View source for Covariant

''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" />''

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" />.

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>

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.

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.

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

<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>

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" />.

====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>