition of the operations of mathematical analysis involves difficulties connected with the concepts of convergence of series, the possibility of term-by-term differentiation and integration, etc.


[1] N.E. Kochin, "Vector calculus and fundamentals of tensor calculus" , Moscow (1965) (In Russian)
[2] P.K. Rashevskii, "A course of differential geometry" , Moscow (1956) (In Russian)



[a1] R. Courant, F. John, "Introduction to calculus and analysis" , 1 , Wiley (Interscience) (1965)
[a2] J.E. Marsden, A.J. Tromba, "Vector calculus" , Freeman (1981)
[a3] J.A. Dieudonné, "Foundations of modern analysis" , Acad. Press (1960) (Translated from French)
[a4] A. Jeffrey, "Mathematics for scientists and engineers" , v. Nostrand-Reinhold (1989) pp. 493ff
How to Cite This Entry:
Vector function. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article