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''of valency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c0268802.png" />''
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A tensor of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c0268803.png" />, an element of the tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c0268804.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c0268805.png" /> copies of the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c0268806.png" /> of the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c0268807.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c0268808.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c0268809.png" /> is itself a vector space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688010.png" /> with respect to the addition of covariant tensors of the same valency and multiplication of them by scalars. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688011.png" /> be finite dimensional, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688012.png" /> be a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688013.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688014.png" /> be the basis dual to it of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688015.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688016.png" /> and the set of all tensors of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688018.png" />, forms a basis for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688019.png" />. Any covariant tensor can be represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688020.png" />. The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688021.png" /> are called the coordinates, or components, of the covariant tensor relative to the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688022.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688023.png" />. Under a change of a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688024.png" /> according to the formulas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688025.png" /> and the corresponding change of the basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688026.png" />, the components of the covariant tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688027.png" /> are changed according to the so-called covariant law
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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/c026880/c02688028.png" /></td> </tr></table>
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''of valency  $  s \geq  1 $''
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688029.png" />, the covariant tensor is called a [[Covariant vector|covariant vector]]; when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688030.png" /> a covariant tensor corresponds in an invariant way with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688031.png" />-linear mapping from the direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688032.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688033.png" /> times) into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688034.png" /> by taking the components of the covariant tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688035.png" /> relative to the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688036.png" /> as the values of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688037.png" />-linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688038.png" /> at the basis vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688039.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688040.png" />, and conversely; for this reason a covariant tensor is sometimes defined as a multilinear functional on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026880/c02688041.png" />.
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A tensor of type  $  ( 0, s) $,
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an element of the tensor product  $  T _ {s} ( E) = E  ^ {*} \otimes \dots \otimes E  ^ {*} $
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of  $  s $
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copies of the dual space  $  E  ^ {*} $
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of the vector space  $  E $
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over a field  $  K $.
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The space  $  T _ {s} ( E) $
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is itself a vector space over  $  K $
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with respect to the addition of covariant tensors of the same valency and multiplication of them by scalars. Let  $  E $
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be finite dimensional, let  $  e _ {1} \dots e _ {n} $
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be a basis of  $  E $
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and let  $  e  ^ {1} \dots e  ^ {n} $
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be the basis dual to it of  $  E  ^ {*} $.
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Then  $  \mathop{\rm dim}  T _ {s} ( E) = n  ^ {s} $
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and the set of all tensors of the form  $  e ^ {i _ {1} } \otimes \dots \otimes e ^ {i _ {s} } $,
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$  1 \leq  i _ {1} \dots i _ {s} \leq  n $,
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forms a basis for  $  T _ {s} ( E) $.
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Any covariant tensor can be represented in the form  $  t = t _ {i _ {1}  \dots i _ {s} } e ^ {i _ {1} } \otimes \dots \otimes e ^ {i _ {s} } $.  
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The numbers  $  t _ {i _ {1}  \dots i _ {s} } $
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are called the coordinates, or components, of the covariant tensor relative to the basis  $  e _ {1} \dots e _ {n} $
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of  $  E $.  
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Under a change of a basis of  $  E $
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according to the formulas  $  e _ {j}  ^  \prime  = a _ {j}  ^ {i} e _ {i} $
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and the corresponding change of the basis of  $  T _ {s} ( E) $,
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the components of the covariant tensor  $  t $
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are changed according to the so-called covariant law
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$$
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t _ {j _ {1}  \dots j _ {s} }  ^  \prime  = \
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a _ {j _ {1}  } ^ {i _ {1} } \dots
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a _ {j _ {s}  } ^ {i _ {s} }
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t _ {i _ {1}  \dots i _ {s} } .
 +
$$
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If  $  s = 1 $,  
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the covariant tensor is called a [[Covariant vector|covariant vector]]; when $  s \geq  2 $
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a covariant tensor corresponds in an invariant way with an $  s $-
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linear mapping from the direct product $  E  ^ {s} = E \times \dots \times E $(
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$  s $
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times) into $  K $
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by taking the components of the covariant tensor $  t $
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relative to the basis $  e _ {1} \dots e _ {n} $
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as the values of the $  r $-
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linear mapping $  \widetilde{t}  $
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at the basis vectors $  ( e _ {i _ {1}  } \dots e _ {i _ {s}  } ) $
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in $  E  ^ {s} $,  
 +
and conversely; for this reason a covariant tensor is sometimes defined as a multilinear functional on $  E  ^ {s} $.
  
 
For references see [[Covariant vector|Covariant vector]].
 
For references see [[Covariant vector|Covariant vector]].

Latest revision as of 17:31, 5 June 2020


of valency $ s \geq 1 $

A tensor of type $ ( 0, s) $, an element of the tensor product $ T _ {s} ( E) = E ^ {*} \otimes \dots \otimes E ^ {*} $ of $ s $ copies of the dual space $ E ^ {*} $ of the vector space $ E $ over a field $ K $. The space $ T _ {s} ( E) $ is itself a vector space over $ K $ with respect to the addition of covariant tensors of the same valency and multiplication of them by scalars. Let $ E $ be finite dimensional, let $ e _ {1} \dots e _ {n} $ be a basis of $ E $ and let $ e ^ {1} \dots e ^ {n} $ be the basis dual to it of $ E ^ {*} $. Then $ \mathop{\rm dim} T _ {s} ( E) = n ^ {s} $ and the set of all tensors of the form $ e ^ {i _ {1} } \otimes \dots \otimes e ^ {i _ {s} } $, $ 1 \leq i _ {1} \dots i _ {s} \leq n $, forms a basis for $ T _ {s} ( E) $. Any covariant tensor can be represented in the form $ t = t _ {i _ {1} \dots i _ {s} } e ^ {i _ {1} } \otimes \dots \otimes e ^ {i _ {s} } $. The numbers $ t _ {i _ {1} \dots i _ {s} } $ are called the coordinates, or components, of the covariant tensor relative to the basis $ e _ {1} \dots e _ {n} $ of $ E $. Under a change of a basis of $ E $ according to the formulas $ e _ {j} ^ \prime = a _ {j} ^ {i} e _ {i} $ and the corresponding change of the basis of $ T _ {s} ( E) $, the components of the covariant tensor $ t $ are changed according to the so-called covariant law

$$ t _ {j _ {1} \dots j _ {s} } ^ \prime = \ a _ {j _ {1} } ^ {i _ {1} } \dots a _ {j _ {s} } ^ {i _ {s} } t _ {i _ {1} \dots i _ {s} } . $$

If $ s = 1 $, the covariant tensor is called a covariant vector; when $ s \geq 2 $ a covariant tensor corresponds in an invariant way with an $ s $- linear mapping from the direct product $ E ^ {s} = E \times \dots \times E $( $ s $ times) into $ K $ by taking the components of the covariant tensor $ t $ relative to the basis $ e _ {1} \dots e _ {n} $ as the values of the $ r $- linear mapping $ \widetilde{t} $ at the basis vectors $ ( e _ {i _ {1} } \dots e _ {i _ {s} } ) $ in $ E ^ {s} $, and conversely; for this reason a covariant tensor is sometimes defined as a multilinear functional on $ E ^ {s} $.

For references see Covariant vector.

How to Cite This Entry:
Covariant tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Covariant_tensor&oldid=13043
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article