Covariance and Contravariance

(redirected from Contravariant)

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.

References in periodicals archive ?
X] exists weakly and is given by categorical tensor; and, (iii) the contravariant functor C(_, A) is fully faithful.
n] is contravariant velocity, which is represented as
A] = 0 on [partial derivative]U (as can be checked for a vector on (50) and for the contravariant metric tensor on (39)).
constitute the contravariant basis at the point [theta](y), where [[delta].
Definition 2 A species (with restrictions) is a contravariant functor P: [set.
n] be an n-dimensional differentiable manifold endowed with a (1,1) tensor field [phi], a contravariant vector field [xi], a covariant vector field [eta] and a Lorentzian metric g of type (0, 2) such that for each point p [member of] M, the tensor [g.
ij] can be expressed in contravariant and covariant components, respectively, as:
mu]v]--definition of the gravitational tensor with contravariant indices by means of the metric tensor [g.
ij](x) the contravariant Cartan space, a dual sequence of the sequence (I) is obtained:
T] and the category of contravariant Pos-functors 1 [right arrow] [Pos.
A Poisson structure on a manifold is defined by a skew symmetric contravariant bilinear form subjected to the Jacobi identity expressed as the vanishing of the Schouten bracket of Poisson tensor with itself [32, 35, 42, 49].