Covariance and Contravariance

(redirected from Co-variant)
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Covariance and Contravariance

 

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

of the variables xi there corresponds a transformation

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

then the systems xi and yi are called contravariant (oppositely transforming), or contragredient.

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