Covariant
of a tensor on a finite-dimensional vector space
A mapping of the space
of tensors of a fixed type over
into a space
of covariant tensors over
such that
for any non-singular linear transformation
of
and any
. This is the definition of the covariant of a tensor with respect to the general linear group
. If
is not arbitrary but belongs to a fixed subgroup
, then one obtains the definition of a covariant of a tensor relative to
, or simply a covariant of
.
In coordinate language, a covariant of a tensor on a finite-dimensional vector space is a set of functions
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of the components of the tensor with the following properties: Under a change of the set of numbers
defined by a non-singular linear transformation
, the set of numbers
changes according to that of a covariant tensor
over
under the transformation
. In similar fashion one defines (by considering instead of one tensor
a finite collection of tensors) a joint covariant of the system of tensors. If instead, one replaces the covariance condition for the tensor
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 and its dual
, cf. "form associated to a tensor" in the article Tensor on a vector space). Suppose that the form
corresponds in this manner to the tensor
and that the form
corresponds to its covariant
. Then
if a form of contravariant vectors only. In the classical theory of invariants
was called the covariant of
. A case that was particularly often considered is when
is a form in one single contravariant vector. The degree of this form is called the order of the covariant. If the coefficients of
are polynomials in the coefficients of
, then the highest of the degrees of these polynomials is called the degree of the covariant.
Example. Let be a form of a degree
, where
are the components of a contravariant vector. The form
corresponds to a symmetric covariant tensor
of valency
with components
. Let
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Then the coefficients of are the components of some covariant tensor
. The tensor
(or the form
) is a covariant of the tensor
(or form
). The form
is called the Hessian of
.
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=12146