Difference between revisions of "Covariant tensor"
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| − | + | ''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|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|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
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