Namespaces
Variants
Actions

Difference between revisions of "Vector function"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v0964501.png" /> of an argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v0964502.png" /> whose values belong to a [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v0964503.png" />.
+
<!--
 +
v0964501.png
 +
$#A+1 = 26 n = 0
 +
$#C+1 = 26 : ~/encyclopedia/old_files/data/V096/V.0906450 Vector function
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
A vector function with values in a finite-dimensional (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v0964504.png" />-dimensional) vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v0964505.png" /> is completely determined by its components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v0964506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v0964507.png" />, with respect to some basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v0964508.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v0964509.png" />:
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v09645010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
A function  $  \mathbf r ( t) $
 +
of an argument  $  t $
 +
whose values belong to a [[Vector space|vector space]]  $  V $.
  
A vector function is said to be continuous, differentiable, etc. (at a point or in a domain) if all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v09645011.png" /> are continuous, differentiable, etc. The following formulas are valid for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v09645012.png" /> of one variable:
+
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 $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v09645013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{1 }
 +
\mathbf r ( t)  = \
 +
\sum _ { j= } 1 ^ { m }  r _ {j} ( t) \mathbf e _ {j} .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v09645014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v09645015.png" /></td> </tr></table>
+
$$ \tag{2 }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v09645016.png" /></td> </tr></table>
+
\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).
 
(Taylor's formula).
  
The set of vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v09645017.png" /> (starting at zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v09645018.png" />) is called the hodograph of the vector function. The first derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v09645019.png" /> of a vector function of one real variable is a vector in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v09645020.png" /> tangent to the hodograph at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v09645021.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v09645022.png" /> describes the motion of a point mass, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v09645023.png" /> denotes the time, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v09645024.png" /> is the instantaneous velocity vector of the point at the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v09645025.png" />. The second derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096450/v09645026.png" /> is the acceleration vector of the point.
+
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.
 
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.
Line 25: Line 79:
 
====References====
 
====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></table>
 
<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></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><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>
 
<table><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>

Revision as of 08:28, 6 June 2020


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)

Comments

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
How to Cite This Entry:
Vector function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_function&oldid=11242
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article