Tensor contravariant

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... 

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

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 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,... 

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. 

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... 

The expression given by BCAH for elements of the constrained mobility within the internal subspace is based on inversion of the projection of the modified mobility within the internal subspace, rather than inversion of the projection (at of the mobility within the entire soft subspace. BCAH first define a tensor given by the projection of the modified friction tensor onto the internal subspace, which they denote by the symbol gat and refer to as a modified covariant metric tensor, which is equivalent to our CaT - They then define an inverse of this quantity within the subspace of internal coordinates, which they denote by g and refer to as a modified contravariant metric tensor, which is equivalent to our for afi = 1,..., / — 3. It is this last quantity that appears in their diffusion equation, given in Eq. (16.2-6) of Ref. 4, in place of our constrained mobility Within the space of internal coordinates, the two quantities are completely equivalent. 

Let us briefly recall a few of the basics of the algebra of tensors. An nth rank tensor in m-dimensional space is an object with n indices and rrf components. For a general tensor a distinction is made between contravari-ant (upper) indices and covariant (lower) indices. A tensor of rank mi + m2 may have mi contravariant indices and m2 covariant indices. The order of the indices is significant. Tensors can be classified according to whether they are... 

In contravariant covariant notation, the field tensors are defined by [101]... 

Such a reader might find relief in differential geometry, the mathematical study of multiple coordinate systems. There are many excellent standard texts, such as Isham s book [I] for a gentle introduction to some basic concepts of differential geometry, try [Si]. A text that discusses covariant and contravariant tensors is Spivak s introduction to differential geometry [Sp, Volume I, Chapter 4]. For a quick introduction aimed at physical calculations, try Joshi s book [Jos]. 

Again, we remind physicists that tensor products of vector spaces are neither as general nor as powerful as the objects called tensors appearing in general relativity. Issues of covariance and contravariance have to do with multiple coordinate systems. Because quantum mechanics is Unear, we do not need the more general notion of tensor in this book, so we do not stop to introduce it. We do, however, offer our condolences and a few references to physicists searching for clarification. See Footnote 4 in this chapter. 

In Kirkwood s original formulation of the Fokker-Planck theory, he took into account the possibility that various constraints might apply, e.g., constant bond length between adjacent beads. This led to the introduction of a chain space of lower dimensionality than the full 3A-dimen-sional configuration space of the entire chain and it led to a complicated machinery of Riemannian geometry, with covariant and contravariant tensors, etc. 

There are several major implications of the Jacobi identity (40), so it is helpful to give some background for its derivation. On the U(l) level, consider the following field tensors in c = 1 units and contravariant covariant notation in Minkowski spacetime ... 

Let us now turn to the internal variables. We can consider that one of the internal variables is a tensor of arbitrary rank and transforms as the coordinates do, that is, contravariantly... 

When we rotate a contravariant nxl column vector (for position, velocity, momentum, electric field, etc.) we premultiply it by an n x n rotation tensor R. When, instead, we transform the coordinate system in which such vectors are defined, then the coordinate system and, for example, the V operator are covariant 1 x n row vectors, which are transformed by the tensor R 1 that is the reciprocal of R. A "dot product" or inner product a b must be the multiplication of a row vector a by a column vector b, to give a single number (scalar) as the result. This will be expanded further in the discussion of special relativity (Section 2.13) and of crystal symmetry (Section 7.10). 

Even though we do not invoke the full machinery of tensor analysis (Butkov, 1968), it is useful to keep the distinction between contravariant and covariant components clear. To avoid conflicting notation we do not use upper and lower indices to denote contravariant and covariant indices. Instead, we will use the suffix ao (lower case letters) on tensors whose indices are all contravariant, and AO (capital letters) on tensors whose indices are all covariant. No special suffix is used in the MO basis. For example, using the two- and four-index trace operators the energy is... 

The contravariant metric tensor gjk is known in the theory of small vibrations as Wilson s G matrix (kinematic matrix). 

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