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)
[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