Coordinates covariant/contravariant

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

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... 

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... 

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. 

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. 

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). 

It is worth the trouble to relate the properties of the covariant measuring vectors to those of the contravariant measuring vectors. We shall consider only molecules, which are not subject to constraints. By using the coordinate representation of Va, it follows that... 

Consider a 3-D domain that can be adequately described by the generalized curvilinear coordinate system (u, v, w) and that its mappings are adequately smooth to allow consistent definitions. Then, any vector F can be decomposed into three components with respect to the contravariant a , a , a or the covariant a , a, a,a, linearly independent basis system as... 

The direct base vectors and the reciprocal coordinates h, k, I transform in a covariant manner. The reciprocal base vectors and the direct coordinates u, v, w transform in a contravariant manner. 

An important concept in continuum mechanics is the objectivity, or admissibility, of the constitutive equation. There are the covariant and contravariant ways of achieving objectivity. The molecular theories the elastic dumbbell model of this chapter, the Rouse model to be studied in the next chapter, and the Zimm model which includes the preaveraged hydrodynamic interaction, all give the result equivalent to the contravariant way. In this appendix, we limit our discussion of continuum mechanics to what is needed for the molecular theories studied in Chapters 6 and 7. More detailed discussions of the subject, particularly about the convected coordinates, can be found in Refs. 5 and 6. 

As observed in (2.159), 8 v/8t, 8cv/8t denotes a part of v excluding the change of basis, which corresponds to the change of v when the observer is moving along the same coordinate system of the deformed body. 8 v/8t is known as the contravariant derivative or upper convected rate, and 8cv/8t is known as the covariant derivative or lower convected rate. ... 

The A are named contravariant. Covariant coordinates can be represented by con-travariant coordinates (and vice versa)... 

There are other time derivative operators that transform a tensor from convected to fixed coordinates, giving rise to equivalent expression of covariant and contravariant forms of a given tensor. One such time derivative operator may be formed by eliminating d-j after Eq. (2B.17) is substituted into Eqs. (2B.15) and (2B.16) and adding... 

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