Covariance and Contravariance

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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.

References in periodicals archive ?
Importantly, these are not considered to be indices that transform as co and contravariant tensors under the metric h.
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.
mu]v]--definition of the gravitational tensor with contravariant indices by means of the metric tensor [g.
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].
It will also be convenient to represent a metric in terms of contravariant indices:
pk] be the contravariant components of the pseudo-Riemannian metric [h.
ij] are contravariant components of the stress tensor resolved with respect to the initial base vectors (X, Y) and referring to the undeformed geometry, X and Y are the initial cartesian coordinates, and dS is the differential arc length along [GAMMA] defined in the undeformed state.
v] is a weight 1 contravariant vector density, its covariant divergence (with the symbol [[nabla].
6) Spec o G is a contravariant graded functor from the category of non-commutative Zariski filtered rings with units and strict epimorphisms to the category of topological (graded) spaces and continuous maps.