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An element of the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c0268901.png" /> dual to an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c0268902.png" />-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c0268903.png" />, that is, a linear functional (linear form) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c0268904.png" />. In the ordered pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c0268905.png" />, an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c0268906.png" /> is called a [[Contravariant vector|contravariant vector]]. Within the general scheme for the construction of tensors, a covariant vector is identified with a covariant tensor of valency 1.
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The coordinate notation for a covariant vector is particularly simple if one chooses in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c0268907.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c0268908.png" /> so-called dual bases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c0268909.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689012.png" />, that is, bases such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689013.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689014.png" /> is the [[Kronecker symbol|Kronecker symbol]]); an arbitrary covariant vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689015.png" /> is then expressible in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689016.png" /> (summation over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689017.png" /> from 1 to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689018.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689019.png" /> is the value of the linear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689020.png" /> at the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689021.png" />. On passing from dual bases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689023.png" /> to dual bases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689025.png" /> according to the formulas
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<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/c/c026/c026890/c02689026.png" /></td> </tr></table>
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An element of the vector space  $  E  ^ {*} $
 +
dual to an  $  n $-
 +
dimensional vector space  $  E $,
 +
that is, a linear functional (linear form) on  $  E $.  
 +
In the ordered pair  $  ( E, E  ^ {*} ) $,
 +
an element of  $  E $
 +
is called a [[Contravariant vector|contravariant vector]]. Within the general scheme for the construction of tensors, a covariant vector is identified with a covariant tensor of valency 1.
  
the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689027.png" /> of the contravariant vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689028.png" /> change according to the contravariant law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689029.png" />, while the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689030.png" /> of the covariant vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689031.png" /> change according to the covariant law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689032.png" /> (i.e. they change in the same way as the basis, whence the terminology  "covariant vectorcovariant" ).
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The coordinate notation for a covariant vector is particularly simple if one chooses in  $  E $
 +
and  $  E  ^ {*} $
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so-called dual bases  $  e _ {1} \dots e _ {n} $
 +
in  $  E $
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and  $  e  ^ {1} \dots e  ^ {n} $
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in  $  E  ^ {*} $,
 +
that is, bases such that  $  ( e  ^ {i} e _ {j} ) = \delta _ {j}  ^ {i} $(
 +
where  $  \delta _ {j}  ^ {i} $
 +
is the [[Kronecker symbol|Kronecker symbol]]); an arbitrary covariant vector  $  \omega \in E  ^ {*} $
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is then expressible in the form  $  \omega = f _ {i} e  ^ {i} $(
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summation over  $  i $
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from 1 to  $  n $),
 +
where  $  f _ {i} $
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is the value of the linear form  $  \omega $
 +
at the vector  $  e _ {i} $.
 +
On passing from dual bases  $  ( e _ {i} ) $
 +
and  $  ( e  ^ {j} ) $
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to dual bases  $  ( \overline{e}\; _ {i  ^  \prime  } ) $
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and  $  ( \overline{e}\; {} ^ {j  ^  \prime  } ) $
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according to the formulas
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 +
$$
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\overline{e}\; _ {i  ^  \prime  }  = \
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p _ {i  ^  \prime  }  ^ {i}
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e _ {i} ,\ \
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\overline{e}\; {} ^ {j  ^  \prime  }  = \
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q _ {i} ^ {j  ^  \prime  }
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e  ^ {i} ,\ \
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p _ {k  ^  \prime  }  ^ {i}
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q _ {j} ^ {k  ^  \prime  }  = \
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\delta _ {j}  ^ {i} ,
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$$
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the coordinates  $  x  ^ {i} $
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of the contravariant vector $  x = x  ^ {i} e _ {i} $
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change according to the contravariant law $  \overline{x}\; {} ^ {i  ^  \prime  } = q _ {i} ^ {i  ^  \prime  } x  ^ {i} $,  
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while the coordinates $  f _ {i} $
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of the covariant vector $  \omega $
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change according to the covariant law $  \overline{f}\; _ {i  ^  \prime  } = p _ {i  ^  \prime  }  ^ {i} f _ {i} $(
 +
i.e. they change in the same way as the basis, whence the terminology  "covariant vectorcovariant" ).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.A. Shirokov,  "Tensor calculus. Tensor algebra" , Kazan'  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.V. Beklemishev,  "A course of analytical geometry and linear algebra" , Moscow  (1971)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Schouten,  "Tensor analysis for physicists" , Cambridge Univ. Press  (1951)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.A. Shirokov,  "Tensor calculus. Tensor algebra" , Kazan'  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.V. Beklemishev,  "A course of analytical geometry and linear algebra" , Moscow  (1971)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Schouten,  "Tensor analysis for physicists" , Cambridge Univ. Press  (1951)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  (1970–1975)  pp. 1–5</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  (1970–1975)  pp. 1–5</TD></TR></table>

Latest revision as of 17:31, 5 June 2020


An element of the vector space $ E ^ {*} $ dual to an $ n $- dimensional vector space $ E $, that is, a linear functional (linear form) on $ E $. In the ordered pair $ ( E, E ^ {*} ) $, an element of $ E $ is called a contravariant vector. Within the general scheme for the construction of tensors, a covariant vector is identified with a covariant tensor of valency 1.

The coordinate notation for a covariant vector is particularly simple if one chooses in $ E $ and $ E ^ {*} $ so-called dual bases $ e _ {1} \dots e _ {n} $ in $ E $ and $ e ^ {1} \dots e ^ {n} $ in $ E ^ {*} $, that is, bases such that $ ( e ^ {i} e _ {j} ) = \delta _ {j} ^ {i} $( where $ \delta _ {j} ^ {i} $ is the Kronecker symbol); an arbitrary covariant vector $ \omega \in E ^ {*} $ is then expressible in the form $ \omega = f _ {i} e ^ {i} $( summation over $ i $ from 1 to $ n $), where $ f _ {i} $ is the value of the linear form $ \omega $ at the vector $ e _ {i} $. On passing from dual bases $ ( e _ {i} ) $ and $ ( e ^ {j} ) $ to dual bases $ ( \overline{e}\; _ {i ^ \prime } ) $ and $ ( \overline{e}\; {} ^ {j ^ \prime } ) $ according to the formulas

$$ \overline{e}\; _ {i ^ \prime } = \ p _ {i ^ \prime } ^ {i} e _ {i} ,\ \ \overline{e}\; {} ^ {j ^ \prime } = \ q _ {i} ^ {j ^ \prime } e ^ {i} ,\ \ p _ {k ^ \prime } ^ {i} q _ {j} ^ {k ^ \prime } = \ \delta _ {j} ^ {i} , $$

the coordinates $ x ^ {i} $ of the contravariant vector $ x = x ^ {i} e _ {i} $ change according to the contravariant law $ \overline{x}\; {} ^ {i ^ \prime } = q _ {i} ^ {i ^ \prime } x ^ {i} $, while the coordinates $ f _ {i} $ of the covariant vector $ \omega $ change according to the covariant law $ \overline{f}\; _ {i ^ \prime } = p _ {i ^ \prime } ^ {i} f _ {i} $( i.e. they change in the same way as the basis, whence the terminology "covariant vectorcovariant" ).

References

[1] P.A. Shirokov, "Tensor calculus. Tensor algebra" , Kazan' (1961) (In Russian)
[2] D.V. Beklemishev, "A course of analytical geometry and linear algebra" , Moscow (1971) (In Russian)
[3] J.A. Schouten, "Tensor analysis for physicists" , Cambridge Univ. Press (1951)

Comments

References

[a1] M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish (1970–1975) pp. 1–5
How to Cite This Entry:
Covariant vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Covariant_vector&oldid=46546
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article