Contravariant

The summation convention for double indices, for example, k in Eq. (113), is assumed, as before. However, we no longer make distinction between covariant and contravariant sets.) We set ourselves the task to find anti-Hermitean operators Xf, such that... 

Contravariant and covairant four vectors are connected through the metric = diag (1,-1,-1,-1) by... 

APPENDIX - SETMMARY OF VECTOR AND TENSOR ANALYSTS 8.2.4 Covariant and contravariant vectors... 

Similar to vectors, based on the transfomiation properties of the second tensors the following three types of covariant, contravariant and mixed components are defined... 

Note that convected derivatives of the stress (and rate of strain) tensors appearing in the rheological relationships derived for non-Newtonian fluids will have different forms depending on whether covariant or contravariant components of these tensors are used. For example, the convected time derivatives of covariant and contravariant stress tensors are expressed as... 

Contravariant vectors, those that transform like the coordinate... 

They are called contravariant, covariant and mixed tensors, respectively. A useful mixed tensor of the second rank is the Kronecker delta... 

Although it is customary to refer to covariant and contravariant vectors, this may be misleading. Any vector can be described in terms of its contravariant or its covariant components with equal validity. There is no reason other than numerical simplicity for the preference of one set of components over the other. 

This result shows that, by its transformation properties, Aljkl is equivalent to a covariant vector of rank two. This process of summing over a pair of contravariant and covariant indices is called contraction. It always reduces the rank of a mixed tensor by two and thus, when applied to a mixed tensor of rank two, the result is a scalar ... 

The quantities /" are the contravariant components of a vector in the coordinate system X. They give an actual vector only when multiplied by the unit vector e = hvev. If the unit vectors along the coordinate lines have a scale inverse to the coordinate, e = ev/hv so that... 

The quantities bij, 6U, and 6 are respectively called the components of covariant, contravariant or mixed tensors of the second order, if they transform according to the formulae... 

The products of the components of two covariant vectors, taken in pairs, form the components of a covariant tensor. If the vectors are contravariant,... 

The fractional coordinates in crystallography, referred to direct and reciprocal cells respectively, are examples of covariant and contravariant vector components. 

The are referred to as the contravariant components of f along these basic unit vectors. It is found that... 

Definition 1.1.1. [Grothendieck (1)] Let HUb(X/T) be the contravariant functor from the category SchlnT of locally noetherian T-schemes to the category Ens of sets, which for locally noetherian T-schemes U,V and a morphism V — U is given by... 

The postcondition of the implementation should imply the postcondition of the specification. (Notice that this is the other way around from preconditions, a feature called contravariance.) So if the implementation claims that this operation adds 3 to the... 

The relativistic one-fermion equation in the free field presented hy Dirac in 1928 [2,3] is usually written in the contravariant form, which illustrates the relativistic equivalence of space and time in an elegant way... 

We are often interested in the rms thermal displacements in A. They correspond to the contravariant components UJk along covariant axes of unit length, rather than along the non-unit length a, b, c axes. The rms displacements are obtained from... 

The potential function may be expanded in terms of increasing order of products of the contravariant displacement coordinates (Dawson 1967, Willis 1969)... 

Since a, af = S0, or in tensor notation, a, aj = S, the reciprocal axes are contravariant and are written as a . As the Miller indices are the coordinates in the reciprocal base system, they must be covariant and are written as ht. Thus, the Miller indices transform like the direct axes, both being covariant. 

Placement of indices as superscripts or subscripts follows the conventions of tensor analysis. Contravariant variables, which transform like coordinates, are indexed by superscripts, and coavariant quantities, which transform like derivatives, are indexed by subscripts. Cartesian and generalized velocities and 2 thus contravariant, while Cartesian and generalized forces, which transform like derivatives of a scalar potential energy, are covariant. 

For each coordinate 2 in the full space, we may define a covariant basis vector 0R /02 and a contravariant basis vector 02 /0R, which obey orthogonality and completeness relations... 

It will be assumed throughout that H and are symmetric positive-definite tensors. We write the mobility as a contravariant tensor, with raised bead indices, to reflect its function the mobility H may be contracted with a covariant force vector Fv (e.g., the derivative of a potential energy) to produce a resulting contravariant velocity Fy. 

Representations of these and other tensors in an arbitrary system of coordinates may be constructed as follows. For each contravariant rank 2 Cartesian tensor T " (such as H ) or covariant tensor S v (such as m v), we define corresponding Riemannian representations... 

A covariant metric tensor g p and contravariant inverse metric tensor in the full space are given by... 

These are covariant and contravariant representations of the Cartesian identity tensor, and inverses of each other. 

For each rank 2 contravariant Riemannian tensors T (with two raised indices) we define a. K x K projection onto the hard subspace... 

---