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''of a tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c0268401.png" /> on a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c0268402.png" />''
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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c0268403.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c0268404.png" /> of tensors of a fixed type over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c0268405.png" /> into a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c0268406.png" /> of covariant tensors over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c0268407.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c0268408.png" /> for any non-singular linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c0268409.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684010.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684011.png" />. This is the definition of the covariant of a tensor with respect to the general linear group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684013.png" /> is not arbitrary but belongs to a fixed subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684014.png" />, then one obtains the definition of a covariant of a tensor relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684015.png" />, or simply a covariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684016.png" />.
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{{TEX|done}}
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''of a tensor  $  t $
 +
on a finite-dimensional vector space  $  V $''
 +
 
 +
A mapping $  \phi $
 +
of the space $  T $
 +
of tensors of a fixed type over $  V $
 +
into a space $  S $
 +
of covariant tensors over $  V $
 +
such that $  \phi ( g ( t) ) = g ( \phi ( t) ) $
 +
for any non-singular linear transformation $  g $
 +
of $  V $
 +
and any $  t \in T $.  
 +
This is the definition of the covariant of a tensor with respect to the general linear group $  \mathop{\rm GL} ( V) $.  
 +
If $  g $
 +
is not arbitrary but belongs to a fixed subgroup $  G \subset  \mathop{\rm GL} ( V ) $,  
 +
then one obtains the definition of a covariant of a tensor relative to $  G $,  
 +
or simply a covariant of $  G $.
  
 
In coordinate language, a covariant of a tensor on a finite-dimensional vector space is a set of functions
 
In coordinate language, a covariant of a tensor on a finite-dimensional vector space is a set of functions
  
<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/c026840/c02684017.png" /></td> </tr></table>
+
$$
 +
s _ {i}  = \phi _ {i} ( t _ {1} \dots t _ {n} ) ,\ \
 +
i = 1 \dots m ,
 +
$$
 +
 
 +
of the components of the tensor  $  t $
 +
with the following properties: Under a change of the set of numbers  $  t _ {1} \dots t _ {n} $
 +
defined by a non-singular linear transformation  $  g \in G $,
 +
the set of numbers  $  s _ {1} \dots s _ {m} $
 +
changes according to that of a covariant tensor  $  s $
 +
over  $  V $
 +
under the transformation  $  g $.
 +
In similar fashion one defines (by considering instead of one tensor  $  t $
 +
a finite collection of tensors) a joint covariant of the system of tensors. If instead, one replaces the covariance condition for the tensor  $  s $
 +
by contravariance, one obtains the notion of a contravariant.
 +
 
 +
The notion of a covariant arose in the classical theory of invariants and is a special case of the notion of a [[Comitant|comitant]]. The components of any tensor can be regarded as the coefficients of an appropriate form in several contravariant and covariant vectors (that is, vectors of  $  V $
 +
and its dual  $  V  ^ {*} $,
 +
cf.  "form associated to a tensor" in the article [[Tensor on a vector space|Tensor on a vector space]]). Suppose that the form  $  f $
 +
corresponds in this manner to the tensor  $  t $
 +
and that the form  $  h $
 +
corresponds to its covariant  $  s $.
 +
Then  $  h $
 +
if a form of contravariant vectors only. In the classical theory of invariants  $  h $
 +
was called the covariant of  $  f $.
 +
A case that was particularly often considered is when  $  h $
 +
is a form in one single contravariant vector. The degree of this form is called the order of the covariant. If the coefficients of  $  h $
 +
are polynomials in the coefficients of  $  f $,
 +
then the highest of the degrees of these polynomials is called the degree of the covariant.
 +
 
 +
Example. Let  $  f = \sum a _ {i _ {1}  \dots i _ {r} } x ^ {i _ {1} } \dots x ^ {i _ {r} } $
 +
be a form of a degree  $  r $,
 +
where  $  x  ^ {1} \dots x  ^ {n} $
 +
are the components of a contravariant vector. The form  $  f $
 +
corresponds to a symmetric covariant tensor  $  t $
 +
of valency  $  r $
 +
with components  $  a _ {i _ {1}  \dots i _ {r} } $.  
 +
Let
  
of the components of the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684018.png" /> with the following properties: Under a change of the set of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684019.png" /> defined by a non-singular linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684020.png" />, the set of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684021.png" /> changes according to that of a covariant tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684022.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684023.png" /> under the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684024.png" />. In similar fashion one defines (by considering instead of one tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684025.png" /> a finite collection of tensors) a joint covariant of the system of tensors. If instead, one replaces the covariance condition for the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684026.png" /> by contravariance, one obtains the notion of a contravariant.
+
$$
 +
=
 +
\frac{1}{r  ^ {n} ( r - 1 ) ^ {n} }
  
The notion of a covariant arose in the classical theory of invariants and is a special case of the notion of a [[Comitant|comitant]]. The components of any tensor can be regarded as the coefficients of an appropriate form in several contravariant and covariant vectors (that is, vectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684027.png" /> and its dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684028.png" />, cf.  "form associated to a tensor"  in the article [[Tensor on a vector space|Tensor on a vector space]]). Suppose that the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684029.png" /> corresponds in this manner to the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684030.png" /> and that the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684031.png" /> corresponds to its covariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684032.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684033.png" /> if a form of contravariant vectors only. In the classical theory of invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684034.png" /> was called the covariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684035.png" />. A case that was particularly often considered is when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684036.png" /> is a form in one single contravariant vector. The degree of this form is called the order of the covariant. If the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684037.png" /> are polynomials in the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684038.png" />, then the highest of the degrees of these polynomials is called the degree of the covariant.
+
\left |  
 +
\begin{array}{ccc}
  
Example. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684039.png" /> be a form of a degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684040.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684041.png" /> are the components of a contravariant vector. The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684042.png" /> corresponds to a symmetric covariant tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684043.png" /> of valency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684044.png" /> with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684045.png" />. Let
+
\frac{\partial  ^ {2} f }{( \partial  x  ^ {1} )  ^ {2} }
 +
  &\dots  &
 +
\frac{\partial  ^ {2} f }{\partial  x  ^ {1} \partial  x  ^ {n} }
 +
  \\
 +
\dots  &\dots  &\dots  \\
  
<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/c026840/c02684046.png" /></td> </tr></table>
+
\frac{\partial  ^ {2} f }{\partial  x  ^ {n} \partial  x  ^ {1} }
 +
  &\dots  &
 +
\frac{\partial  ^ {2} f }{( \partial  x  ^ {n} )  ^ {2} }
 +
  \\
 +
\end{array}
 +
\right | .
 +
$$
  
Then the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684047.png" /> are the components of some covariant tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684048.png" />. The tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684049.png" /> (or the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684050.png" />) is a covariant of the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684051.png" /> (or form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684052.png" />). The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684053.png" /> is called the Hessian of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026840/c02684054.png" />.
+
Then the coefficients of $  h $
 +
are the components of some covariant tensor $  s $.  
 +
The tensor $  s $(
 +
or the form $  h $)  
 +
is a covariant of the tensor $  t $(
 +
or form $  f  $).  
 +
The form $  h $
 +
is called the Hessian of $  f $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.B. Gurevich,  "Foundations of the theory of algebraic invariants" , Noordhoff  (1964)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.B. Gurevich,  "Foundations of the theory of algebraic invariants" , Noordhoff  (1964)  (Translated from Russian)</TD></TR></table>

Latest revision as of 17:31, 5 June 2020


of a tensor $ t $ on a finite-dimensional vector space $ V $

A mapping $ \phi $ of the space $ T $ of tensors of a fixed type over $ V $ into a space $ S $ of covariant tensors over $ V $ such that $ \phi ( g ( t) ) = g ( \phi ( t) ) $ for any non-singular linear transformation $ g $ of $ V $ and any $ t \in T $. This is the definition of the covariant of a tensor with respect to the general linear group $ \mathop{\rm GL} ( V) $. If $ g $ is not arbitrary but belongs to a fixed subgroup $ G \subset \mathop{\rm GL} ( V ) $, then one obtains the definition of a covariant of a tensor relative to $ G $, or simply a covariant of $ G $.

In coordinate language, a covariant of a tensor on a finite-dimensional vector space is a set of functions

$$ s _ {i} = \phi _ {i} ( t _ {1} \dots t _ {n} ) ,\ \ i = 1 \dots m , $$

of the components of the tensor $ t $ with the following properties: Under a change of the set of numbers $ t _ {1} \dots t _ {n} $ defined by a non-singular linear transformation $ g \in G $, the set of numbers $ s _ {1} \dots s _ {m} $ changes according to that of a covariant tensor $ s $ over $ V $ under the transformation $ g $. In similar fashion one defines (by considering instead of one tensor $ t $ a finite collection of tensors) a joint covariant of the system of tensors. If instead, one replaces the covariance condition for the tensor $ s $ by contravariance, one obtains the notion of a contravariant.

The notion of a covariant arose in the classical theory of invariants and is a special case of the notion of a comitant. The components of any tensor can be regarded as the coefficients of an appropriate form in several contravariant and covariant vectors (that is, vectors of $ V $ and its dual $ V ^ {*} $, cf. "form associated to a tensor" in the article Tensor on a vector space). Suppose that the form $ f $ corresponds in this manner to the tensor $ t $ and that the form $ h $ corresponds to its covariant $ s $. Then $ h $ if a form of contravariant vectors only. In the classical theory of invariants $ h $ was called the covariant of $ f $. A case that was particularly often considered is when $ h $ is a form in one single contravariant vector. The degree of this form is called the order of the covariant. If the coefficients of $ h $ are polynomials in the coefficients of $ f $, then the highest of the degrees of these polynomials is called the degree of the covariant.

Example. Let $ f = \sum a _ {i _ {1} \dots i _ {r} } x ^ {i _ {1} } \dots x ^ {i _ {r} } $ be a form of a degree $ r $, where $ x ^ {1} \dots x ^ {n} $ are the components of a contravariant vector. The form $ f $ corresponds to a symmetric covariant tensor $ t $ of valency $ r $ with components $ a _ {i _ {1} \dots i _ {r} } $. Let

$$ h = \frac{1}{r ^ {n} ( r - 1 ) ^ {n} } \left | \begin{array}{ccc} \frac{\partial ^ {2} f }{( \partial x ^ {1} ) ^ {2} } &\dots & \frac{\partial ^ {2} f }{\partial x ^ {1} \partial x ^ {n} } \\ \dots &\dots &\dots \\ \frac{\partial ^ {2} f }{\partial x ^ {n} \partial x ^ {1} } &\dots & \frac{\partial ^ {2} f }{( \partial x ^ {n} ) ^ {2} } \\ \end{array} \right | . $$

Then the coefficients of $ h $ are the components of some covariant tensor $ s $. The tensor $ s $( or the form $ h $) is a covariant of the tensor $ t $( or form $ f $). The form $ h $ is called the Hessian of $ f $.

References

[1] G.B. Gurevich, "Foundations of the theory of algebraic invariants" , Noordhoff (1964) (Translated from Russian)
How to Cite This Entry:
Covariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Covariant&oldid=46542
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article