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.