Covariance and Contravariance

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Covariance and Contravariance

 

concepts that play an important role in linear algebra and tensor calculus. Let two systems of n variables x₁, x₂, . . ., x_n and y₁, y₂, . . . , y_n (numbers or vectors) be subject to a homogeneous linear transformation such that to each transformation of x₁, x₂, . . , x_n there corresponds a definite transformation of y₁, y₂ . . . , y_n If to the transformation

of the variables x_i there corresponds a transformation

of the variables y_i, then the systems x_i and y_i are called covariant (similarly transforming), or cogredient. If to the transformation of the x_i defined by formula (1) there corresponds a transformation of the variables y_i given by the formula

then the systems x_i and y_i are called contravariant (oppositely transforming), or contragredient.

The concepts of covariant and contravariant tensors are a generalization of these concepts.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.