Vector function

A function $\mathbf r ( t)$ of an argument $t$ whose values belong to a vector space $V$.

A vector function with values in a finite-dimensional ( $m$- dimensional) vector space $V$ is completely determined by its components $r _ {j} ( t)$, $1 \leq j \leq m$, with respect to some basis $e _ {1} \dots e _ {m}$ of $V$:

$$\tag{1 } \mathbf r ( t) = \ \sum_{j=1} ^ { m } r _ {j} ( t) \mathbf e _ {j} .$$

A vector function is said to be continuous, differentiable, etc. (at a point or in a domain) if all functions $r _ {j} ( t)$ are continuous, differentiable, etc. The following formulas are valid for a function $\mathbf r ( t)$ of one variable:

$$\tag{2 } \frac{d}{dt} \mathbf r ( t) = \ \lim\limits _ {h \rightarrow 0 } \frac{\mathbf r ( t+ h)- \mathbf r ( t) }{h} = \ \sum_{j=1}^ { m } r _ {j} ^ \prime ( t ) \mathbf e _ {j} ,$$

$$\tag{3 } \int\limits _ { t _ {0} } ^ { {t _ 1 } } \mathbf r ( t) dt = \sum_{j=1}^ { m } \left ( \int\limits _ { t _ {0} } ^ { {t _ 1 } } r _ {j} ( t) dt \right ) \mathbf e _ {j} ,$$

$$\mathbf r ( t) = \mathbf r ( t _ {0} ) + \sum_{k=1}^ { N } \frac{1}{k!} \mathbf r ^ {(} k) ( t _ {0} ) ( t- t _ {0} ) ^ {k} +$$

$$+ \frac{1}{N!} \int\limits _ {t _ {0} } ^ { t } ( t- \tau ) ^ {N} {\mathbf r } ^ {(} N+ 1) ( \tau ) d \tau$$

(Taylor's formula).

The set of vectors $\mathbf r ( t)$( starting at zero in $V$) is called the hodograph of the vector function. The first derivative $\dot{\mathbf r} ( t)$ of a vector function of one real variable is a vector in $V$ tangent to the hodograph at the point $\mathbf r ( t)$. If $\mathbf r ( t)$ describes the motion of a point mass, where $t$ denotes the time, then $\dot{\mathbf r} ( t)$ is the instantaneous velocity vector of the point at the time $t$. The second derivative $\dot{\mathbf r} dot ( t)$ 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.

References

[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)

References

[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