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A function  $  \mathbf r ( t) $
of an argument  $  t $
whose values belong to a [[Vector space|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|Vector analysis]]; [[Gradient|Gradient]]; [[Divergence|Divergence]]; [[Curl|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.E. Kochin,   "Vector calculus and fundamentals of tensor calculus" , Moscow  (1965)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.K. Rashevskii,   "A course of differential geometry" , Moscow  (1956)  (In Russian)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Courant,   F. John,   "Introduction to calculus and analysis" , '''1''' , Wiley (Interscience)  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.E. Marsden,   A.J. Tromba,   "Vector calculus" , Freeman  (1981)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.A. Dieudonné,   "Foundations of modern analysis" , Acad. Press  (1960)  (Translated from French)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Jeffrey,   "Mathematics for scientists and engineers" , v. Nostrand-Reinhold  (1989)  pp. 493ff</TD></TR></table>