Tensor covariant

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... 

The matrix gp, represents the components of a covariant second-order tensor called the metric tensor , because it defines distance measurement with respect to coordinates To illustrate the application of this definition in the... 

In a non-Abelian theory (where the Hamiltonian contains noncommuting matrices and the solutions are vector or spinor functions, with N in Eq. (90) >1) we also start with a vector potential Ab. [In the manner of Eq. (94), this can be decomposed into components Ab, in which the superscripta labels the matrices in the theory). Next, we define the field intensity tensor through a covariant curl by... 

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,... 

The generalized Fisher theorems derived in this section are statements about the space variation of the vectors of the relative and absolute space-specific rates of growth. These vectors have a simple natural (biological, chemical, physical) interpretation They express the capacity of a species of type u to fill out space in genetic language, they are space-specific fitness functions. In addition, the covariance matrix of the vector of the relative space-specific rates of growth, gap, [Eq. (25)] is a Riemannian metric tensor that enters the expression of a Fisher information metric [Eqs. (24) and (26)]. These results may serve as a basis for solving inverse problems for reaction transport systems. 

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 covariant rank 2 tensor S v (with two subscripted indices) we define a projection onto the soft subspace... 

Fixman has shown [2] that, for any covariant symmetric tensor S ap defined in the full space, with an inverse = (5 ) in the full space, the determinants S and f of the projections of S and T onto the soft and hard subspaces, respectively, are related by... 

Other definitions may be constructed by the following generalization of the relationship between the dynamical reciprocal vectors and the mobility tensor Given any invertible symmetric covariant Cartesian tensor S v with an inverse we may take... 

In this section, we develop some useful relationships involving the determinants and inverses of projected tensors. Let S ap be the Riemannian representation of an arbitrary symmetric covariant tensor with a Cartesian representation S v We may write the Riemannian representation in block matrix form, using the indices a,b to denote blocks in which a or p mns over the soft coordinates and i,j to represent hard coordinates, as... 

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... 

The development just given illustrates the fact that the topology of the vacuum determines the nature of the gauge transformation, field tensor, and field equations, as inferred in Section (I). The covariant derivative plays a central role in each case for example, the homogeneous field equation of 0(3) electrodynamics is a Jacobi identity made up of covariant derivatives in an internal 0(3) symmetry gauge group. The equivalent of the Jacobi identity in general relativity is the Bianchi identity. 

Recall that in general gauge field theory, for any gauge group, the field tensor is defined through the commutator of covariant derivatives. In condensed notation [6]... 

These concepts of 0(3) electrodynamics also completely resolve the problem that, in Maxwell-Heaviside electrodynamics, the energy momentum of radiation is defined through an integral over the conventional tensor and for this reason is not manifestly covariant. To make it so requires the use of special hypersurfaces as attempted, for example, by Fermi and Rohrlich [40]. The 0(3) energy momentum (78), in contrast, is generally covariant in 0(3) electrodynamics [11-20]. 

Potential differences are primary in gauge theory, because they define both the covariant derivative and the field tensor. In Whittaker s theory [27,28], potentials can exist without the presence of fields, but the converse is not true. This conclusion can be demonstrated as follows. Equation (479b) is invariant under... 

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

The only common factor is that the charge-current 4-tensor transforms in the same way. The vector representation develops a time-like component under Lorentz transformation, while the tensor representation does not. However, the underlying equations in both cases are the Maxwell-Heaviside equations, which transform covariantly in both cases and obviously in the same way for both vector and tensor representations. 

Ansatzes (53)-(55) are given in explicitly covariant form. This fact enables us to perform symmetry reduction of Eqs. (46) in a unified way. First, we give without derivation three important identities for the tensor [35] ... 

The curvature tensor is defined in terms of covariant derivatives of the spin-affine connections fip, and according to Section ( ), has its equivalent in 0(3) electrodynamics. 

The curvature tensor can be written as a commutator of covariant derivatives... 

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