Difference between revisions of "Covariant"
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+ | $#C+1 = 54 : ~/encyclopedia/old_files/data/C026/C.0206840 Covariant | ||
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− | A mapping | + | {{TEX|auto}} |
+ | {{TEX|done}} | ||
+ | |||
+ | ''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 | ||
− | + | $$ | |
+ | s _ {i} = \phi _ {i} ( t _ {1} \dots t _ {n} ) ,\ \ | ||
+ | i = 1 \dots m , | ||
+ | $$ | ||
+ | |||
+ | 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 | ||
− | + | $$ | |
+ | h = | ||
+ | \frac{1}{r ^ {n} ( r - 1 ) ^ {n} } | ||
− | + | \left | | |
+ | \begin{array}{ccc} | ||
− | + | \frac{\partial ^ {2} f }{( \partial x ^ {1} ) ^ {2} } | |
+ | &\dots & | ||
+ | \frac{\partial ^ {2} f }{\partial x ^ {1} \partial x ^ {n} } | ||
+ | \\ | ||
+ | \dots &\dots &\dots \\ | ||
− | + | \frac{\partial ^ {2} f }{\partial x ^ {n} \partial x ^ {1} } | |
+ | &\dots & | ||
+ | \frac{\partial ^ {2} f }{( \partial x ^ {n} ) ^ {2} } | ||
+ | \\ | ||
+ | \end{array} | ||
+ | \right | . | ||
+ | $$ | ||
− | Then the coefficients of | + | 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) |
Covariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Covariant&oldid=46542